Cohomology of the Second Voronoi

Compactification of A_{4}

Klaus Hulek and Orsola Tommasi

Received: January 3, 2012

Communicated by Gavril Farkas

Abstract. In this paper we compute the cohomology groups of the second Voronoi and the perfect cone compactification A_{4}^{Vor} and A_{4}^{perf} respectively, of the moduli space of abelian fourfolds in degree ≤ 9. The main tool is the investigation of the strata of A_{4}^{Vor} corresponding to semi-abelic varieties with constant torus rank.

2010 Mathematics Subject Classification: Primary 14K10; Secondary 14F25, 14C15, 14D22 Keywords and Phrases: Abelian varieties, Voronoi compactification, perfect cone compactification, Chow ring, cohomology ring

1. Introduction and plan

The moduli space A_{g} of principally polarized abelian varieties of genus g are much studied objects in algebraic geometry. Although much progress has been made in understanding the geometry of these spaces, we still know relatively little about the cohomology or the Chow groups of A_{g} and its compactifications. These are difficult questions even for low genus. Mumford in his seminal paper [Mu2] computed the Chow ring of \overline{M}_{2}, or what is the same, of the second Voronoi compactification A_{2}^{Vor}. It was also in this paper that he laid the foundations for the study of the Chow ring of \overline{M}_{g} in general. Lee and Weintraub [LW] have investigated the cohomology of certain level covers of A_{2}^{Vor}. The cohomology of A_{3} and of the Satake compactification A_{3}^{Sat} were determined by Hain [Ha], while the Chow group of the second Voronoi compactification A_{3}^{Vor} had earlier been computed by van der Geer [vdG1]. The authors of this paper proved in [HT] that the Chow ring and the cohomology ring of A_{g}^{Vor} are isomorphic for g = 2, 3. Very little is known about the topology of A_{g} and its compactifications in general. A positive exception is given by the subring generated by the Chern classes λ_{i} of the Hodge bundle in the Chow ring or the cohomology ring of A_{g}. By [vdG2] this subring is known explicitly; in particular, it is generated by the odd degree Hodge classes λ_{2i+1}. Furthermore, by a classical result by Borel [Bo] the Hodge classes λ_{2i+1} generate the stable cohomology of A_{g}, which is defined in terms of the natural maps A_{g′} ↪ A_{g} for g′ < g given by multiplication with a fixed abelian variety of dimension g − g′. Note that by the construction of A_{g} as a quotient of Siegel space, the rational cohomology of A_{g} coincides with the group cohomology of the symplectic group Sp(2g, ℤ). In this paper we investigate the case of genus 4 of whose cohomology very little is known. There are two reasons why we believe that it is worth the effort to undertake this study. One is that the spaces A_{2} and A_{3} are very close to the moduli spaces M_{2} and M_{3} whose cohomology is rather well understood, whereas A_{4} is the first Siegel moduli space where the Torelli map fails to be birational and thus one might expect new phenomena. The second reason is that, as our discussion of the boundary strata shows, the complexity of the structure of A_{g} and its compactifications grows dramatically with g. At the moment calculations on the cohomology of A_{5}, or any of its compactifications, seems far out of reach and thus A_{4} is the only remaining low genus case where the computation of the entire cohomology currently seems within reach. In this paper we thus investigate the cohomology of toroidal compactifications of A_{4}. In general there are several meaningful compactifications of A_{g}. Besides the second Voronoi compactification A_{g}^{Vor} there is the perfect compactification A_{g}^{perf}, given by the second Voronoi decomposition and the perfect cone (or first Voronoi) decomposition respectively, as well as the Igusa compactification A_{g}^{Igu}. It was shown by Alexeev [Al] and Olsson [Ol] that (at least up to normalization) A_{g}^{Vor} represents a geometric functor given by stable semi-abelic varieties. On the other hand A_{g}^{perf} is, as was proved by Shepherd-Barron [S-B], a canonical model in the sense of Mori theory, i.e. its canonical bundle is ample, if g ≥ 12. Finally, Igusa constructed A_{g}^{Igu} as a partial blow-up of A_{g}^{Sat} and it was shown by Namikawa [Nam] that Igusa’s model is the toroidal compactification defined by the central cone decomposition. In genus g ≤ 3 all of the above toroidal compactifications coincide. In genus 4 the Igusa and the perfect cone decomposition coincide and the second Voronoi compactification A_{4}^{Vor} is a blowup of A_{4}^{perf}. However, for general g all three compactifications are different. The main result of our paper is the determination of the Betti numbers of A_{4}^{perf} of degree less than or equal to 9 and of all Betti numbers of A_{4}^{Vor} with the exception of the middle Betti number b_{10}. This reduces the problem of the computation of the cohomology of A_{4}^{Vor} to the computation of the Euler number of A_{4}, which is an independent problem in its own right. Indeed, one can compute the Euler number of level covers A_{4}(n) for n ≥ 3 by Hirzebruch– Mumford proportionality. From this one could compute e(A_{4}) if one had a complete classification of torsion elements in the group Sp(8, ℤ). Although this is not known, it does not seem an impossible task to obtain such a classification. This is, however, a hard problem which requires different methods from the the ones used in this paper; therefore, we will not approach it here.

Our approach involves computing a spectral sequence converging to the cohomology of A_{4}^{Vor}. As a by-product, this spectral sequence determines the long exact sequence relating the cohomology with compact support of A_{4}^{Vor} with that of A_{4} and of its boundary. For this reason, a natural application of our analysis is to obtain evidence on the existence of non-trivial cohomology classes of A_{4} in different degrees, by giving lower bounds for the Betti numbers of A_{4}. However, all contributions from the cohomology of A_{4} to our computation can be explained using either the Hodge classes or a certain non-algebraic class related to the one in H^{6}(A_{3}; ℚ) described in [Ha, Thm 1]. This gives rise to the following question:

Question. Is the rational cohomology of A_{4} generated by the Hodge classes λ_{1}, λ_{3} and by one non-algebraic cohomology class of Hodge weight 18 arising in degree 12?

The starting point of our investigations is the fact that every toroidal compactification A_{g}^{tor} admits a map φ_{g}: A_{g}^{tor} ⟶ A_{g}^{Sat}. We recall that

A_{g}^{Sat} = A_{g} ⊔ A_{g−1} ⊔ … ⊔ A_{0},

which allows us to construct a stratification of A_{g}^{tor} by considering the closed loci β_{i}^{(g)} = β_{i} = φ_{g}^{−1}(A_{g−i}^{Sat}) and their open parts β_{i}^{0} = β_{i} \ β_{i+1} = φ_{g}^{−1}(A_{g−i}). Each stratum β_{i}^{0} is itself the disjoint union of locally closed substrata that are quotients of torus bundles over the product of a certain number of copies of the universal family X_{g−i} over A_{g−i} by finite groups. The strategy is then to compute the cohomology with compact support of each of these substrata using Leray spectral sequences and then to glue these strata by Gysin spectral sequences to compute the cohomology with compact support of β_{i}^{0}. Although this is a natural approach, we are not aware that it has been used in this form before apart from [HT] where it was applied to the case of genus 3. The reader will however notice that the complexity encountered in the present case is of a very different level: we need a full understanding of the Voronoi decomposition in genus 4, which in this case can no longer be deduced from the knowledge of the basic cone alone. The use of Leray spectral sequences requires to know the cohomology with compact support of A_{g−i} not only with constant coefficients, but also with coefficients in certain symplectic local systems of low weight. In the case of i = 1, 2 we deduce this information from results on the cohomology of moduli spaces of pointed curves. Passing from the moduli space of curves to the moduli space of abelian varieties produces a small ambiguity, which does not influence our final result, mainly because it disappears at the level of Euler characteristics. Up to this ambiguity, we are able to obtain complete results for the cohomology with compact support of all strata contained in the boundary as well as of the closure \overline{J}_{4} of the Jacobian locus in A_{4} and we believe that this is of some independent interest. In the case of the cohomology of A_{4} itself, there are two facts which help us. The first is that the complement in A_{4} of the closure of the locus of jacobians has a smooth affine variety as coarse moduli space. This implies that its cohomology with compact support is trivial if the degree is smaller than 10 and thus that the cohomology with compact support of A_{4} agrees with that of \overline{J}_{4} in degree ≤ 9. The second is that A_{4}^{Vor} is (globally) the quotient of a smooth projective scheme by a finite group. This implies that its cohomology satisfies Poincaré duality, and, more specifically, that its cohomology in degree k carries a pure Hodge structure of weight k. In the case of A_{4}^{Vor}, putting the cohomological information from all strata β_{i}^{0} together yields Table 1, from which we can deduce Theorem 1 by using the Gysin spectral sequence associated to the stratification given by the β_{i}. A consequence of this spectral sequence is that the cohomology groups with compact support of A_{4} in degree ≤ 9 are sufficient to determine all cohomology groups of A_{4}^{Vor} of degree ≠ 10. In turn, the cohomology groups of A_{4}^{Vor} in degree ≥ 11 so obtained can be used to gain information on the cohomology with compact support of A_{4} in degree ≥ 11, thus leading (using Poincaré duality) to the question formulated above. Finally, we obtain the Betti numbers for A_{4}^{perf} in Theorem 2 by using the fact that A_{4}^{Vor} is a blow-up of A_{4}^{perf} in one point. The plan of the paper is as follows. In §2 we prove the main results using cohomological informations on the strata β_{i}^{0}. The geometrical study of each of the five strata β_{i}^{0} is performed in the following five sections. Finally, in the appendix we collect and prove all results on the cohomology of local systems on A_{2} and A_{3} used in Section 4 and 5.

Acknowledgments. Partial support from DFG under grants Hu 337/6-1 and Hu 337/6-2 is gratefully acknowledged. During the preparation of this paper, the second author has been partially supported by the programme Wege in die Forschung II of Leibniz Universität Hannover. We would like to thank MSRI for hospitality during the first part of the preparation of this paper. Finally, we thank Frank Vallentin for untiringly answering our questions about the second Voronoi decomposition.

Notation.

A_{g} moduli stack of principally polarized abelian varieties of

genus g

X_{g}  universal family over A_{g}

𝕍_{λ_{1},...,λ_{g}} rational local system on A_{g} induced by the Sp(2g, ℚ)-

representation indexed by the partition (λ_{1}, …, λ_{g})

A_{g}^{Sat}  Satake compactification of A_{g}
A_{g}^{Vor}  Voronoi compactification of A_{g}
X_{g}^{Vor}  universal family over A_{g}^{Vor}
A_{g}^{perf}  perfect cone compactification of A_{g}
A_{g}^{Igu}  Igusa compactification of A_{g}

M_{g,n} moduli stack of non-singular curves of genus g with n marked points M_{g} := M_{g,0}

𝔖_{d}  symmetric group in d letters
Sym_{≥0}^{2}(ℝ^{g}) space of real positive semidefinite quadratic forms in ℝ^{g}
⟨φ_{1}, …, φ_{r}⟩ convex cone generated by the half rays ℝ_{≥0}φ_{1}, …, ℝ_{≥0}φ_{r}

For every g, we denote by φ_{g}: A_{g}^{Vor} → A_{g}^{Sat} (respectively, ψ_{g}: A_{g}^{perf} → A_{g}^{Sat}) the natural map from the Voronoi (respectively, perfect cone) to the Satake compactification. Let π_{g}: X_{g}^{Vor} → A_{g}^{Vor} be the universal family, q_{g}: X_{g}^{Vor} → X_{g}^{Vor}/±1 the quotient map from the universal family to the universal Kummer family and k_{g}: X_{g}^{Vor}/ ± 1 → A_{g}^{Vor} the universal Kummer morphism. For 0 ≤ i ≤ g, we set β_{i}^{0} = φ_{g}^{−1}(A_{g−i}) ⊂ A_{g}^{Vor}, β_{i} = φ_{g}^{−1}(A_{g−i}^{Sat}) ⊂ A_{g}^{Vor} and β_{i}^{perf} = ψ_{g}^{−1}(A_{g−i}^{Sat}) ⊂ A_{g}^{perf}. We denote the Torelli map in genus g by τ_{g}: M_{g} → A_{g} its image, the Jacobian locus, by J_{g} = τ_{g}(M_{g}) and closure of the image in A_{g} by \overline{J_{g}}. Throughout the paper, we work over the field ℂ of complex numbers. All cohomology groups we consider will have rational coefficients. Since the rational cohomology of a Deligne–Mumford stack coincides with the rational cohomology of its coarse moduli space, we will sometimes abuse notation and denote stack and coarse moduli space with the same symbol. In this paper, we make extensive use of mixed Hodge structures, focussing mainly on their weight filtration. We will denote by ℚ(−k) the Hodge structure of Tate of weight 2k. For two mixed Hodge structures A, B we will denote by A ⊕ B their direct sum and by A + B any extension

0 → B → E → A → 0.

Furthermore, we will denote Tate twists of mixed Hodge structures by A(−k) = A ⊗ ℚ(−k).

2. Main theorems

Theorem 1. The cohomology of A_{4}^{Vor} vanishes in odd degree and is algebraic in all even degrees, with the only possible exception of degree 10. The Betti numbers are given by

\begin{matrix}i & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 \\ b_{i} & 1 & 3 & 5 & 11 & 17 & 10 + e(A_{4}) & 17 & 11 & 5 & 3 & 1 \end{matrix}

where e(A_{4}) denotes the Euler number of A_{4}.

The only missing information needed to compute all Betti numbers of A_{4}^{Vor} is the Euler number. As we shall see, we are able to compute the Euler numbers of all strata β_{i}^{0} for i ≥ 1, and thus, as already mentioned, it would suffice to compute the Euler number of the space A_{4} itself (see the introduction for comments on this).

Theorem 2. The Betti numbers of A_{4}^{perf} in degree ≤ 9 are given by

\begin{matrix}i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ b_{i} & 1 & 0 & 2 & 0 & 3 & 0 & 8 & 0 & 14 & 0 \end{matrix}

Moreover, all cohomology classes of degree ≤ 9 are algebraic.

Corollary 3. The rational cohomology of A_{4} vanishes in degree 1, 11 and all degrees ≥ 13. Furthermore one has H^{0}(A_{4}; ℚ) = ℚ(−10), H^{2}(A_{4}; ℚ) = ℚ(−9) and H^{12}(A_{4}; ℚ) = ℚ(−6) + ℚ(−9). For the remaining Betti numbers a lower bound is given in the table

\begin{aligned}i & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & ≥ 13 \\ b_{i} & 1 & 0 & 1 & ≥ 0 & ≥ 1 & ≥ 0 & ≥ 2 & ≥ 0 & ≥ 1 & ≥ 0 & ≥ 1 & 0 & 2 & 0 \end{aligned}

Note that the vanishing of b_{1} also follows independently from simple connectedness of A_{4} [HR, Theorem 4.1] and that b_{2} = 1 corresponds to the fact that the Picard group is generated by the Hodge line bundle (over ℚ).

Proof of Theorem 1. To compute the cohomology of A_{4}^{Vor}, we study the Gysin spectral sequence E_{r}^{p,q} ⇒ H^{p+q}(A_{4}^{Vor}; ℚ) associated with the filtration {T_{i}}_{i=1,...,6} such that

• T_{i} = β_{5−i}, i = 1, …, 4;
• T_{5} = \overline{J_{4}} ∪ T_{4};
• T_{6} = A_{4}^{Vor}.

The E_{1} term of this spectral sequence has the form E_{1}^{p,q} = H_{c}^{p+q}(T_{p} \ T_{p−1}; ℚ). For p = 1, …, 4 the strata T_{p} \ T_{p−1} coincide with the strata of A_{4}^{Vor} of semiabelic varieties of torus rank 5 − p; their cohomology with compact support is computed in the next sections by combining combinatorial information on the toroidal compactification with the geometry of fibrations on moduli spaces of abelian varieties (see Propositions 7, 8, 24 and Theorem 26). The stratum T_{5} \ T_{4} is the closure inside A_{4} of the locus of jacobians. Its cohomology with compact support is computed in Lemma 6. The only remaining stratum is the open stratum T_{6} \ T_{5}. Let J_{4}^{Sat} be the closure of J_{4} in A_{4}^{Sat}. Since this contains the entire boundary of A_{4}^{Sat} it follows that

T_{6} \ T_{5} = A_{4} \ \overline{J}_{4} = A_{4}^{Sat} \ J_{4}^{Sat}.

The latter set is affine since it is the complement of an ample hypersurface on A_{4}^{Sat} (see [HaHu]). In particular, its cohomology with compact support can be non-trivial only if the degree lies between 10 and 20. From this it follows that the E_{1} term of the Gysin spectral sequence associated with the filtration {T_{i}} is as given in Table 1. For the sake of simplicity, in that table we have denoted H_{c}^{3}(A_{2}; 𝕍_{2,2}) and H_{c}^{4}(A_{2}; 𝕍_{2,2}) with the same symbol H, even though a priori they are only isomorphic after passing to the associated graded piece with respect to the weight filtration. (Furthermore, the results in in [T4] imply H = 0.)

Table 1. E_{1} term of the Gysin spectral sequence associated with the filtration T_{i}.

q

17  ℚ(−9)  0  0  0  0  0
16  0  0  0  0  0  0
15  ℚ(−8)  0  0  0  0  0
14  0  0  0  ℚ(−9)  0  ℚ(−10)
13  ℚ(−7)^{⊕2}  0  ℚ(−8)  0  ℚ(−9)  ?
12  0  ℚ(−7)  0  ℚ(−8)^{⊕2}  0  ?
11  ℚ(−6)^{⊕4}  0  ℚ(−7)^{⊕3}  0  ℚ(−8)  ?
10  0  ℚ(−6)^{⊕3}  0  ℚ(−7)^{⊕3}  0  ?
9  ℚ(−5)^{⊕6}  0  ℚ(−6)^{⊕5}  ℚ(−6)^{⊕ϵ}  ℚ(−7)^{⊕2}  ?
8  0  ℚ(−5)^{⊕4}  ℚ(−3)  ℚ(−6)^{⊕(4+ϵ)} + ℚ(−3)  0  ?
7  ℚ(−4)^{⊕7}  0  ℚ(−5)^{⊕5} + H(−1)  ℚ(−5)^{⊕ϵ}  ℚ(−6)  ?
6  0  ℚ(−4)^{⊕4}  ℚ(−2) + H(−1)  ℚ(−5)^{⊕(3+ϵ)} + ℚ(−2)  0  ?
5 ℚ(−3)^{⊕6} + ℚ(−1) ℚ(−1) ℚ(−4)^{⊕3} + ℚ(−2) + H  ℚ(−2)  ℚ(−5)  ?
4  0  ℚ(−3)^{⊕2}  ℚ(−1) + H  ℚ(−4)^{⊕2} + ℚ(−1)  0  ?
3  ℚ(−2)^{⊕3}  ℚ  ℚ(−3)^{⊕2}  ℚ(−1)  ℚ(−4) + ℚ(−1)  0
2  0  ℚ(−2)  0  ℚ(−3) + ℚ  0  0
1  ℚ(−1)^{⊕2}  0  ℚ(−2)  0  0  0
0  0  ℚ(−1)  0  0  0  0
−1  ℚ  0  0  0  0  0
1  2  3  4  5  6  p

H = H_{c}^{3}(A_{2}; 𝕍_{2,2}) ≅ H_{c}^{4}(A_{2}; 𝕍_{2,2}) (up to grading), ϵ := rank H_{c}^{9}(A_{3}; 𝕍_{1,1,0}).

Documenta Mathematica 17 (2012) 195–244

Table 2. E_{∞}^{p,q} in the range p + q ≤ 9.
q
8  0
7 ℚ(−4)^{⊕7}  0
6  0  ℚ(−4)^{⊕4}  0
5 ℚ(−3)^{⊕6}  0  ℚ(−4)^{⊕3}  0
4  0  ℚ(−3)^{⊕2}  0  ℚ(−4)^{⊕2}  0
3 ℚ(−2)^{⊕3}  0  ℚ(−3)^{⊕2}  0  ℚ(−4) 0
2  0  ℚ(−2)  0  ℚ(−3)  0  0
1 ℚ(−1)^{⊕2}  0  ℚ(−2)  0  0  0
0  0  ℚ(−1)  0  0  0  0
−1  ℚ  0  0  0  0  0
1  2  3  4  5  6 p

Since the terms in the sixth column are only known for q ≤ 3, in the following we will only deal with the terms of the spectral sequence that are independent of them, that is, the E_{r}^{p,q} terms with p + q ≤ 8. Let us recall that A_{4}^{Vor} is a smooth Deligne–Mumford stack which is globally the quotient of a smooth proper variety by a finite group. From this it follows that the cohomology groups of A_{4}^{Vor} carry pure Hodge structures of weight equal to the degree. Therefore, the Hodge structures on E_{∞}^{p,q} have to be pure of weight p + q. This means that for all p, q, the graded pieces of E_{1}^{p,q} of weight different from p+q are killed by differentials. In particular, if we restrict to the range p+q ≤ 9, this gives that the E_{∞} terms are as given in Table 2. Of course, this does not describe precisely at which E_{r} the spectral sequence degenerates, or what exactly is the rank of the differentials. For instance, if one assumes H = 0 (which in view of the results in [T4] is indeed the case), a natural thing to expect is that the d_{1}-differentials E_{1}^{1,5} → E_{1}^{2,5}, E_{1}^{3,5} → E_{1}^{4,5}, E_{1}^{3,4} → E_{1}^{4,4} and E_{1}^{4,3} → E_{1}^{5,3}, as well as the d_{2}-differential E_{2}^{2,3} → E_{2}^{4,2} have rank 1, but this is not the only possibility. The claim on the cohomology of A_{4}^{Vor} in degree ≤ 9 follows from the E_{∞} term in Table 2. The claim on the cohomology in degree ≥ 11 follows by Poincaré duality. Finally, the computation relating the middle Betti number b_{10} to the Euler number e(A_{4}) follows from the additivity of the Betti numbers of the stratification {T_{•}}. □

Remark 4. The fact that the mixed Hodge structures on the E_{1}^{p,q} in Table 1 are compatible with obtaining E_{∞}^{p,q} terms that carry pure Hodge stuctures of the correct weight provide an important check on the correctness of our computations.

Proof of Theorem 2. The proof is analogous to that of Theorem 1. Rather than working with the filtration {T_{i}}, we will consider the stratification {T_{i}^{perf}} defined analogously by T_{i}^{perf} = β_{5−i}^{perf} for 1 ≤ i ≤ 4 and T_{5}^{perf} = \overline{J}_{4} ∪ T_{4}^{perf}, T_{6}^{perf} = A_{4}^{perf}. The closed stratum T_{1} is the the locus β_{4}^{perf} of torus rank 4 inside A_{4}^{perf}. Hence E_{1}^{1,q} = H^{q+1}(β_{4}^{perf}; ℚ) can be obtained from Theorem 25. Since the exceptional divisor of the blow-up map q: A_{4}^{Vor} → A_{4}^{perf} is contained in T_{1}, we have (A_{4}^{perf} \ q(T_{1})) ≅ (A_{4}^{Vor} \ T_{1}). In particular, the Gysin spectral sequence associated with the stratification of A_{4}^{perf} has E_{1}^{p,q} terms that coincide with those of Table 1 for p ≥ 2. Moreover, also the rank of all differentials E_{r}^{p,q} → E_{r}^{p+r,q−r+1} coincide with those for the filtration {T_{i}} as long as no E_{r}^{1,q}-terms are involved. This already implies the claim for all degrees different from 6. In degree 6, it is necessary to decide whether the class of Hodge weight 2 in E_{1}^{5,1} is killed by differentials of the spectral sequence or not. If we consider the map A_{4}^{Vor} ⊃ β_{4} → β_{4}^{perf} ⊂ A_{4}^{perf}, we have that the weight 2 class on β_{4}^{perf} lies in the image of the weight 2 class in the cohomology of β_{4}, which was killed by differentials for purity reasons on A_{4}^{Vor}. This implies that this must be the case also on A_{4}^{perf}. From this the claim follows. □

Proof of Corollary 3. The lower bounds in the claim are those given by the dimension of the subring generated by λ_{1} and λ_{3}. The vanishing of the cohomology of A_{4} in all degrees i ≥ 13 and in degree 11, as well as H^{12}(A_{4}; ℚ) = ℚ(−6)+ℚ(−9) follow directly from the last two columns of Table 1 by Poincaré duality. □

Remark 5. Comparing Table 1 with the cohomology of A_{4}^{Vor} of degree ≥ 11 suggests that the cohomology of the open stratum A_{4} \ \overline{J}_{4} could vanish in all positive degrees, with the exception of degree 10 on which Poincaré duality yields no information. The reason is the following. One can reinterpret the first 4 columns of Table 1 as the E_{1} terms of a Gysin spectral sequence converging to the cohomology of the boundary ∂A_{4}^{Vor} = A_{4}^{Vor} \ A_{4}. Then the remaining information from that Table is equivalent to the study of the long exact sequence

(2.1)  H_{c}^{k}(A_{4}; ℚ) → H^{k}(A_{4}^{Vor}; ℚ) → H^{k}(∂A_{4}^{Vor}; ℚ) → H_{c}^{k+1}(A_{4}; ℚ)

associated with the closed inclusion of the boundary in A_{4}^{Vor}. One can compare the information on the cohomology of the boundary coming from Table 1 with the Betti numbers of A_{4}^{Vor} in degree ≥ 11 from Theorem 1. Then one sees that even in this range the results are compatible with the vanishing of the cohomology of A_{4} \\overline{J}_{4}, or, equivalently, with the hypothesis that the cohomology of A_{4} is the minimal possible, i.e. generated by λ_{1}, λ_{3} and α ∈ H^{9,9}(H^{12}(A_{4}; ℚ)) from Corollary 3. This would imply that the cohomology of A_{4} coincides with the stable cohomology in degree ≤ 10, while a priori this is known only in degree ≤ g − 2 = 2. Furthermore, (2.1) gives strong restrictions on the possible existence of cohomology classes on A_{4} that are not in the subring generated by the Hodge classes. This follows from the purity of the cohomology of A_{4}^{Vor} combined with Table 1, which ensures that the cohomology of ∂A_{4}^{Vor} is very close to be pure itself. Practically, this forces non-trivial cohomology classes from A_{4} \ \overline{J}_{4} to appear in pairs of isomorphic Hodge structures, occurring as graded pieces of H^{k}(A_{4} \ \overline{J}_{4}; ℚ) and H^{k+1}(A_{4} \ \overline{J}_{4}; ℚ). One could also use Table 1 to prove the vanishing of H^{k}(A_{4} \ \overline{J}_{4}; ℚ). Then one needs to prove that all (algebraic) classes of weight 10 − k that occur in E_{1}^{p,q} with p+q = 10−k give rise to cohomology classes in H_{c}^{10−k}(A_{4}^{Vor}; ℚ) that are linearly independent. This is known for divisors (ensuring the vanishing of H^{k}(A_{4} \ \overline{J}_{4}; ℚ) for k = 1, 2). It would be interesting to investigate it for classes of higher codimension. Note that, if one knew that H^{k}(A_{4} \ \overline{J}_{4}; ℚ) vanishes for all 1 ≤ k ≤ 9, then this would yield the following result for the Betti numbers of A_{4}^{perf} in degree ≥ 11:

\begin{matrix}i & 12 & 14 & 16 & 18 & 20 \\ b_{i} & 14 & 9 & 4 & 2 & 1 \end{matrix}

as well as the vanishing of all odd Betti numbers of A_{4}^{perf}.

3. Torus rank 0

We start by considering T_{5} \ T_{4}, which is the Zariski closure \overline{J}_{4} of the locus of jacobians J_{4} = τ_{4}(M_{4}) inside A_{4}.

Lemma 6. The only non-zero Betti numbers with compact support of \overline{J}_{4} are as follows:

\begin{matrix}i & 18 & 16 & 14 & 12 & 10 & 8 \\ b_{i} & 1 & 1 & 2 & 1 & 1 & 2 \end{matrix}

In particular, all odd Betti numbers vanish. Furthermore, all cohomology groups with compact support are generated by algebraic classes, with the only exception of H_{c}^{8}(\overline{J}_{4}; ℚ), which is an extension of ℚ(−4) by ℚ(−1).

Proof. We compute the cohomology with compact support of \overline{J}_{4} by recalling that the Zariski closure of the locus of jacobians in A_{4} is the union of the image of the Torelli map and the locus of abelian fourfolds that are products of abelian varieties of dimension ≤ 3. This allows to cover \overline{J_{4}} by the following locally closed disjoint strata:

S_{1} = Sym^{4} A_{1}, S_{2} = τ_{2}(M_{2}) × Sym^{2} A_{1}, S_{3} = Sym^{2} τ(M_{2}),
S_{4} = τ_{3}(M_{3}) × A_{1}, S_{5} = τ_{4}(M_{4}).

Furthermore, the Torelli map in all genera induces an isomorphism in cohomology with rational coefficients between M_{g} and its image τ_{g}(M_{g}). This allows to compute the cohomology with compact support of all strata from previously known results on the cohomology of M_{g} with g ≤ 4 ([Mu2],[Lo],[T1]). These yield that the E_{1} term E_{1}^{p,q} = H_{c}^{p+q}(S_{p}; ℚ) of the Gysin exact sequence of the filtration associated with the stratification S_{j} is as in Table 3. In view of Table 3, to calculate the cohomology with compact support of \overline{J_{4}} it is sufficient to know the rank of the differential

d : H_{c}^{12}(\overline{J} _{4}^{red}; ℚ) ≅ ℚ(−6)^{⊕2} ⟶ H_{c}^{13}(J_{4}; ℚ) ≅ ℚ(−6)

Table 3. E_{1} term of the Gysin spectral sequence converging to the cohomology with compact support of the closure of the locus of jacobians in A_{4}

q
13  0  0  0  0  ℚ(−9)
12  0  0  0  0  0
11  0  0  0  0  ℚ(−8)
10  0  0  0  ℚ(−7)  0
9  0  0  ℚ(−6)  0  ℚ(−7)
8  0  ℚ(−5)  0  ℚ(−6) ℚ(−6)
7 ℚ(−4)  0  0  0  0
6  0  0  0  0  0
5  0  0  0  0  0
4  0  0  0  ℚ(−1)  0
1  2  3  4  5  p

in the Gysin long exact sequence associated with the closed inclusion of the locus \overline{J} _{4}^{red} = \overline{J}_{4} \ J_{4} = \overline{S}_{3} ∪ \overline{S}_{4} ⊂ A_{4} of reducible abelian fourfolds in the Zariski closure in A_{4} of the locus of jacobians J_{4} = τ_{4}(M_{4}). We observe that H_{c}^{12}(\overline{J} _{4}^{red}; ℚ) is generated by two 6-dimensional algebraic cycles C_{1} and C_{2}, where C_{1} is the fundamental class of \overline{S_{3}} and C_{2} the fundamental class of \overline{τ(H_{3})} × A_{1}, where H_{3} is the hyperelliptic locus. Therefore, the surjectivity of d is equivalent to the existence of a relation between C_{1} and C_{2} viewed as elements of the Chow group of \overline{J}_{4}. Let us denote by M_{4}^{ct} the moduli space of stable genus 4 curves of compact type, i.e. such that that their generalized Jacobian is compact. Then the Torelli map extends to a proper morphism

τ^{ct} : M_{4}^{ct} ⟶ \overline{J}_{4}.

From the geometric description of the map τ^{ct} it follows that the image under τ^{ct} of the Chow group of dimension 6 cycles supported on the boundary M_{4}^{ct} \ M_{4} coincides with ⟨C_{1}, C_{2}⟩. Indeed, let D_{1} be the closure of the locus of stable curves consisting of two genus 2 curves intersecting in a Weierstrass point and let D_{2} be the closure of the locus of stable curves consisting of elliptic curves intersecting a hyperelliptic genus 3 curve in a Weierstrass point. Then D_{1} and D_{2} map to C_{1} and C_{2} respectively. It is known that the dimension 6 classes in M_{4}^{ct} fulfill a relation, given by the restriction of the relation on \overline{M}_{4} of [Y, Prop. 2]. When pushed forward via τ^{ct}, this relation gives a non-trivial relation between C_{1} and C_{2}. Thus the differential d has to be surjective and the claim follows. □

Table 4. E_{2} term of the Leray spectral sequence converging to H_{c}^{•}(β_{1}^{0}; ℚ).

q

6  0  ℚ(−6) + ℚ(−3) 0  ℚ(−7)  0  ℚ(−8) 0 ℚ(−9)
5  0  0  0  0  0  0  0  0
4 ℚ(−2) ℚ(−5) + ℚ(−2) 0 ℚ(−6)^{⊕(1+ϵ)} ℚ(−6)^{⊕ϵ} ℚ(−7) 0 ℚ(−8)
3 0 0 0 0 0 0 0 0
2 ℚ(−1) ℚ(−4) + ℚ(−1) 0 ℚ(−5)^{⊕(1+ϵ)} ℚ(−5)^{⊕ϵ} ℚ(−6) 0 ℚ(−7)
1  0  0  0  0  0  0  0  0
0  0  ℚ(−3) + ℚ  0  ℚ(−4)  0  ℚ(−5) 0 ℚ(−6)
5  6  7  8  9  10  11  12  p
ϵ = rank H_{c}^{9}(A_{3}; 𝕍_{1,1,0}) ∈ {0, 1}.

4. Torus rank 1

Next, we deal with the locus β_{0}^{1} of semi-abelic varieties of torus rank 1.

Proposition 7. The rational cohomology with compact support of β_{1}^{0} is as follows: the non-zero Betti numbers are

\begin{matrix}i & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 16 & 18 \\ b_{i} & 2 & 1 & 3 & 1 & 4 + ϵ & ϵ & 5 + ϵ & ϵ & 3 & 2 & 1 \end{matrix}

where ϵ = rank H_{c}^{9}(A_{3}; 𝕍_{1,1,0}). The cohomology groups of even degree 2k are algebraic for k ≥ 7; for k ≤ 6 they are extensions of pure Hodge structures of the form H_{c}^{2k}(β_{2}^{0}; ℚ) = ℚ(−k)^{⊕(b_{2k}−1)} +ℚ(k −3). The Hodge structures in odd degree are given by H_{c}^{2k+1}(β_{2}^{0}; ℚ) = ℚ(2 − k) for k = 7, 9 and H_{c}^{2k+1}(β_{2}^{0}; ℚ) = ℚ(−k)^{⊕ϵ} for k = 11, 13.

Proof. To compute the cohomology with compact support of β_{1}^{0} we will use the map k_{3} : β_{1}^{0} → A_{3} realizing β_{1}^{0} as the universal Kummer variety over A_{3}. The fibre of β_{1}^{0} over a point parametrizing an abelian surface S is K := S/ ± 1. Note that the cohomology of K vanishes in odd degree because of the Kummer involution. The cohomology of K is one-dimensional in degree 0 and 6 and induces trivial local systems on A_{3}. The cohomology group H^{2}(K; ℚ) ≅ ⋀^{2} H^{1}(S; ℚ) is 15-dimensional and induces the local system 𝕍_{1,1,0} ⊕ ℚ(−1) on A_{3}. By Poincaré duality we have H^{4}(K; ℚ) ≅ H^{2}(K; ℚ) ⊗ ℚ(−1), inducing the local system 𝕍_{1,1,0}(−1) ⊕ ℚ(−2) on A_{3}. The cohomology with compact support of A_{3} in the local system 𝕍_{1,1,0} is calculated in Lemma 35. We refer to Theorem 31 for the cohomology with compact support of A_{3} with constant coefficients, which was calculated by Hain in [Ha]. These results allow to compute E_{2}^{p,q} = H_{c}^{p}(A_{3}; R_{!}^{q}k_{3}_{∗}(ℚ)) for the Leray spectral sequence E_{•}^{p,q} ⇒ H_{c}^{p+q}(β_{1}^{0}; ℚ) associated with k_{3} : β_{1}^{0} → A_{3}. This E_{2} term is given in Table 4. Note that all differentials of this Leray spectral sequence vanish for Hodgetheoretic reasons, so that E_{2} = E_{∞}. Specifically, all differentials must involve one E_{2}^{p,q} term with p + q odd, but there are only two such terms, namely E_{2}^{5,2} and E_{2}^{5,4}. It follows from Table 4 that for both these terms, any differential d_{k} with k ≥ 2 which involves one of them will map either to 0 or to a E_{2} term that carries a pure Hodge structure of different weight. In both cases the differential has to be 0. □

5. Torus rank 2

In this section we compute the cohomology with compact support of the stratum β_{2}^{0} of A_{4}^{Vor} of rank 2 degenerations of abelian fourfolds. For this purpose we recall first the known global construction of β_{0}^{2} as the quotient of a ℙ^{1}-bundle of a fibre product of the universal family over A_{2}. Furthermore, let us recall that the restriction of the Voronoi fan in genus g to Sym_{≥0}^{2}(ℝ^{g′}) for genus g ≥ g′ coincides with the Voronoi fan in genus g′. This implies that the geometric constructions of the fibrations β_{2}^{0} → A_{2} and β_{3}^{0} → A_{1} we give in this section and in the following one, respectively, are actually independent of the choice of g = 4 but extend to analogous descriptions of the fibres of fibrations β_{2}^{0} → A_{g−2} and β_{3}^{0} → A_{g−3} that exist for β_{2}^{0}, β_{3}^{0} ⊂ A_{g}^{Vor} independently of g. In particular, the geometric construction of β_{2}^{0} explained in this section coincides with the construction used in [HT, §4] to compute the cohomology with compact support of the corresponding locus in A_{3}^{Vor}.

Proposition 8. The rational cohomology with compact support of β_{2}^{0} is as follows: the non-zero Betti numbers are

\begin{matrix}i & 4 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 14 & 16 \\ b_{i} & 1 & 2 & 1 + r & 4 + r & 1 + r & 5 + r & 1 & 5 & 3 & 1 \end{matrix}

where r = rank H_{c}^{3}(A_{2}; 𝕍_{2,2}). If we assume r = 0, then all cohomology groups of even degree are algebraic, with the exception of H_{c}^{8}(β_{2}^{0}; ℚ) = ℚ(−4)^{⊕3} + ℚ(−2) which is an extension of Hodge structures of Tate type. The Hodge structure in odd degree 2k + 1 with k = 3, 4, 5 is pure of Tate type of weight 2k − 4.

Remark 9. It follows from [T4] that r = 0.

The proof of this Proposition will given in §5.4 after some preliminary steps. In the previous section, we calculated the cohomology with compact support of β_{1}^{0} using the map k_{3} : β_{1}^{0} → A_{3} given by the universal Kummer variety. This map extends to the stratum β_{2}^{0} of degenerations of abelian fourfolds of torus rank 2, giving a map k_{3} : (β_{1} \ β_{3}) → A_{3}^{Vor}. Under this map, the elements of A_{4}^{Vor} with torus rank 2 are mapped to elements of A_{3}^{Vor} of torus rank 1. If we denote by β′_{t}^{0} the stratum of A_{3}^{Vor} of semi-abelian varieties of torus rank exactly t, we get a commutative diagram

[figure: A_{4}^{Vor}; A_{3}^{Vor}; A_{2}^{Vor}; β_{2}^{0}; k_{3}; _{β′_{1}_{0}}; k_{2}; A_{2}; π_{3}^{−1}(β′_{1}^{0}); q_{3}; π_{3}; X_{2}; q_{2}; π_{2}]

The map π_{3} is the restriction of the universal family over A_{3}^{Vor}. In particular, the fibres of π_{3} over points of β′_{1}^{0} are rank 1 degenerations of abelian threefolds, i.e. compactified ℂ^{∗}-bundles over abelian surfaces. A geometric description of these compactified ℂ^{∗}-bundles is given in [Mu1]. They are obtained by taking the ℙ^{1}-bundle associated to the ℂ^{∗}-bundle and then gluing the 0- and the ∞-section with a shift, defined by a point of the underlying abelian surface that is uniquely determined by the line bundle associated to the ℂ^{∗}-bundle. In particular, this shift is 0 for the fibres of the π_{3} over the 0-section of the Kummer fibration β′_{1}^{0} ≅ (X_{2}/ ± 1) ⟶^{k_{2}} A_{2}, which are thus products of a nodal curve and an abelian surface. We want to describe the situation in more detail. For this, consider the universal Poincaré bundle P → X_{2} ×_{A_{2}} X̂_{2}, normalized so that the restriction to the zero section X̂_{2} → X_{2} ×_{A_{2}} X̂_{2} is trivial. Let \overline{U} = ℙ(P ⊕ O_{X_{2}×_{A_{2}} X̂_{2}}) be the associated ℙ^{1}-bundle. Using the principal polarization we can naturally identify X̂_{2} and X_{2}, which we will do from now on. We denote by Δ the union of the 0-section and the ∞-section of this bundle. Set U = \overline{U} \ Δ, which is simply the ℂ^{∗}-bundle given by the universal Poincaré bundle P with the 0-section removed and denote the bundle map by f: U → X_{2} ×_{A_{2}} X_{2}. Then there is a map ρ: \overline{U} → β_{2}^{0} with finite fibres. Note that the two components of Δ are identified under the map ρ. The restriction of ρ to both U and to Δ is given by a finite group action, although the group is not the same in the two cases (see the discussion below).

5.1. Geometry of the ℂ^{∗}-bundle. We now consider the situation over a fixed point [S] ∈ A_{2}. For a fixed degree 0 line bundle L_{0} on S the preimage f^{−1}(S × {L_{0}}) is a semi-abelian threefold, namely the ℂ^{∗}-bundle given by the extension corresponding to L_{0} ∈ Ŝ. This semi-abelian threefold admits a Kummer involution ι which acts as x ↦ −x on the base S and by t ↦ 1/t on the fibre over the zero section. The Kummer involution ι is defined universally on U. Consider the two involutions i_{1}, i_{2} on X_{2} ×_{A_{2}} X_{2} defined by

i_{1}(S, p, q) = (S, −p, −q) and i_{2}(S, p, q) = (S, q, p)

for every abelian surface S and every p, q ∈ S. These two involutions lift to involutions j_{1} and j_{2} on U that act trivially on the fibre of f: U → X_{2} ×_{A_{2}} X_{2} over the zero section.

The following lemma can also be proved directly from the toroidal construction of A_{4}^{Vor} using the approach of [S-B, Lemma 2.4].

Lemma 10. The diagram

(5.1)  \xymatrix{U \ar[r]_{f} \ar[rr]^{g} \ar[d]^{ρ|_{U}} & X_{2} ×_{A_{2}} X_{2} \ar[d]^{ρ′} \ar[r] & A_{2} \\ β_{2}^{0} \ ρ(Δ) \ar[r] & Sym_{A_{2}}^{2}(X_{2}/ ± 1) \ar@{-->}[ur]}

where ρ′: X_{2} ×_{A_{2}} X_{2} → Sym_{A_{2}}^{2}(X_{2}/ ± 1) is the natural map, is commutative. Moreover ρ|_{U}: U → ρ(U) = β_{2}^{0} \ ρ(Δ) is the quotient of U by the subgroup of the automorphism group of U generated by ι, j_{1} and j_{2}.

Proof. Since the map ρ′ in the diagram (5.1) has degree 8 and ι, j_{1}, j_{2} generate a subgroup of order 8 of the automorphism group of U, it suffices to show that the map ρ|_{U} factors through each of the involutions ι and j_{1}, j_{2}. Recall that the elements of β_{2}^{0} correspond to rank 2 degenerations of abelian fourfolds. More precisely, every point of ρ(U) corresponds to a degenerate abelian fourfold X whose normalization is a ℙ^{1} × ℙ^{1}-bundle, namely the compactification of a product of two ℂ^{∗}-bundles on the abelian surface S given by k_{1} ∘ k_{2}([X]). The degenerate abelian threefold itself is given by identifying the 0-sections and the ∞-sections of the ℙ^{1} × ℙ^{1}-bundle. This identification is determined by a complex parameter, namely the point on a fibre of f: U → X_{2} ×_{A_{2}} X_{2}. Since a degree 0 line bundle L_{0} and its inverse define isomorphic semi-abelian threefolds and since the role of the two line bundles is symmetric, the map ρ|_{U} factors through ι and j_{2}. Since j_{1} is the commutator of ι and j_{2} the map ρ|_{U} also factors through j_{1}. □

We will compute the cohomology with compact support of β_{2}^{0} by considering the Leray spectral sequence associated with the fibration k_{2} ∘ k_{3} : β_{2}^{0} → A_{2}. This requires to compute the cohomology with compact support of the fibre (k_{2}∘k_{3})^{−1}([S]) over a point [S] ∈ A_{2}. To this end, we decompose (k_{2}∘k_{3})^{−1}([S]) into an open part given by its intersection with ρ(U) and a closed part given by its complement.

5.2. Cohomology of the open part of the fibre. The fibration g: U → A_{2} obtained by composing the ℂ^{∗}-bundle f: U → X_{2} ×_{A_{2}} X_{2} with the natural map X_{2} ×_{A_{2}} X_{2} → A_{2} plays an important role in the study of the restriction of k_{2} ∘ k_{3} to ρ(U). Namely, the fibre of (k_{2} ∘ k_{3})|_{ρ(U)} over [S] ∈ A_{2} coincides with the quotient of the fibre of g under the automorphism group generated by j_{1}, j_{2} and ι. Therefore, the cohomology of the fibre of k_{2} ∘ k_{3} restricted to ρ(U) is the part of the cohomology of g^{−1}([S]) that is invariant under j_{1}, j_{2} and ι. We start by computing the actions of i_{1}, i_{2} and of the involution κ : (p, q) ↦ (−p, q) induced by the Kummer involution of semi-abelian threefolds of torus

k H^{k}(S × S; ℚ)^{(i_{1},i_{2})}  κ-invariant  κ-alternating
8 ℚ(−4)  ℚ(−4)  0
6 (⋀^{2} Λ)(−2)^{⊕2}  (⋀^{2} Λ)(−2)  (⋀^{2} Λ)(−2)
4 ℚ(−2) ⊕ Λ^{⊗2}(−1) ⊕ Sym^{2}(⋀^{2} Λ) ℚ(−2) ⊕ Sym^{2}(⋀^{2} Λ) Λ^{⊗2}(−1)
2 ⋀^{2} Λ^{⊕2}  ⋀^{2} Λ  ⋀^{2} Λ
0 ℚ  ℚ  0
Table 5. Cohomology of S × S/(i_{1}, i_{2})

q

1 0 0 ⋀^{2} Λ(−1) 0  Λ^{⊗2}(−2)  0 (⋀^{2} Λ)(−3) 0  0
0 ℚ 0 (⋀^{2} Λ) 0 ℚ(−2) ⊕ Sym^{2}(⋀^{2} Λ) 0 (⋀^{2} Λ)(−2) 0 ℚ(−4)
0 1  2  3  4  5  6  7  8 p

Table 6. E_{2} term of the spectral sequence converging to H^{k}(g^{−1}([S]); ℚ)^{(i_{1},i_{2},ι)}

rank 1 on the cohomology of S × S. Recall that the cohomology of S is isomorphic to the exterior algebra generated by the 6-dimensional space Λ := H^{1}(S; ℚ) and that H^{•}(S × S; ℚ) ≅ H^{•}(S; ℚ)^{⊗2} by the Künneth formula. Using this description, one can calculate the part of the cohomology of S × S which is invariant under i_{1} and i_{2}. In particular, since all cohomology in odd degree is alternating under the involution i_{1}, the only non-trivial invariant cohomology groups are in even degree. We give the description of the invariant cohomology groups in the second column of Table 5. One then proceeds to investigate their structure with respect to κ. For instance one can use the isomorphism H^{k}(S × S; ℚ)^{(i_{1},i_{2},κ)} ≅ H^{k}(S × S/(i_{1}, i_{2}, κ); ℚ), together with the fact that the quotient of S × S by the subgroup generated by i_{1}, i_{2} and κ is the second symmetric product of S/±1. In this way one proves that the behaviour of the cohomology with respect to κ is as given in the last two columns of Table 5.

Lemma 11. The (i_{1}, i_{2}, ι)-invariant part of the Leray spectral sequence associated with the ℂ^{∗}-bundle g^{−1}([S]) → S × S gives rise to a spectral sequence E_{•}^{p,q} ⇒ H^{p+q}((k_{2} ∘ k_{3}|_{ρ(U)})^{−1}([S]); ℚ) which behaves as follows:

- E_{2}^{p,q} vanishes for q ≠ 0, 1;

- E_{2}^{p,0} is the part of H^{k}(S × S; ℚ) which is invariant under i_{1}, i_{2} and κ;

- E_{2}^{p,1} is the part of H^{k}(S × S; ℚ) which is invariant under i_{1}, i_{2} and alternating under κ, tensored with the Tate Hodge structure ℚ(−1).

Furthermore, the E_{∞} term of this spectral sequence, together with its structure as Sp(4, ℚ)-representation, is as given in Table 7.

q
1 0 0  0  0  𝕍_{2,0}(−2)  0 𝕍_{1,1}(−3) 0 0
0 ℚ 0 ℚ(−1) ⊕ 𝕍_{1,1} 0 ℚ(−2)^{⊕2} ⊕ 𝕍_{2,2} 0  0  0 0
0 1  2  3  4  5  6  7 8 p

Table 7. E_{∞} term of the spectral sequence converging to H^{k}(g^{−1}([S]); ℚ)^{(i_{1},i_{2},ι)}

Proof. Let us consider the ℂ^{∗}-bundle f_{S} := f|_{g^{−1}([S])} : g^{−1}([S]) → S × S. The Leray spectral sequence in cohomology associated with f_{S} converges to the cohomology of g^{−1}([S]) and has E_{2} term E_{2}^{p,q} ≅ H^{p}(S × S; ℚ) ⊗ H^{q}(ℂ^{∗}; ℚ). However, we are only interested in the part of the cohomology of g^{−1}([S]) which is invariant under j_{1}, j_{2} and ι. Since the actions of j_{1}, j_{2} and ι respect the map g^{−1}([S]) → S × S, they act also on the terms of the Leray spectral sequence associated with f_{S}. In particular, the spectral sequence whose E_{r} terms are the (j_{1}, j_{2}, ι)-invariant part of the terms of the Leray spectral sequence associated with f_{S} converges to H^{k}(g^{−1}([S]); ℚ)^{j_{1},j_{2},ι}. In particular, the E_{2} term of this spectral sequence is given by the (j_{1}, j_{2}, ι)- invariant part of H^{p}(S × S; ℚ) ⊗ H^{q}(ℂ^{∗}; ℚ). We have already determined the behaviour of the projection of these involutions to S × S in Table 5, so it remains only to determine their action on the fibre ℂ^{∗}. Since j_{1} and j_{2} both fix the fibre of f over the origin, they act trivially on the cohomology of ℂ^{∗}. Instead, the Kummer involution ι acts as the identity on H^{0}(ℂ^{∗}; ℚ) and as the alternating representation on H^{1}(ℂ^{∗}; ℚ). From this the first part of the claim follows. For the convenience of the reader, we have written the E_{2} term of the spectral sequence in Table 6. Notice that this spectral sequence has only two non-trivial rows. Therefore, it could be written equivalently as a long exact sequence. In particular, the only differentials one needs to study are the d_{2}-differentials. These differentials are given by restriction of the differentials of the Leray spectral sequence associated with the ℂ^{∗}-bundle f_{S}. Recall that f_{S} is the ℂ^{∗}-bundle obtaining by subtracting the 0-section from the Poincaré bundle over S × S. Therefore (see e.g. [Hu, XVI.7.5]) the d_{2}-differentials are given by taking the intersection product with the first Chern class of the Poincaré bundle, which is known to be equal to [diag(S)] − [S × {0}] − [{0} × S], where [·] denotes the fundamental class and diag: S → S × S is the diagonal map. An explicit computation of the intersections of this class with the κ-alternating classes in H^{k}(S × S; ℚ)^{i_{1},i_{2}} yields the description of E_{3} = E_{∞} given in Table 7. Here we have used the fact that Sym^{2} Λ is the irreducible Sp(4, ℚ) -representation 𝕍_{2,0}, whereas ⋀^{2} Λ decomposes into irreducible Sp(4, ℚ)-representations as ℚ(−1) ⊕ 𝕍_{1,1} and Sym^{2}(⋀^{2} Λ) decomposes as ℚ(−2)^{⊕2} ⊕ 𝕍_{1,1}(−1) ⊕ 𝕍_{2,2}. In the notation, Tate twists are only relevant for the Hodge structure.

□

5.3. Geometry of ρ(Δ). The map ρ identifies the two components of Δ, each of which is isomorphic to X_{2} ×_{A_{2}} X_{2}. In particular, the space ρ(Δ) can be realized as a finite quotient of X_{2} ×_{A_{2}} X_{2}. This can be read off from the construction of the toroidal compactification, as in [S-B, Lemma 2.4]. See also [HS1, Section I] for an outline of this construction. Also note that the stratum Δ corresponds to the stratum in the partial compactification in the direction of the 2-dimensional cusp associated with a maximal-dimensional cone in the second Voronoi decomposition for g = 4. A detailed description can be found in [HKW, Part I, Chapter 3]. Specifically, the stratum ρ(Δ) corresponds to the GL(2, ℤ)-orbit of the cone ⟨x_{1}^{2}, x_{2}^{2}, (x_{1} − x_{2})^{2}⟩. Hence, the map X_{2} ×_{A_{2}} X_{2} → ρ(Δ) is the quotient map with respect to the stabilizer G of the cone ⟨x_{1}^{2}, x_{2}^{2}, (x_{1} − x_{2})^{2}⟩ in Sym^{2}(ℤ^{2}). This is generated by three involutions: the multiplication map by −1, the involution interchanging x_{1} and x_{2} and the involution x_{1} ↦ x_{1}, x_{2} ↦ x_{1} − x_{2}. These generators of G act on X_{2} ×_{A_{2}} X_{2} by the following three involutions: the involution i_{1} which acts by (x, y) ↦ (−x, −y) on each fibre S × S, the involution i_{2} which interchanges the two factors of X_{2} ×_{A_{2}} X_{2} and finally the involution i_{3} which acts by (x, y) ↦ (x + y, −y). From this description, it follows that there is a fibration g′: ρ(Δ) → A_{2} whose fibre over [S] ∈ A_{2} is isomorphic to the quotient of S × S by the subgroup of Aut(S × S) generated by the three involutions i_{1}, i_{2} and i_{3} introduced above. If we write Λ := H^{1}(S×{0}; ℚ) and Λ′ := H^{1}({0}×S; ℚ), then the cohomology of S × S is the exterior algebra of H^{1}(S × S; ℚ) = Λ ⊕ Λ′. If we denote by f_{1}, …, f_{4}, resp. f_{5}, …, f_{8} the generators of Λ, resp. Λ′, the three involutions act of H^{1}(S × S; ℚ) as follows:

(5.2)  f_{i} ↦ −f_{i}, i = 1, …, 8,

(5.3)  f_{i} ↔ f_{i+4}, i = 1, …, 4,

(5.4)  f_{i} ↦ f_{i}, f_{i+4} ↦ f_{i} − f_{i+4}, i = 1, …, 4.

Then one proceeds to determine the invariant part of the exterior algebra of Λ⊕ Λ′ under these involution. Moreover, to determine the local systems R_{!}^{q}g′_{∗}(ℚ) that appear in the Leray spectral sequence associated with g′: ρ(Δ) → A_{2}, one needs to investigate the structure of the invariant subspaces as representations of Sp(4, ℚ). An explicit calculation of the invariant classes yields the results which we summarize in the following lemma.

Lemma 12. The rational cohomology groups the fibre of g′: ρ(Δ) → A_{2} over a point [S] ∈ A_{2}, with their mixed Hodge structures and structure as Sp(4, ℚ)- representations, are given by

H^{k}(g′^{−1}([S]); ℚ) = \{ \begin{matrix}ℚ & k = 0, \\ ⋀^{2} 𝕍_{1,0} = ℚ(−1) ⊕ 𝕍_{1,1} & k = 2, \\ ℚ(−2)^{⊕2} ⊕ 𝕍_{1,1}(−1) ⊕ 𝕍_{2,2} & k = 4, \\ (⋀^{2} 𝕍_{1,0})(−2) = ℚ(−3) ⊕ 𝕍_{1,1}(−2) & k = 6, \\ ℚ(−4) & k = 8, \\ 0 & otherwise. \end{matrix}

5.4. Proof of Proposition 8. We will prove Proposition 8 by investigating the Leray spectral sequence associated with the fibration k_{2} ∘ k_{3} : β_{2}^{0} → A_{2}. As explained at the beginning of this section, the fibre of k_{2} ∘ k_{3} over a point [S] ∈ A_{2} is the disjoint union of an open part, which is (k_{2} ∘ k_{3}|_{ρ(U)})^{−1}([S]), and a closed part, which is the fibre of g′: ρ(Δ) → A_{2}. The cohomology of the fibre of k_{2} ∘ k_{3}|_{ρ(U)} was determined in Lemma 11, whereas the cohomology of the fibre of g′ was computed in Lemma 12. Notice that (k_{2} ∘ k_{3}|_{ρ(U)})^{−1}([S]) = g^{−1}([S])/(j_{1}, j_{2}, ι) is the finite quotient of a smooth quasi-projective variety, so that we can use Poincaré duality to obtain its cohomology with compact support from its cohomology. Furthermore, since g^{′−1}([S]) = S^{2}/(i_{1}, i_{2}, i_{3}) is compact, its cohomology with compact support coincides with its cohomology. To compute the cohomology with compact support of the fibre of k_{2}∘k_{3} one can use the Gysin long exact sequence associated with the inclusion g^{′−1}([S]) ↪ (k_{2} ∘ k_{3})^{−1}([S]):

(5.5) ⋯ → H_{c}^{k}(g^{−1}([S]); ℚ)^{(j_{1},j_{2},ι)} → H_{c}^{k}((k_{2} ∘ k_{3})^{−1}([S]); ℚ) →

H_{c}^{k}(S × S; ℚ)^{(i_{1},i_{2},i_{3})} ⟶^{δ_{k}} H_{c}^{k+1}(g^{−1}([S]); ℚ)^{(j_{1},j_{2},ι)} → ⋯

Notice that all differentials δ_{k} in (5.5) have to respect the structure of the cohomology groups as representations of Sp(4, ℚ). In this specific case, this implies that all δ_{k} with k ≠ 2 vanish, whereas

δ_{2} : ℚ(−1) ⊕ 𝕍_{1,1} ⟶ 𝕍_{1,1}

is surjective by Lemma 13 below. The above determines the cohomology with compact support of the fibre of k_{2} ∘ k_{3}. In particular, it also determines the local systems R_{!}^{q}(k_{2} ∘ k_{3})_{∗}(ℚ) occurring in the Leray spectral sequence in cohomology with compact support associated with the fibration k_{2} ∘ k_{3}. These local systems are given in the first column of Table 8. Recall that the E_{2} term of the Leray spectral sequence E_{r}^{p,q} ⇒ H_{c}^{p+q}(β_{2}^{0}; ℚ) associated with k_{2} ∘ k_{3} are of the form E_{2}^{p,q} = H_{c}^{p}(A_{2}; R_{!}^{q}(k_{2} ∘ k_{3})_{∗}(ℚ)). From the decomposition into symplectic local systems of the R_{!}^{q}(k_{2} ∘ k_{3})_{∗}(ℚ), one gets the E_{2} term of the Leray spectral sequence as in Table 8. Here we used the description of the cohomology with compact support of A_{2} with coefficients in the local systems 𝕍_{1,1}, 𝕍_{2,0} and 𝕍_{2,2} from Lemma 34 and 36.

R_{!}^{q}(k_{2} ∘ k_{3})_{∗}(ℚ) q
ℚ(−5) 10  0  ℚ(−7)  0 ℚ(−8)
0 9  0  0  0  0
ℚ(−4)^{⊕2}⊕𝕍_{1,1}(−3) 8  ℚ(−3)  ℚ(−6)^{⊕2}  0 ℚ(−7)^{⊕2}
0 7  0  0  0  0
ℚ(−3)^{⊕3}⊕𝕍_{1,1}(−2) 6 ℚ(−2) ⊕ H(−1) ℚ(−5)^{⊕3} ⊕ H(−1) 0 ℚ(−6)^{⊕3}
⊕𝕍_{2,2}(−1)
𝕍_{2,0}(−1) 5  ℚ(−2)  0  0  0
ℚ(−2)^{⊕2}⊕𝕍_{1,1}(−1) 4  ℚ(−1) ⊕ H  ℚ(−4)^{⊕2} ⊕ H  0 ℚ(−5)^{⊕2}
⊕𝕍_{2,2}
0 3  0  0  0  0
ℚ(−1) 2  0  ℚ(−3)  0 ℚ(−4)
0 1  0  0  0  0
ℚ 0  0  ℚ(−2)  0 ℚ(−3)
3  4  5  6  p

Table 8. E_{2} term of the Leray spectral sequence converging to the cohomology with compact support of β_{2}^{0}. We denote H = H_{c}^{3}(A_{2}; 𝕍_{2,2}) ≅ H_{c}^{4}(A_{2}; 𝕍_{2,2}) (up to grading).

To prove the claim, it remains to show that the Leray spectral sequence degenerates at E_{2}. From the shape of the spectral sequence, it follows that all d_{2} differentials, and all differentials d_{r} with r ≥ 4 are necessarily trivial. The only differentials one needs to investigate are the d_{3}-differentials E_{3,q}^{3} → E_{6,q−2}^{3}. These are necessarily 0 by Hodge-theoretic reasons, because morphisms of Hodge structures between pure Hodge structures of different weights are necessarily trivial. From this the claim follows. □

Lemma 13. The differential

δ_{2} : H_{c}^{2}(S × S; ℚ)^{(i_{1},i_{2},i_{3})} → H_{c}^{3}(g^{−1}([S]); ℚ)^{(j_{1},j_{2},ι)}

is surjective.

Proof. We shall prove the claim by an explicit computation on the generators of the groups involved. Since in the proofs of Lemma 11 and Lemma 12 we described the cohomology of the fibres of E rather than those of the cohomology with compact support, to compute the rank of δ_{2} we shall compute the rank of the map induced by δ_{2} on cohomology by Poincaré duality

δ_{2}^{∗} : H^{6}(S × S; ℚ)^{(i_{1},i_{2},i_{3})} ⊗ ℚ(−1) ⟶ H^{7}(g^{−1}([S]); ℚ)^{(j_{1},j_{2},ι)},

which can be described explicitly as the composition of the map

\begin{matrix}H^{6}(S × S; ℚ)^{(i_{1},i_{2},i_{3})} & ⟶ & H^{7}(g^{−1}([S]); ℚ) \\ α & ⟼ & Q ⊗ α, \end{matrix}

where Q denotes the image of the generator of H^{1}(ℂ^{∗}; ℚ) inside the cohomology of g^{−1}([S]), and the symmetrization with respect to the group G generated by j_{1}, j_{2} and ι. A direct computation yields that the classes

v_{i,j,k,l} = f_{i} ∧ f_{j} ∧ f_{i+4} ∧ f_{j+4} ∧ (2f_{k} ∧ f_{l} + 2f_{k+4} ∧ f_{l+4} + f_{k} ∧ f_{l+4} + f_{k+4} ∧ f_{l})

with {i, j, k, l} = {1, 2, 3, 4} form a basis of H^{6}(S × S; ℚ)^{(i_{1},i_{2},i_{3})}. Here f_{1}, …, f_{8} denote the basis of f_{1}, …, f_{8} described in Section 5.3. Then we have

δ_{2}^{∗}(v_{i,j,k,l}) = Q ⊗ f_{i} ∧ f_{j} ∧ f_{i+4} ∧ f_{j+4} ∧ (f_{k} ∧ f_{l+4} + f_{k+4} ∧ f_{l})

and these classes generate H^{7}(g^{−1}([S]); ℚ). From this the claim follows. □

6. Torus rank 3

In this section we compute the cohomology with compact support of the stratum with torus rank 3. As in the previous section, our strategy is based on a detailed geometric analysis of the fibration β_{3}^{0} → A_{1} whose toric part is actually independent of the choice of g = 4.

6.1. Description of the geometry. We first note that the spaces A_{4}^{Igu} and A_{4}^{Vor} only differ over A_{0} and hence β_{3}^{perf} \ β_{4}^{perf} = β_{3}^{Vor} \ β_{4}^{Vor} =: β_{3}^{0}. In this section we want to compute H_{c}^{•}(β_{0}^{3}; ℚ). For this we first give a geometric description. In order to compactify A_{4} we start with the lattice ℤ^{4}. The choice of a toroidal compactification corresponds to the choice of an admissible fan Σ_{4} in the cone of semi-positive forms in Sym^{2}(ℤ^{4}). One possible choice for such a fan is given by the perfect cone decomposition Σ_{4}^{perf}. A cusp of A_{4} corresponds to the choice of an isotropic subspace U ⊂ ℚ^{4}. In our case, for the stratum over A_{1} we take U = ⟨e_{1}, e_{2}, e_{3}⟩ where the e_{i} (1 ≤ i ≤ 4) are the standard basis of ℤ^{4}. This defines an embedding Sym^{2}(ℤ^{3}) ⊂ Sym^{2}(ℤ^{4}) and, by restriction of Σ_{4}^{perf}, also a fan in Sym^{2}(ℤ^{3}) which is nothing but Σ_{3}^{perf}. The stratum β_{3}^{0} itself consists of different strata which are in 1 : 1-correspondence with the GL(3, ℤ)-orbits of the cones σ in Σ_{3}^{perf} whose interior contains rank 3 matrices. Up to the action of GL(3, ℤ) there is a unique minimal cone with this property, namely the cone σ^{(3)} = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}⟩. Beyond that there are (again up to group action) 4 further cones. In dimension 4 there are two cones, namely σ_{I}^{(4)} = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, (x_{2} − x_{3})^{2}⟩ and σ_{II}^{(4)} = ⟨x_{1}^{2}, x_{2}^{2}, (x_{2} − x_{3})^{2}, (x_{1} − x_{3})^{2}⟩. In dimensions 5 and 6 there are one cone each, namely σ^{(5)} = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, (x_{2}−x_{3})^{2}, (x_{1}−x_{3})^{2}⟩ and σ^{(6)} = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, (x_{2} − x_{3})^{2}, (x_{1} − x_{3})^{2}, (x_{1} − x_{2})^{2}⟩. Note that all cones are contained in σ^{(6)}. In fact the perfect cone decomposition in genus 3 (where it coincides with the second Voronoi decomposition) is obtained by taking the GL(3, ℤ)-orbit of σ^{(6)} and all its faces. To describe the various strata let X_{1} → A_{1} be the universal elliptic curve and let X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} → A_{1} be the triple product with itself over A_{1}. Let T = Sym^{2}(ℤ^{3}) ⊗ ℂ^{∗} be the 6-dimensional torus associated with Sym^{2}(ℤ^{3}). Every cone σ in Σ_{3}^{perf} is basic (i.e. the generators of the rays are part of a ℤ-basis of Sym^{2}(ℤ^{3})) and defines a subtorus T^{σ} ⊂ T of rank dim(σ). We can now give a description of β_{3}^{0}.

Proposition 14. The variety β_{3}^{0} admits a stratification into strata as follows:

(i) there are 6 strata of β_{3}^{0}, corresponding to the cones σ^{(3)}, σ_{I}^{(4)}, σ_{II}^{(4)}, σ^{(5)}

and σ^{(6)}.

(ii) Each stratum is the finite quotient of a torus bundle over X_{1}×_{A_{1}} X_{1}×_{A_{1}}

X_{1} → A_{1} with fibre T/T^{σ}.

Proof. See [S-B, Lemma 2.4]. □

We shall now compute the cohomology with compact support for each of these strata and then use a spectral sequence argument to compute the cohomology with compact support of β_{3}^{0}. We denote the substratum of β_{3}^{0} associated with a cone σ by β_{3}^{0}(σ) and the total space of the torus bundle by T (σ). Before we state the results we have to give a brief outline of the construction of the stratum β_{3}^{0}(σ) with a view towards describing suitable coordinates in which our calculations can be done. Consider a point in Siegel space of genus 4:

τ = (\begin{matrix}τ_{1,1} & τ_{1,2} & τ_{1,3} & τ_{1,4} \\ τ_{1,2} & τ_{2,2} & τ_{2,3} & τ_{2,4} \\ τ_{1,3} & τ_{2,3} & τ_{3,3} & τ_{3,4} \\ τ_{1,4} & τ_{2,4} & τ_{3,4} & τ_{4,4} \end{matrix}) ∈ ℍ_{4}.

Going to the cusp over A_{1} means sending the top left hand 3 × 3 block of this matrix to i∞. We shall make this more precise. We consider the basis of Sym^{2}(ℤ^{3}) given by U_{i,j}^{∗} = (2 − δ_{i,j})x_{i}x_{j}. Let t_{i,j} (1 ≤ i, j ≤ 3) be the dual basis. Setting

t_{i,j} = e^{2π\sqrt{−1}τ_{i,j}} (1 ≤ i, j ≤ 3)

defines a map

(6.1)  ℍ_{4} → T × ℂ^{3} × ℍ_{1}
τ ↦ ((t_{i,j}), τ_{1,4}, τ_{2,4}, τ_{3,4}, τ_{4,4}).

This corresponds to taking the partial quotient X(U) = P′(U)\ℍ_{4} with respect to the center P′(U) of the unipotent radical of the parabolic subgroup P (U) associated with the cusp U. We denote P′′(U) = P (U)/P′(U). The partial quotient X(U) can be considered as an open set of the trivial torus bundle X (U) (with fibre T) over ℂ^{3} × ℍ_{1}. Using the fan Σ_{3}^{perf} one constructs X_{Σ_{3}^{perf}} (U) by taking a fibrewise toric embedding. Let X_{Σ_{3}^{perf}} (U) be the interior of the closure of X(U) in X_{Σ_{3}^{perf}} (U). The action of the group P′′(U) on X(U) extends to an action on X_{Σ_{3}^{perf}} (U) and one obtains the partial compactification in the direction of the cusp U by Y_{Σ_{3}^{perf}} (U) = P′′(U)\X_{Σ_{3}^{perf}} (U).

Every cone σ ∈ Σ_{3}^{perf} defines an affine toric variety X_{σ}. Since all cones σ are basic one has X_{σ} = ℂ^{k} × (ℂ^{∗})^{6−k} where k is the number of generators of σ. Every inclusion σ ⊂ σ′ induces an inclusion X_{σ} ⊂ X_{σ′}. Note that X_{(0)} = T and, in particular we obtain an inclusion X_{(0)} = T ⊂ X_{σ^{(6)}} ≅ ℂ^{6}. Let T_{1}, …, T_{6} be the coordinates on X_{σ^{(6)}} ≅ ℂ^{6} corresponding to the generators of σ^{(6)} which form a basis of Sym^{2}(ℤ^{3}). Computing the dual basis of this basis one finds that this inclusion is given by

(6.2)  \begin{matrix}T_{1} = t_{1,1}t_{1,3}t_{1,2}, & T_{2} = t_{2,2}t_{2,3}t_{1,2}, & T_{3} = t_{3,3}t_{1,3}t_{2,3}, \\ T_{4} = t_{2,3}^{−1}, & T_{5} = t_{1,3}^{−1}, & T_{6} = t_{1,2}^{−1}. \end{matrix}

The relation to the strata β_{3}^{0}(σ) is then the following. The coordinate τ_{4,4} defines a point in A_{1} and the coordinates τ_{1,4}, τ_{2,4}, τ_{3,4} define a point in the fibre of X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} → A_{1} over [τ_{4,4}] ∈ A_{1} which is E_{τ_{4,4}} × E_{τ_{4,4}} × E_{τ_{4,4}}, where E_{τ_{4,4}} = ℂ/(ℤ + ℤτ_{4,4}) is the elliptic curve defined by τ_{4,4}. The fibres of β_{3}^{0}(σ) → X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} are isomorphic to the torus T/T^{σ}. Finally, we have to make some comments on the structure of the parabolic subgroup P (U). This group is generated by four types of matrices. The first type are block matrices of the form

g_{1} = (\begin{matrix}1 & 0 & S & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , where S = ^{t}S ∈ Sym^{2}(ℤ^{3}).

These matrices generate the center P′(U) of the unipotent radical and act by

(\begin{matrix}τ_{1,1} & τ_{1,2} & τ_{1,3} & τ_{1,4} \\ τ_{1,2} & τ_{2,2} & τ_{2,3} & τ_{2,4} \\ τ_{1,3} & τ_{2,3} & τ_{3,3} & τ_{3,4} \\ τ_{1,4} & τ_{2,4} & τ_{3,4} & τ_{4,4} \end{matrix}) → (\begin{matrix}τ_{1,1} + s_{1,1} & τ_{1,2} + s_{1,2} & τ_{1,3} + s_{1,3} & τ_{1,4} \\ τ_{1,2} + s_{1,2} & τ_{2,2} + s_{2,2} & τ_{2,3} + s_{2,3} & τ_{2,4} \\ τ_{1,3} + s_{1,3} & τ_{2,3} + s_{2,3} & τ_{3,3} + s_{3,3} & τ_{3,4} \\ τ_{1,4} & τ_{2,4} & τ_{3,4} & τ_{4,4} \end{matrix})

giving rise to the partial quotient ℍ_{4} → T × ℂ^{3} × ℍ_{1} described above. The second set of generators is of the form

g_{2} = (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & a & 0 & b \\ 0 & 0 & 1 & 0 \\ 0 & c & 0 & d \end{matrix}) , where (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, ℤ),

resp.

g_{3} = (\begin{matrix}1 & M & 0 & N \\ 0 & 1 & ^{t}N & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & −^{t}M & 1 \end{matrix}) , where M, N ∈ ℤ^{3}.

Note that the elements of type g_{2}, g_{3} generate a Jacobi group, which, in particular, acts on the base ℂ^{3} × ℍ_{1} of the partial quotient by P′(U) given by the map ℍ_{4} → ℂ^{3} × ℍ_{1} giving rise to the triple product X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1}.

Finally we have matrices of the form

g_{4} = (\begin{matrix}^{t}Q^{−1} & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & Q & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) , where Q ∈ GL(3, ℤ).

These matrices are of particular importance to us as they operate on the space Sym^{2}(ℤ^{3}) by

GL(3, ℤ) ∋ g : M ↦ ^{t}Q^{−1}MQ^{−1}.

6.2. The cohomology of β_{3}^{0}(σ^{(3)}). In this section we will prove

Lemma 15. The rational cohomology groups with compact support of β_{3}^{0}(σ^{(3)}) are given by

H_{c}^{k}(β_{3}^{0}(σ^{(3)}); ℚ) = \{ \begin{matrix}ℚ(−k/2) & k = 12, 14 \\ ℚ((5 − k)/2) & k = 9, 7 \\ 0 & otherwise. \end{matrix}

We start by giving an explicit description of the torus bundle T (σ^{(3)}) defined by the cone σ^{(3)}.

Lemma 16. Let q_{σ^{(3)}} : T (σ^{(3)}) → X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} be the rank 3 torus bundle associated with σ^{(3)}. Then over each fibre E × E × E of X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} we have

T (σ^{(3)})|_{E×E×E} ≅ p_{2,3}^{∗}(P^{0}) ⊕ p_{1,3}^{∗}(P^{0}) ⊕ p_{1,2}^{∗}(P^{0})

where P^{0} is the Poincaré bundle over the product E × E with the 0-section removed, and p_{i,j}: E × E × E → E × E is the projection to the ith and jth factor.

Proof. We first recall the following description of the Poincaré bundle over E × E where E = ℂ/(ℤ + ℤτ). Consider the action of the group ℤ^{4} on the trivial rank-1 bundle on ℂ × ℂ given by

(6.3)  (n_{1}, n_{2}, m_{1}, m_{2}): (z_{1}, z_{2}, w) ↦
(z_{1} + n_{1} + m_{1}τ, z_{2} + n_{2} + m_{2}τ, we^{−2πi(m_{1}z_{2}+m_{2}z_{1}+m_{1}m_{2}τ)})

(where the z_{i} are the coordinates on the base and w is the fibre coordinate). We claim that the quotient line bundle on E × E is the Poincaré bundle. For this it is enough to see that this line bundle is trivial on E × {0} and {0} × E (which is obvious) and that it is isomorphic to O_{E}(O − P) on E × {P }. The latter can be checked by comparing the transformation behaviour of (6.3) to the transformation behaviour of the theta function ϑ(z, τ) in one variable (see e.g. [La, 15.1.3.]). We have to compare this to our situation. In this case we have an action of the group generated by the matrices g_{3} with M, N ∈ ℤ^{3}. For N = (n_{1}, n_{2}, n_{3}) we have τ_{i,4} ↦ τ_{i,4} + n_{i} and for M = (m_{1}, m_{2}, m_{3}) we have τ_{i,j} ↦ τ_{i,j} + m_{j}τ_{i,4} + m_{i}τ_{j,4} + m_{i}m_{j}τ_{4,4} for 1 ≤ i, j ≤ 3 and τ_{i,4} ↦ τ_{i,4} + m_{i}τ_{4,4}. Recall that the entries τ_{i,4} for i = 1, 2, 3 are coordinates on the factors of E × E × E and that it follows from (6.2) that we can choose t_{i,j}^{−1} with t_{i,j} = e^{2πiτ_{i,j}} for (i, j) = (1, 2), (1, 3), (2, 3) as coordinates on the torus T (σ^{(3)}). Comparing this to the transformation (6.3) gives the claim. □

Proof of Lemma 15. Recall that the stratum β_{3}^{0}(σ^{(3)}) is a finite quotient of the rank 3 torus bundle q_{σ^{(3)}} : T (σ^{(3)}) → X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1}. This enables us to calculate its rational cohomology by exploiting Leray spectral sequences. Notice that the base of q_{σ^{(3)}} is the total space of the fibration p: X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} → A_{1}. Over a point [E] ∈ A_{1}, the fibre of p is p^{−1}([E]) ≅ E × E × E and the fibre of p ∘ q_{σ^{(3)}} over [E] is the total space of the rank 3 torus bundle q_{σ^{(3)}}|_{E×E×E} : T (σ^{(3)})|_{E×E×E} → E × E × E described in Lemma 16. The cohomology of (p∘q_{σ^{(3)}})^{−1}([E]) can be computed by the Leray spectral sequence associated with this rank 3 torus bundle:

(6.4) E_{2}^{p,q}(q_{σ^{(3)}}) = H^{q}(T/T^{(σ^{(3)})}; ℚ) ⊗ H^{p}(E × E × E; ℚ)

⟹ H^{p+q}((p ∘ q_{σ^{(3)}})^{−1}([E]); ℚ).

Note that the cohomology of E × E × E (respectively, the torus T/T^{(σ^{(3)})}) is an exterior algebra generated by H^{1}(E × E × E; ℚ) (resp. H^{1}(T/T^{(σ^{(3)})}; ℚ)). We denote by Q_{1}, Q_{2} and Q_{3}, respectively, the generators of H^{1}(T/T^{σ^{(3)}}; ℚ) ≅ H^{1}((ℂ^{∗})^{3}; ℚ) ≅ ℚ^{3} defined by integrating along the loop around 0 defined, respectively, by |t_{2,3}^{−1}| = 1, |t_{1,3}^{−1}| = 1 or |t_{1,2}^{−1}| = 1. We can write each copy of E as a quotient E = ℂ/(ℤe_{2i−1} + ℤe_{2i}); i = 1, 2, 3. Then e_{1}, …, e_{6} give rise to a basis of the first homology group of E × E × E. We will denote by f_{1}, …, f_{6} the elements of the basis of H^{1}(E ×E ×E; ℚ) dual to e_{1}, …, e_{6}. Notice that the transformation behaviour of the f_{2i−1} and of the f_{2i} for 1 ≤ i ≤ 3 agrees with the transformation behaviour of the coordinates {τ_{i,4}| 1 ≤ i ≤ 3} of ℂ^{3} ≅ (ℤe_{1} + ℤe_{2} + ⋯ + ℤe_{6}) ⊗_{ℤ} ℂ (and that of the differentials dτ_{i,4} which give rise to classes in cohomology). As we are interested in the quotient of T (σ^{(3)}) by the finite group G(σ^{(3)}), we shall compute the invariant cohomology with respect to this group. This is done in Lemma 17 for the invariant cohomology of the fibre T (σ^{(3)})|_{E×E×E} = (p ∘ q_{σ^{(3)}})^{−1}([E]) using a Leray spectral sequence argument. It remains to determine the local systems R_{!}^{i}(p ∘ q_{σ^{(3)}})_{∗}(ℚ) over A_{1} defined by the fibration p ∘ q_{σ^{(3)}} : β_{3}^{0}(σ^{(3)}) = T (σ^{(3)}) → A_{1}. This is quite straightforward, since the cohomology with compact support of the fibre is constant in degrees 12 and 10, and since Sym^{2} H^{1}(E; ℚ) induces the symplectic local system 𝕍_{2} on A_{1}. Recall that the cohomology with compact support of A_{1} with constant coefficients is concentrated in degree 2, and that the only non-trivial cohomology group of A_{1} with coefficients in 𝕍_{2} is H_{c}^{1}(A_{1}; 𝕍_{2}) = ℚ (see e.g. [G1, Thm. 5.3]). In particular, it then follows from Lemma 17 that the Leray spectral sequence associated with p ∘ q_{σ^{(3)}} has only two columns containing non-trivial E_{2} terms, so it has to degenerate at E_{2}. From this the claim follows. □

Lemma 17. For every [E] ∈ A_{1}, the rational cohomology with compact support of the fibre of β_{3}^{0}(σ^{(3)}) → A_{1}, with its Hodge structures, coincides with the G(σ^{(3)})-invariant part of the cohomology with compact support of the rank 3 torus bundle T (σ^{(3)})|_{E×E×E} and is given by

(H_{c}^{k}(T (σ^{(3)})|_{E×E×E}; ℚ))^{G(σ^{(3)})} = \{ \begin{matrix}ℚ(−6) & k = 12, \\ ℚ(−5) & k = 10, \\ Sym^{2}(H^{1}(E; ℚ)) ⊗ ℚ(−2) & k = 8, \\ Sym^{2}(H^{1}(E; ℚ)) ⊗ ℚ(−1) & k = 6, \\ 0 & otherwise. \end{matrix}

Proof. The stabilizer G(σ^{(3)}) of σ^{(3)} in GL(3, ℤ) is an extension of the symmetric group 𝔖_{3} (permuting the coordinates x_{1}, x_{2}, x_{3}) by (ℤ/2ℤ)^{3} (acting by involutions (x_{1}, x_{2}, x_{3}, x_{4}) ↦ (±x_{1}, ±x_{2}, ±x_{3}, x_{4})). The interchange of two coordinates (say, x_{i} and x_{j}) acts on H^{1}(E × E × E; ℚ) by interchanging f_{2i−1} with f_{2j−1}, f_{2i} with f_{2j} and leaving all other generators invariant. The action on H^{1}(T/T^{(σ^{(3)})}; ℚ) interchanges Q_{i} and Q_{j} and leaves the third generator invariant. The automorphism mapping x_{i} to −x_{i} acts on H^{1}(E × E × E; ℚ) as multiplication by −1 on the generators f_{2i−1}, f_{2i} and on H^{1}(T/T^{(σ^{(3)})}; ℚ) as multiplication by −1 on Q_{k} with k ≠ i. All other generators are invariant. We can compute the G(σ^{(3)})-invariant part of the rational cohomology of the rank 3 torus bundle T (σ^{(3)})|_{E×E×E} by restricting to the G(σ^{(3)})- invariant part of the Leray spectral sequence (6.4) associated with q_{σ^{(3)}}. This yields a spectral sequence E_{2}^{p,q} converging to the G(σ^{(3)})-invariant part of H^{p+q}(T (σ^{(3)})|_{E×E×E}; ℚ). A computation of the part of the tensor product ⋀^{•} H^{1}(E × E × E; ℚ) ⊗ ⋀^{•} H^{1}(T/T^{(σ^{(3)})}; ℚ) which is invariant under G(σ^{(3)}) yields that E_{2}^{p,q} is nonzero only for (p, q) ∈ {(2, 0), (2, 1), (4, 0), ((2, 2), (4, 1), (6, 0)}. A precise description of the generators of the non-trivial E_{2} terms is given in Table 9. Note that the spaces for p = q = 2 and p = 4, q = 2 are both isomorphic to Sym^{2} H^{1}(E; ℚ) as Sp(2, ℚ)-representations. Next, one investigates the differentials of the spectral sequence. As differentials have to occur between E_{2}^{p,q} terms such that the two p + q have different parity, an inspection of the spectral sequence quickly reveals that all differentials have to be trivial, with the possible exception of

(6.5)  d_{2}^{2,1} : E_{2}^{2,1} → E_{2}^{4,0}

and

(6.6)  d_{2}^{4,1} : E_{2}^{4,1} → E_{2}^{6,0}.

We can determine their rank by exploiting the description of the restriction to E × E × E of the torus bundle T (σ^{(3)}) given in Lemma 16 as a direct sum of pull-backs of the Poincaré bundle with the 0-section removed. This description implies that one can employ the usual description of d_{2} differentials of ℂ^{∗}- bundles to investigate d_{2}^{2,1} and d_{2}^{4,1}. In particular, each of these differentials is

p q dim. generators
0 0 1 1
2 0 1 ∑_{i} f_{2i−1} ∧ f_{2i}
2 1 1 ∑_{i<j,k≠i,j} Q_{k} ⊗ (f_{2i−1} ∧ f_{2j} − f_{2i} ∧ f_{2j−1})
2 2 3 ∑_{i<j,k≠i,j} Q_{i} ∧ Q_{j} ⊗ W_{k}^{(m)}, m = 1, 2, 3
4 0 1 ∑_{i<j} f_{2i−1} ∧ f_{2i} ∧ f_{2j−1} ∧ f_{2j}
4 1 1 ∑_{i<j,k≠i,j} Q_{k} ⊗ (f_{2i−1} ∧ f_{2j} − f_{2i} ∧ f_{2j−1}) ∧ f_{2k−1} ∧ f_{2k}
4 2 3 ∑_{i<j,k≠i,j} Q_{i} ∧ Q_{j} ⊗ W_{k}^{(m)} ∧ f_{2k−1} ∧ f_{2k}, m = 1, 2, 3
6 0 1 f_{1} ∧ f_{2} ∧ f_{3} ∧ f_{4} ∧ f_{5} ∧ f_{6}.

All indices i, j, k are between 1 and 3. For indices i < j we set W_{k}^{(1)} = f_{2i−1} ∧ f_{2j} + f_{2i} ∧ f_{2j−1}, W_{k}^{(2)} = f_{2i−1} ∧ f_{2j−1} and W_{k}^{(3)} = f_{2i} ∧ f_{2j} for k ≠ i, j.

Table 9. Description of the generators of the E_{2} terms of the G(σ^{(3)})-invariant part of the spectral sequence associated with q_{σ^{(3)}}.

given by formally replacing each generator Q_{k} of H^{1}(T/T^{σ^{(3)}}; ℚ) by the first Chern class of the bundle p_{i,j}^{∗}(P), where 1 ≤ i < j ≤ 3 are chosen such that {i, j, k} = {1, 2, 3}. Recall that on the product E × E the Poincaré bundle P ≅ O_{E×E}(E × {0} + {0} × E − Δ), where Δ is the diagonal. From this one concludes that c_{1}(P) = f_{1} ∧f_{2} +f_{3} ∧f_{4} −(f_{1} +f_{3})∧(f_{2} +f_{4}) = f_{2} ∧f_{3} −f_{1} ∧f_{4}. It is then a straightforward calculation to prove that both differentials are isomorphisms. It remains to pass from cohomology to cohomology with compact support, which we can do by Poincaré duality, using the fact that T (σ^{(3)})|_{E×E×E} is smooth of complex dimension 6. Finally, we can identify the G(σ^{(3)})-invariant part of the cohomology with compact support of T (σ^{(3)})|_{E×E×E} with the cohomology with compact support of its finite quotient (T (σ^{(3)})|_{E×E×E}) /G(σ^{(3)}), which coincides with the fibre of β_{3}^{0}(σ^{(3)}) → A_{1} over [E]. □

6.3. The cohomology of β_{3}^{0}(σ_{I}^{(4)}). In this section we will prove Lemma 18. The rational cohomology groups with compact support of β_{3}^{0}(σ_{I}^{(4)}) are given by

H_{c}^{k}(β_{3}^{0}(σ_{I}^{(4)}); ℚ) = \{ \begin{matrix}ℚ(−6) & k = 12 \\ ℚ(−5)^{⊕2} & k = 10 \\ ℚ(−4) + ℚ(−2) & k = 8 \\ ℚ & k = 5 \\ 0 & otherwise. \end{matrix}

Proof. We shall make again use of the twofold fibre structure of this stratum. The stratum β_{3}^{0}(σ_{I}^{(4)}) is a finite quotient of a rank 2 torus bundle q_{σ_{I}^{(4)}} : T (σ_{I}^{(4)}) → X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} with fibres isomorphic to T/T^{(σ_{I}^{(4)})}.

Note that the generators of σ_{I}^{(4)} correspond to the first four generators of the cone σ^{(6)}. Comparing this to the embedding described in (6.2) we find that we can choose t_{1,3}^{−1}, t_{1,2}^{−1} as coordinates on T/T^{(σ_{I}^{(4)})}. As before we denote p: X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} → A_{1}. As we are interested in the quotient of T (σ_{I}^{(4)}) by the finite group G(σ_{I}^{(4)}), we shall compute the invariant cohomology with respect to this group. Thus we first have to describe the automorphism group G(σ_{I}^{(4)}) of the cone σ_{I}^{(4)}, i.e. all elements of the form g_{3} ∈ GL(3, ℤ) which fix this cone. We have already discussed this in [HT, Section 3]. The result is that the automorphism group is generated by the following four transformations:

(6.7)  x_{1} ↦ x_{1}, x_{2} ↦ x_{2} − x_{3}, x_{3} ↦ −x_{3}
(6.8)  x_{1} ↦ −x_{1}, x_{2}, x_{3} ↦ x_{2}, x_{3}
(6.9)  x_{1} ↦ x_{1}, x_{2} ↔ x_{3}.
(6.10)  x_{i} ↦ −x_{i}; i = 1, 2, 3.

Note that these automorphisms act trivially on the base of the fibration X_{1} ×_{A_{1}} X_{1} ×_{A_{1}} X_{1} → A_{1}. Again we shall determine the invariant cohomology of the fibre (q_{σ_{I}^{(4)}} ∘p)^{−1}([E])

using the Leray spectral sequence with terms E_{2}^{p,q} = H^{q}(T/T^{(σ_{I}^{(4)})}, ℚ)⊗H^{p}(E× E × E, ℚ). The result is given by:

Lemma 19. For every [E] ∈ A_{1}, the rational cohomology with compact support of the fibre of β_{3}^{0}(σ_{I}^{(4)}) → A_{1}, with its Hodge structures, is given by

(H_{c}^{k}(T (σ_{I}^{(4)})|_{E×E×E}; ℚ))^{G(σ_{I}^{(4)})} = \{ \begin{matrix}ℚ(−5) & k = 10, \\ ℚ(−4)^{⊕2} & k = 8 \\ Sym^{2}(H^{1}(E; ℚ)) ⊗ ℚ(−2) & k = 7, \\ ℚ(−3) & k = 6 \\ Sym^{2}(H^{1}(E; ℚ)) & k = 4, \\ 0 & otherwise. \end{matrix}

Proof. We denote the generators of H^{1}(T/T^{(σ_{I}^{(4)})}; ℚ) ≅ H^{1}((ℂ^{∗})^{2}; ℚ) ≅ ℚ^{2} corresponding to t_{1,3}^{−1}, t_{1,2}^{−1} by Q_{2}, Q_{3}. The f_{i}, i = 1, …, 6 are, as before, a basis of the cohomology of the triple product E × E × E. We must now compute the action on (co)homology of the automorphisms of σ_{I}^{(4)}. As a non-trivial example we shall do this in detail in the case of the transformation given in (6.7), the computations in the other cases are analogous. The action of this transformation on Siegel space is given by:

(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & −1 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix}τ_{1,1} & τ_{1,2} & τ_{1,3} & τ_{1,4} \\ τ_{1,2} & τ_{2,2} & τ_{2,3} & τ_{2,4} \\ τ_{1,3} & τ_{2,3} & τ_{3,3} & τ_{3,4} \\ τ_{1,4} & τ_{2,4} & τ_{3,4} & τ_{4,4} \end{matrix}) (\begin{matrix}1 & 0 & 0 & 0 \\ 0 & 1 & −1 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) =

= (\begin{matrix}τ_{1,1} & τ_{1,2} & −τ_{1,2} − τ_{1,3} & τ_{1,4} \\ τ_{1,2} & τ_{2,2} & −τ_{2,2} − τ_{2,3} & τ_{2,4} \\ −τ_{1,2} − τ_{1,3} & −τ_{2,2} − τ_{2,3} & τ_{2,2} + 2τ_{2,3} + τ_{3,3} & −τ_{2,4} − τ_{3,4} \\ τ_{1,4} & τ_{2,4} & −τ_{2,4} − τ_{3,4} & τ_{4,4} \end{matrix}) .

From this we conclude that under this transformation:

(6.11)  Q_{2} ↦ −Q_{2} − Q_{3}; Q_{3} ↦ Q_{3};
f_{i} ↦ f_{i}, i = 1, …, 4 f_{i} ↦ −f_{i−2} − f_{i}; i = 5, 6,

Note that the latter coincides with the transformation behaviour of the differentials dτ_{i,4}, i = 1, 2, 3, and the former with the transformation behaviour of −τ_{1,3}, −τ_{1,2}. An analogous computation for the other automorphisms gives the following results:

(6.12)  Q_{2}, Q_{3} ↦ −Q_{2}, −Q_{3};
f_{1}, f_{2} ↦ −f_{1}, −f_{2}, f_{i} ↦ f_{i}, i = 3, …, 6.
(6.13)  Q_{2}, Q_{3} ↦ Q_{3}, Q_{2};
f_{1}, f_{2} ↦ f_{1}, f_{2}, f_{3} ↔ f_{5}, f_{4} ↔ f_{6},
(6.14)  Q_{2}, Q_{3} ↦ Q_{2}, Q_{3};
f_{i} ↦ −f_{i}, i = 1, …, 6.

Now we must compute the invariant cohomology with respect to G(σ_{I}^{(4)}). This can either be done by a (lengthy) computation by hand or a standard computer algebra system. The E_{2}^{p,0} terms of the spectral sequence can be computed as follows. The invariant part E_{2}^{2,0} of the cohomology group H^{0}(T/T^{σ_{I}^{(4)}}; ℚ)⊗H^{2}(E×E×E; ℚ) is two-dimensional and generated by the tensors

I_{1} = f_{1} ∧ f_{2}, I_{2} = 2(f_{3} ∧ f_{4} + f_{5} ∧ f_{6}) + (f_{3} ∧ f_{6} + f_{5} ∧ f_{4}).

The term E_{2}^{4,0} is also two-dimensional, with generators I_{1} ∧ I_{2} and I_{2} ∧ I_{2}. The terms E_{2}^{0,0} ≅ H^{0}(T/T^{σ_{I}^{(4)}}; ℚ)⊗H^{0}(E ×E ×E; ℚ) and E_{2}^{6,0} ≅ H^{0}(T/T^{σ_{I}^{(4)}}; ℚ)⊗ H^{6}(E × E × E; ℚ) are one-dimensional and generated by fundamental classes.

Table 10. E_{2} term of the spectral sequence converging to the cohomology of β_{3}^{0}(σ_{I}^{(4)})

q

2 0 0  𝕍_{2}(−2)  0  𝕍_{2}(−3)  0  0
1 0 0 𝕍_{2}(−1) ⊕ ℚ(−2) 0 𝕍_{2}(−2) ⊕ ℚ(−3) 0  0
0 ℚ 0  ℚ(−1)^{⊕2}  0  ℚ(−2)^{⊕2}  0 ℚ(−3)
0 1  2  3  4  5  6  p

The term E_{2}^{2,1}, which is the invariant part of H^{1}(T/T^{σ_{I}^{(4)}}; ℚ)⊗H^{2}(E×E×E; ℚ) is 4-dimensional, with generators

g_{i,j} = ((Q_{2} + 2Q_{3}) ⊗ f_{j+2} + (2Q_{2} + Q_{3}) ⊗ f_{j+4}) ∧ f_{i}, i, j = 1, 2.

In particular, it is isomorphic to H^{1}(E; ℚ) ⊗ H^{1}(E; ℚ) = Sym^{2}(H^{1}(E; ℚ)) ⊕ ⋀^{2} H^{1}(E; ℚ). The term E_{2}^{4,1} is also four-dimensional and generated by (g_{i,j} ∧ I_{2}). All other E_{2}^{p,1} vanish. Finally, the only non-trivial terms of the form E_{2}^{p,2} are those with p = 2 and p = 4. The subspace E_{2}^{2,2} ⊂ H^{2}(T/T^{σ_{I}^{(4)}}; ℚ) ⊗ H^{2}(E × E × E; ℚ) is isomorphic to Sym^{2}(H^{1}(E; ℚ)) and is generated by the invariant tensors

Q_{2} ∧ Q_{3} ⊗ f_{3} ∧ f_{5}, Q_{2} ∧ Q_{3} ⊗ (f_{3} ∧ f_{6} + f_{4} ∧ f_{5}), Q_{2} ∧ Q_{3} ⊗ f_{4} ∧ f_{6}.

Finally, the subspace E_{2}^{4,2} is 4-dimensional and equal to E_{2}^{2,2} ∧ I_{1}. In terms of local systems this gives rise to the Table 10. We claim that that the differentials d_{2}^{p,q} : E_{2}^{p,q} → E_{2}^{p+2,q−1} for (p, q) = (2, 1), (2, 2) and (4, 1) are of maximal rank. Indeed, by Schur’s lemma it is enough to prove that they are non-zero. To check this it is enough to recall that the torus bundle is isomorphic to p_{1,3}^{∗}(P^{0}) ⊕ p_{1,2}^{∗}(P^{0}). In particular, for every class α ∈ E_{2}^{p,1} we obtain d_{2}^{p,1}(α) by replacing Q_{2} with c_{1}(p_{1,3}^{∗}(P)) = −(f_{1} ∧ f_{6} + f_{5} ∧ f_{2}) and Q_{3} with c_{1}(p_{1,3}^{∗}(P)) = −(f_{1} ∧ f_{4} + f_{3} ∧ f_{2}) in the expression of α. Analogously, for every class β ∈ E_{2}^{2,2} we get d_{2}^{2,2}(β) by replacing Q_{2}∧Q_{3} with Q_{2}⊗c_{1}(p_{1,2}^{∗}(P))− Q_{3}⊗c_{1}(p_{1,3}^{∗}(P)). Then the claim follows from a straightforward calculation. □

To complete the proof of Lemma 18 is now an easy consequence of the Leray spectral sequence of the fibration p ∘ q_{σ_{I}^{(4)}} : T (σ_{I}^{(4)}) → A_{1}. Looking at the weights of the Hodge structures, we see immediately that all differentials must vanish and thus the result follows. □

6.4. The cohomology of β_{3}^{0}(σ_{II}^{(4)}). Before we can describe the cohomology of this stratum we must identify the toric bundle T (σ_{II}^{(4)}).

Lemma 20. Let p_{1,2} : E × E × E → E × E be the projection onto the first two factors and let q : E × E × E → E × E be the map given by q(x, y, z) = (x + y + z, z). Then

T (σ_{II}^{(4)})|_{E×E×E} ≅ p_{1,2}^{∗}(P^{0}) ⊕ q^{∗}((P^{−1})^{0}).

where P^{0} is the Poincaré bundle over the product E × E with the 0-section removed.

Proof. Since the generators of the cone σ_{II}^{(4)} correspond to T_{1}, T_{2}, T_{4}, T_{5} we can take T_{6} = t_{1,2}^{−1} and T_{3} = t_{3,3}t_{1,3}t_{2,3} as coordinates on the torus T/T^{(σ_{II}^{(4)})}. In Lemma 17 we had seen that the action of the group generated by the matrices g_{3} with M, N ∈ ℤ^{3} is as follows. For N = (n_{1}, n_{2}, n_{3}) we have τ_{i,4} ↦ τ_{i,4} + n_{i} and for M = (m_{1}, m_{2}, m_{3}) we have τ_{i,j} ↦ τ_{i,j} + m_{j}τ_{i,4} + m_{i}τ_{j,4} + m_{i}m_{j}τ_{4,4} for 1 ≤ i, j ≤ 3 and τ_{i,4} ↦ τ_{i,4} + m_{i}τ_{4,4}. In particular

τ_{1,2} ↦ τ_{1,2} + m_{2}τ_{1,4} + m_{1}τ_{2,4} + m_{1}m_{2}τ_{4,4}

whereas

(τ_{1,3} + τ_{2,3} + τ_{3,3}) ↦ (τ_{1,3} + τ_{2,3} + τ_{3,3})+
m_{3}(τ_{1,4} + τ_{2,4} + τ_{3,4}) + (m_{1} + m_{2} + m_{3})τ_{3,4} + m_{3}(m_{1} + m_{2} + m_{3})τ_{4,4}.

A comparison with the transformation behaviour for the Poincaré bundle described in Lemma 17 gives the claim. □

Lemma 21. The rational cohomology groups with compact support of β_{3}^{0}(σ_{II}^{(4)}) are given by

H_{c}^{k}(β_{3}^{0}(σ_{II}^{(4)}); ℚ) = ℚ(−k/2), k = 10, 12.

Proof. As in the previous case we first have to describe the automorphism G(σ_{II}^{(4)}) of the cone σ_{II}^{(4)}. This group is the symmetric group S_{4} permuting the generators of the cone together with the map x_{i} ↦ −x_{i}. Hence we can work with the following generators:

(6.15)  x_{i} ↦ −x_{i}, i = 1, …, 6
(6.16)  x_{1} ↔ x_{2}, x_{3} ↦ x_{1} + x_{2} − x_{3}
(6.17)  x_{1} ↦ x_{1} − x_{3}, x_{2} ↦ −x_{2}, x_{3} ↦ −x_{3}
(6.18)  x_{1} ↦ x_{3} − x_{2}, x_{2} ↦ −x_{2}, x_{3} ↦ x_{1} − x_{2}.

We now have to compute the induced action of these automorphisms on the cohomology groups H^{•}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{•}(E × E × E; ℚ). To this end, we denote by Q_{3} (respectively, R) the generator of H^{1}(T/T^{(σ_{II}^{(4)})}; ℚ) corresponding to the parameter T_{6} = t_{1,2}^{−1} (respectively, to T_{3} = t_{1,3}t_{2,3}t_{3,3}). It is immediately clear that in the case of (6.15) the action is given by

(6.19)  Q_{3}, R ↦ Q_{3}, R;
f_{i} ↦ −f_{i}, i = 1, …, 6.

We note that this implies that there can be no non-trivial invariant cohomology classes involving terms of odd degree in H^{•}(E × E × E). Next we claim that the action on cohomology of (6.16) is given by

(6.20)  Q_{3} ↦ Q_{3} − R, R ↦ −R

f_{i} ↦ f_{i+2} +f_{i+4}, i = 1, 2; f_{i} ↦ f_{i−2} +f_{i+2}, i = 3, 4; f_{i} ↦ −f_{i}, i = 5, 6. To see this we compute

(\begin{matrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) (\begin{matrix}τ_{1,1} & τ_{1,2} & τ_{1,3} & τ_{1,4} \\ τ_{1,2} & τ_{2,2} & τ_{2,3} & τ_{2,4} \\ τ_{1,3} & τ_{2,3} & τ_{3,3} & τ_{3,4} \\ τ_{1,4} & τ_{2,4} & τ_{3,4} & τ_{4,4} \end{matrix}) (\begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 1 & −1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) =

= (\begin{matrix}∗ & τ_{1,2} + τ_{2,3} + τ_{1,3} + τ_{3,3} & −τ_{2,3} − τ_{3,3} & τ_{2,4} + τ_{3,4} \\ ∗ & ∗ & −τ_{1,3} − τ_{3,3} & τ_{1,4} + τ_{3,4} \\ ∗ & ∗ & τ_{3,3} & −τ_{3,4} \\ ∗ & ∗ & ∗ & τ_{4,4} \end{matrix}) .

This immediately gives the claim for the f_{i}. For Q_{3}, R we observe that the action induced on the homology is dual to the action on the subspace ⟨−τ_{1,2}, τ_{1,3} + τ_{2,3} + τ_{3,3}. Since cohomology is dual to homology, the action on Q_{3}, R agrees with that on −τ_{1,2}, τ_{1,3} + τ_{2,3} + τ_{3,3}. A similar calculation gives the following results in the remaining cases:

(6.21)  Q_{3} ↦ −Q_{3}, R ↦ −Q_{3} + R;

f_{i} ↦ f_{i}, i = 1, 2; f_{i} ↦ −f_{i}, i = 3, 4; f_{i} ↦ −f_{i−4} − f_{i}, i = 5, 6.

(6.22)  Q_{3} ↔ R;

f_{i} ↦ f_{i+4}, i = 1, 2; f_{i} ↦ −f_{i−2} −f_{i} −f_{i+2}, i = 3, 4; f_{i} ↦ f_{i−4}, i = 5, 6.

It is now straightforward to compute the invariants under G(σ_{II}^{(4)}). In the cohomology group H^{0}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{2}(E × E × E; ℚ) we find one invariant tensor, namely

I_{1} = 3(f_{1} ∧ f_{2} + f_{3} ∧ f_{4}) + 2φ + ψ,

where we denoted φ = (f_{1} + f_{3} + f_{5}) ∧ f_{6} + f_{5} ∧ (f_{2} + f_{4} + f_{6}) and ψ = f_{1} ∧ f_{4} + f_{3} ∧ f_{2}. In H^{1}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{2}(E × E × E; ℚ) we also obtain one invariant tensor, namely

I_{2} = −R ⊗ (2φ + ψ) + Q ⊗ (φ + 2ψ).

The invariant class in H^{0}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{4}(E × E × E; ℚ) is I_{1} ∧ I_{1} and in H^{1}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{4}(E × E × E; ℚ) it is I_{2} ∧ I_{1}. This together with the fundamental classes in H^{0}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{0}(E × E × E; ℚ) and H^{0}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{6}(E × E × E; ℚ) are the only invariants. As before we now look at the Leray spectral sequence in cohomology associated with p∘q_{σ_{II}^{(4)}} : T (σ_{II}^{(4)}) → A_{1}. Since all representations are trivial we thus obtain Table 11. Hence, we have two differentials which could be non-zero, namely Table 11. E_{2} term of the spectral sequence converging to the cohomology of β_{3}^{0}(σ_{II}^{(4)})

q
2 0 0  0  0  0  0  0
1 0 0 ℚ(−2) 0 ℚ(−3) 0  0
0 ℚ 0 ℚ(−1) 0 ℚ(−2) 0 ℚ(−3)
0 1  2  3  4  5  6  p

d_{2}^{2,1} : H^{1}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{2}(E × E × E; ℚ) → H^{0}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{4}(E × E × E; ℚ),

resp.

d_{2}^{4,1} : H^{1}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{4}(E × E × E; ℚ) → H^{0}(T/T^{(σ_{II}^{(4)})}; ℚ) ⊗ H^{6}(E × E × E; ℚ).

Indeed we claim that they do not vanish. For this we use the description of T (σ_{II}^{(4)})|_{E×E×E} given in Lemma 20. It follows from this description that this bundle splits into the product of two factors with Euler classes −φ and ψ. The claim that the first differential is non-zero is now equivalent to

φ ∧ (2φ + ψ) + ψ ∧ (φ + 2ψ) ≠ 0.

For the second differential we must check that

(φ ∧ (2φ + ψ) + ψ ∧ (φ + 2ψ)) ∧ I_{1} ≠ 0.

This can be checked by direct calculation. At the same time this proves that the first differential does not vanish. The claim of the lemma now follows immediately after converting to cohomology with compact support. □

6.5. The cohomology of β_{3}^{0}(σ^{(5)}).

Lemma 22. The rational cohomology groups with compact support of β_{3}^{0}(σ^{(5)}) are given by

H_{c}^{k}(β_{3}^{0}(σ^{(5)}); ℚ) = \{ \begin{matrix}ℚ(−k/2) & k = 6, 10 \\ ℚ(−k/2)^{⊕2} & k = 8 \\ 0 & otherwise. \end{matrix}

Proof. We first have to compute the automorphism group G(σ^{(5)}). It is not hard to see that this group is generated by the transformations

(6.23)  x_{i} ↦ −x_{i}, i = 1, 2, 3
(6.24)  x_{1} ↔ x_{2}, x_{3} ↦ x_{3}
(6.25)  x_{1} ↦ x_{1} − x_{3}, x_{2} ↦ x_{2} − x_{3}, x_{3} ↦ −x_{3}
(6.26)  x_{1} ↦ x_{1} − x_{3}, x_{2} ↦ −x_{2}, x_{3} ↦ −x_{3}.

Table 12. E_{2} term of the spectral sequence converging to the cohomology of β_{3}^{0}(σ^{(5)})

q
1 0 0 ℚ(−2) 0 ℚ(−3) 0  0
0 ℚ 0 ℚ(−1)^{⊕2} 0 ℚ(−2)^{⊕2} 0 ℚ(−3)
0 1  2  3  4  5  6  p

A computation analogous to that in Lemma 18 shows that this results in the following action on cohomology, where again we denoted by f_{i} the generators of the cohomology of E × E × E and by Q_{3} the generator of the cohomology of the fibre of the torus bundle:

(6.27)  f_{i} ↦ −f_{i}, i = 1, …, 6; Q_{3} ↦ Q_{3}
(6.28)  f_{i} ↔ f_{i+2}, i = 1, 2; f_{j} ↦ f_{j}, j = 5, 6; Q_{3} ↦ Q_{3}
(6.29) f_{i} ↦ f_{i}, i = 1, …, 4; f_{k} ↦ −f_{k−4} − f_{k−2} − f_{k}, k = 5, 6; Q_{3} ↦ Q_{3}

(6.30)  \begin{matrix}f_{i} ↦ f_{i}, i = 1, 2; f_{j} ↦ −f_{j}, j = 3, 4; \\ f_{k} ↦ −f_{k−4} − f_{k}, k = 5, 6; Q_{3} ↦ −Q_{3}. \end{matrix}

Next, we compute the invariant cohomology in H^{0}(ℂ^{∗}; ℚ) ⊗ H^{2k}(E × E × E; ℚ). Clearly this is 1-dimensional for k = 0, 6. By duality it is enough to do the computation for k = 2. Here we find a 2-dimensional invariant subspace generated by i_{1} := f_{1} ∧ f_{2} + f_{3} ∧ f_{4} and i_{2} := f_{1} ∧ f_{4} + f_{3} ∧ f_{2} + 2(f_{1} + f_{3} + f_{5}) ∧ f_{6} + 2f_{5} ∧ (f_{2} + f_{4} + f_{6}). In this situation we also have invariant cohomology in H^{1}(ℂ^{∗}; ℚ)⊗H^{2}(E ×E × E; ℚ). This is 1-dimensional and generated by Q_{3}⊗(f_{1}∧f_{4}+f_{3}∧f_{2}). By duality we also have a 1-dimensional invariant subspace in H^{1}(ℂ^{∗})⊗H^{4}(E ×E ×E). A standard calculation shows that this is generated by Q_{3} ∧(f_{1} ∧f_{4} +f_{3} ∧f_{2})∧i_{2}. In this case the differentials in the Leray spectral sequence are not automatically 0. The situation is described in Table 12. here are two differentials which we have to consider. These are:

d_{2}^{2,1} : H^{1}(ℂ^{∗}; ℚ) ⊗ H^{2}(E × E × E; ℚ) → H^{0}(ℂ^{∗}; ℚ) ⊗ H^{4}(E × E × E; ℚ),

resp.

d_{2}^{4,1} : H^{1}(ℂ^{∗}; ℚ) ⊗ H^{4}(E × E × E; ℚ) → H^{0}(ℂ^{∗}; ℚ) ⊗ H^{6}(E × E × E; ℚ).

We claim that both differentials are non-zero, i.e. they have rank 1. We first treat d_{2}^{1,2}. The differential is given by taking the cup-product with the first Chern class of the vector bundle spanned by the torus bundle T (σ^{(5)})|_{E×E×E}.

As in previous cases on can see that T (σ^{(5)})|_{E×E×E} ≅ p_{12}^{∗}(P^{0}). This shows that

d_{2}^{1,2} : Q_{3} ⊗ (f_{1} ∧ f_{4} + f_{3} ∧ f_{2}) ↦
(f_{1} ∧ f_{4} + f_{3} ∧ f_{2}) ∧ (f_{1} ∧ f_{4} + f_{3} ∧ f_{2}) = 2f_{1} ∧ f_{2} ∧ f_{3} ∧ f_{4} ≠ 0.

The argument for d_{2}^{1,4} is analogous. Finally we use the duality H_{c}^{k}(β_{3}^{0}(σ^{(5)}); ℚ) = H^{10−k}(β_{3}^{0}(σ^{(5)}); ℚ)^{∗} ⊗ ℚ(−5) (which holds on finite smooth covers) to obtain the claim.

□

6.6. The cohomology of β_{3}^{0}(σ^{(6)}).

Lemma 23. The rational cohomology groups with compact support of β_{3}^{0}(σ^{(6)}) are given by

H_{c}^{k}(β_{3}^{0}(σ^{(6)}); ℚ) = \{ \begin{matrix}ℚ(−k/2) & k = 2, 4, 6, 8 \\ 0 & otherwise. \end{matrix}

Proof. The proof of this lemma is analogous to the other cases. We first note that the automorphism group of G(σ^{(6)}) is generated by the symmetric group in three variables permuting the coordinates x_{i} (i = 1, 2, 3) and the transformations (6.23) and (6.25) already considered in the previous section. In this case the torus rank is 0 and hence it suffices to compute the action on the cohomology of the triple product E ×E ×E. In view of the transformation (6.23) there is no invariant in odd degree. By duality it is enough to compute the invariant cohomology in H^{2}(E×E×E; ℚ). A straightforward calculation shows that this is 1-dimensional with generator 2v_{1}+v_{2}+v_{3}, with v_{1} = f_{1}∧f_{2}+f_{3}∧f_{4}+f_{5}∧f_{6}, v_{2} = f_{1} ∧ f_{4} + f_{3} ∧ f_{6} + f_{5} ∧ f_{2} and v_{3} = f_{1} ∧ f_{6} + f_{3} ∧ f_{2} + f_{5} ∧ f_{4}. □

6.7. The cohomology of β_{3}^{0}. In this section, we will use the computations on the strata of β_{3}^{0} to prove the following result.

Proposition 24. The rational cohomology with compact support of β_{3}^{0} is as follows: the non-zero Betti numbers are

\begin{matrix}i & 2 & 4 & 5 & 6 & 7 & 8 & 10 & 12 & 14 \\ b_{i} & 1 & 1 & 1 & 2 & 1 & 4 & 4 & 3 & 1 \end{matrix}

One has H_{c}^{7}(β_{3}^{0}; ℚ) = ℚ(−1) and H_{c}^{5}(β_{3}^{0}; ℚ) = ℚ. Furthermore all cohomology groups of even degree are algebraic.

Proof. We consider the Gysin spectral sequence associated with the stratification of β_{3}^{0} given by the locally closed strata W_{1} = β_{3}^{0}(σ^{(6)}), W_{2} = β_{3}^{0}(σ^{(5)}), W_{3} = β_{3}^{0}(σ_{I}^{(4)}) ∪ β_{3}^{0}(σ_{II}^{(4)}) and W_{4} = β_{3}^{0}(σ^{(3)}). We set Y_{p} = \overline{W}_{p}. This is the spectral sequence E_{•}^{p,q} ⇒ H_{c}^{p+q}(β_{3}^{(0)}; ℚ) with E_{1}^{p,q} = H_{c}^{p+q}(Y_{p} \ Y_{p−1}; ℚ) = H_{c}^{p+q}(W_{p}; ℚ). The cohomology with compact support of the strata W_{i} was computed in the Lemmas 15, 18, 21, 22 and 23. In view of these results, the E_{1} term of the Gysin spectral sequence is as given in Table 13. We consider the differentials d_{r}^{p,q}: E_{r}^{p,q} → E_{r}^{p+r,q−r+1}. Inspection of Table 13 shows that the only possible non-zero differential is d_{1}^{3,5} : E_{1}^{3,5} = H_{c}^{8}(β_{3}^{0}(σ_{I}^{(4)}); ℚ) → E_{1}^{4,5} = Table 13. Gysin spectral sequence converging to the cohomology with compact support of β_{3}^{0}

q
10  0  0  0  ℚ(−7)
9  0  0  ℚ(−6)^{⊕2}  0
8  0  ℚ(−5)  0  ℚ(−6)
7 ℚ(−4)  0  ℚ(−5)^{⊕3}  0
6  0  ℚ(−4)^{⊕2}  0  0
5 ℚ(−3)  0  ℚ(−4) + ℚ(−2) ℚ(−2)
4  0  ℚ(−3)  0  0
3 ℚ(−2)  0  0  ℚ(−1)
2  0  0  ℚ  0
1 ℚ(−1)  0  0  0
1  2  3  4  p

H_{c}^{9}(β_{3}^{0}(σ^{(3)}); ℚ). We can interpret this differential as arising from the Gysin long exact sequence associated with the inclusion of β_{3}^{0}(σ_{I}^{(4)} in the partial compactification β_{3}^{0}(σ^{(3)}) ∪ β_{3}^{0}(σ_{I}^{(4)}) of β_{3}^{0}(σ^{(3)}). Let us denote by Ψ_{E} the fibre of the fibration β_{3}^{0}(σ^{(3)}) ∪ β_{3}^{0}(σ_{I}^{(4)}) → A_{1} over a point [E] ∈ A_{1}. Thanks to the Leray spectral sequence associated with that fibration, all we need to know is that the cohomology with compact support of Ψ_{E} vanishes in degree 7. This requires to prove that the differential

d_{7} : H_{c}^{7}((p ∘ q_{σ_{I}^{(4)}})^{−1}([E]); ℚ) → H_{c}^{8}((p ∘ q_{σ^{(3)}})^{−1}([E]); ℚ)

in the Gysin long exact sequence associated with (p ∘ q_{σ_{I}^{(4)}})^{−1}([E]) ⊂ Ψ_{E} is an isomorphism. Since in the proofs of Lemma 17 and Lemma 19 we described the generators of the cohomology of the fibres of E rather than those of the cohomology with compact support, we shall analyze the map induced by d_{7} on cohomology by Poincaré duality, whose rank coincides with that of d_{7}. Let us recall that the inclusion of {0}×(ℂ^{∗})^{2} in ℂ×(ℂ^{∗})^{2} induces a Gysin long exact sequence whose differentials define the maps

\begin{matrix}H^{k}((ℂ^{∗})^{2}; ℚ) ⊗ ℚ(−1) & ⟶ & H^{k+1}((ℂ^{∗})^{3}; ℚ) \\ T_{i} & ⟼ & T_{i} ∧ T_{1} & i = 2, 3. \end{matrix}

As a consequence, the differential H^{3}((p ∘ q_{σ_{I}^{(4)}})^{−1}([E]); ℚ) ⊗ ℚ(−1) → H^{4}((p ∘ q_{σ^{(3)}})^{−1}([E]); ℚ) maps each of the generators g_{i,j} described in the proof of Lemma 19 to the class obtained by replacing T_{2} by T_{2} ∧ T_{1} in the expression, and then symmetrizing for the action of the group G(σ^{(3)}). This yields:

g_{i,j} ⟼ \frac{2}{3} ∑_{0≤k,l≤2} Q_{k+1} ∧ Q_{l+1} ⊗ f_{2k+i} ∧ f_{2l+j},

hence in particular the differential is surjective. From this the claim follows.

□

7. Torus rank 4

In this section we compute the cohomology of the closed strata β_{4} ⊂ A_{4}^{Vor} and β_{4}^{perf} ⊂ A_{4}^{perf} of torus rank 4 in the second Voronoi and the perfect cone compactification, respectively. We shall first state the main results:

Theorem 25. The cohomology groups with rational coefficients of the closed stratum β_{4}^{perf} ⊂ A_{4}^{perf} of the perfect cone compactification of the moduli space of abelian varieties of dimension 4 are non-zero only in even degree. The only non-zero Betti numbers are b_{0} = b_{2} = b_{4} = 1, b_{6} = b_{8} = 4, b_{10} = 3 and b_{12} = 1. The cohomology is algebraic in all degrees different from 6, whereas H^{6}(β_{4}^{perf}; ℚ) is an extension of ℚ(−3)^{⊕3} by ℚ(−1).

The closed stratum β_{4} ⊂ A_{4}^{Vor} has two irreducible components: a nine-dimen-sional component E, which is the exceptional divisor of the blow-up q: A_{4}^{Vor} → A_{4}^{perf}, and a six-dimensional component, which is the proper transform of β_{4}^{perf} under q.

Theorem 26. (1) The rational cohomology of E is all algebraic. The only non-zero Betti numbers are b_{0} = b_{2} = b_{16} = b_{18} = 1, b_{4} = b_{14} = 2 and b_{6} = b_{8} = b_{10} = b_{12} = 3.

(2) The rational cohomology of β_{4} is non-trivial only in even degree. The non-zero Betti numbers are

\begin{matrix}i & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 \\ b_{i} & 1 & 2 & 3 & 7 & 7 & 6 & 4 & 2 & 1 & 1 \end{matrix}

All cohomology groups are algebraic, with the exception of H^{6}(β_{4}; ℚ), which is an extension of ℚ(−3)^{⊕6} by ℚ(−1).

7.1. Cone decompositions. It is in this section that we require full information about the perfect cone or first Voronoi and the second Voronoi decomposition in Sym_{≥0}^{2}(ℝ^{4}). Details concerning these decompositions can be found in [ER1], [ER2], [Val] and [Vor]. We start by recalling the perfect cone decomposition. The starting point is two 10-dimensional cones, namely the principal cone Π_{1}(4) and the second perfect cone Π_{2}(4). These cones are given by

\begin{aligned}Π_{1}(4) = & ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}, (x_{1} − x_{2})^{2}, (x_{1} − x_{3})^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{3})^{2}, \\ & (x_{2} − x_{4})^{2}, (x_{3} − x_{4})^{2}⟩ \end{aligned}

and

\begin{aligned}Π_{2}(4) = & ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}, (x_{1} − x_{3})^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{3})^{2}, (x_{2} − x_{4})^{2}, \\ & (x_{3} − x_{4})^{2}, (x_{1} + x_{2} − x_{3})^{2}, (x_{1} + x_{2} − x_{4})^{2}, (x_{1} + x_{2} − x_{3} − x_{4})^{2}⟩ \end{aligned}

respectively. The perfect cone decomposition consists of all GL(4, ℤ)-translates of these cones and their faces. While the cone Π_{1}(4) is basic, the cone Π_{2}(4) is not, hence it defines a singular point P_{sing} ∈ A_{4}^{perf}. Nevertheless, all 9dimensional faces of Π_{2}(4) are basic. Modulo the action of GL(4, ℤ) these 9-dimensional faces define two orbits. Traditionally these are called RT (red triangle) and BF (black face) respectively (see [ER2]). In genus 4 and 5 (but not in general) the second Voronoi decomposition is a subdivision of the perfect cone decomposition. In our case it is the refinement of the perfect cone decomposition obtained by adding all cones that arise as spans of the 9-dimensional faces of Π_{2}(4) with the central ray generated by

(7.1) e = \frac{1}{3} [x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2} + (x_{1} − x_{3})^{2} + (x_{1} − x_{4})^{2} + (x_{2} − x_{3})^{2} + (x_{2} − x_{4})^{2}

+(x_{3} − x_{4})^{2} + (x_{1} + x_{2} − x_{3})^{2} + (x_{1} + x_{2} − x_{4})^{2} + (x_{1} + x_{2} − x_{3} − x_{4})^{2}] .

In particular, all perfect cones, with the exception of Π_{2}(4), belong to the second Voronoi decomposition. Geometrically this means that A_{4}^{Vor} is a blowup of A_{4}^{perf} = A_{4}^{Igu} in the singular point P_{sing}. Since all cones on the second Voronoi decomposition are basic A_{4}^{Vor} is smooth (as a stack). Moreover, the exceptional divisor E is irreducible and smooth (again as a stack). (For a discussion of this see also [HS2]. A description of representatives of all GL(4, ℤ)-orbits of cones in the second Voronoi, and hence also the perfect cone decomposition, can be found in [Val, Chapter 4]. For cones with extremal rays spanned by quadratic forms of rank 1 the list is given in [Val, S.4.4.4]. Note that in this list K_{5} denotes the cone Π_{2}(4), and the 9-dimensional cones K_{5} − 1 and K_{3,3} correspond to the equivalence classes BF, respectively, RT of [ER2]. The remaining cones are listed in [Val, S.4.4.5]. The following list gives the number of GL(4, ℤ)-orbits of cones in each dimension for the two decompositions.

dimension  1 2 3 4 5 6 7 8 9 10
# perfect cones  1 1 2 3 4 5 4 2 2 2
# second Voronoi cones 2 2 4 7 9 11 11 7 4 3

From this we see that the perfect cone decomposition has 26 different cones, whereas the second Voronoi decomposition has 60 different cones. The lists in [Val] also allow us to write down generators for the extremal rays of representatives in all cases.

7.2. Plan for computation. We briefly recall the structure of β_{4} and β_{4}^{perf} which comes from the toroidal construction. More generally, let β_{4}^{Σ} be the stratum of any admissible fan Σ (in our case either the perfect cone or the second Voronoi fan), then each cone σ ∈ Σ defines a torus orbit T_{σ} of dimension 10 − k where k is the dimension of σ. Let G_{σ} ⊂ GL(4, ℤ) be the stabilizer of σ with respect to the natural action of GL(4, ℤ) on Sym_{≥0}^{2}(ℝ^{4}). Then G_{σ} acts on T_{σ} and β_{4}^{Σ} is the disjoint union of the quotients Z_{σ} = T_{σ}/G_{σ} where σ runs through a set of representatives of all cones in Σ which contain a form of rank 4 in their interior. We then define a stratification by defining S_{p} as the union of all Z_{σ} where dim σ ≥ 10 − p. In particular, S_{p} \ S_{p−1} is the union of all Z_{σ} with dim σ = 10 − p. The Gysin spectral sequence E_{•}^{p,q} ⇒ H_{c}^{p+q}(β_{4}^{perf}; ℚ) = H^{p+q}(β_{4}^{perf}; ℚ) associated with the filtration S_{p} has E_{1} term given by

E_{1}^{p,q} = H_{c}^{p+q}(S_{p} \ S_{p−1}; ℚ).

Since S_{p} \ S_{p−1} is the disjoint union of the Z_{σ} with dim σ = 10 − p it follows that

H_{c}^{•}(S_{p} \ S_{p−1}; ℚ) = ⨁_{dim σ} H_{c}^{•}(Z_{σ}; ℚ).

In our situation we have considerably more information. In particular we know that, with the exception of Π_{2}(4), all cones in both the perfect cone and the second Voronoi decomposition, are basic. In particular all strata S_{p} with p ≤ 9 are locally quotients of a smooth variety by a finite group. Moreover T_{σ} = (ℂ^{∗})^{10−dim σ} and Z_{σ} = (ℂ^{∗})^{10−dim σ}/G_{σ}. The torus orbit of Π_{2}(4) is a point. Thus we have to compute for each cone σ the cohomology of the torus T_{σ} with respect to G_{σ}. Recall that H^{•}((ℂ^{∗})^{k}; ℚ) is the exterior algebra generated by the k-dimensional vector space H^{1}((ℂ^{∗})^{k}; ℚ). Moreover, a basis of the vector space H^{1}((ℂ^{∗})^{k}; ℚ) can be obtained by taking the Alexander dual classes of the fundamental classes of the components

{(y_{1}, …, y_{k})|y_{i} = 0}, i = 1, …, k

of the complement of (ℂ^{∗})^{k} in ℂ^{k}. This means that, once the generators of the cone σ and of the group G_{σ} are known, the computation of the cohomology of Z_{σ} reduces to a linear algebra problem, which can be solved using computational tools. In our case, the generators of the stabilizers G_{σ} were calculated with Magma ([BCP]) and the invariant part of the algebra ⋀^{•} H^{1}((ℂ^{∗})^{k}; ℚ) with Singular ([GPS]).

7.3. Perfect cones. We shall now perform the programme outlined above for the perfect cone compactification, which coincides with the Igusa compactification in genus 4. We have already mentioned that a list of representatives of all cones in the perfect cone decomposition, together with their generators, can be found in [Val, Ch. 4]. This enables us to compute the stabilizer groups G_{σ} as well as the invariant cohomology of the torus orbits T_{σ} = (ℂ^{∗})^{k} where k = 10 − dim σ. The results so obtained are listed in Table 14, where the notation for the cones is the one of [Val, §4]. The information on the cohomology of the strata is given in the form of Hodge Euler characteristics, i.e. what is given is the Euler characteristic of H_{c}^{•}(Z_{σ}; ℚ) in the Grothendieck group of Hodge structures. The symbol L denotes the class of the weight 2 Tate Hodge structure ℚ(−1) in the Grothendieck group. The relationship between cohomology and cohomology with compact support is given by Poincaré duality:

H_{c}^{l}(Z_{σ}; ℚ) = Hom(H^{2k−l}(Z_{σ}; ℚ), ℚ(−k)),

which holds since the Z_{σ} are finite quotients of the smooth varieties T_{σ}.

Table 14. GL(4, ℤ)-orbits of perfect cones

Σ  dim Σ e_{Hdg}(Z_{Σ})
K_{5} = Π_{1}(4) 10 1
Π_{2}(4)  10 1
K_{5} − 1  9  L
K_{3,3}  9  L − 1
K_{5} − 2  8  L^{2}
K_{5} − 1 − 1 8  L^{2} − L
K_{5} − 2 − 1 7  L^{3} − L^{2}
C_{2221}  7  L^{3}
K_{5} − 3  7  L^{3}

Σ  dim Σ e_{Hdg}(Z_{Σ})
K_{4} + 1  7  L^{3}
C_{222}  6  L^{4} − L^{3}
C_{321}  6  L^{4} + 1
C_{221} + 1  6  L^{4}
C_{3} + C_{3}  6  L^{4}
C_{5}  5  L^{5} − 1
C_{4} + 1  5  L^{5}
C_{3} + 1 + 1  5  L^{5} + L
1 + 1 + 1 + 1 4  L^{6}

Table 15. E_{1} term of the spectral sequence converging to H^{•}(β_{4}^{perf}; ℚ).

q

6 0  0  0  0  0  0  ℚ(−6)
5 0  0  0  0  0  ℚ(−5)^{⊕3} 0
4 0  0  0  0  ℚ(−4)^{⊕4}  0  0
3 0  0  0  ℚ(−3)^{⊕4} ℚ(−3)  0  0
2 0  0  ℚ(−2)^{⊕2} ℚ(−2)  0  0  0
1 0 ℚ(−1)^{⊕2} ℚ(−1)  0  0  ℚ(−1)  0
0 ℚ^{⊕2}  ℚ  0  0  ℚ  ℚ  0
0  1  2  3  4  5  6  p

In view of the information on the cohomology of the Z_{σ} given in Table 14, this yields that the E_{1} terms of the spectral sequence E_{•}^{p,q} ⇒ H^{p+q}(β_{4}^{perf}; ℚ) are as shown in Table 15. To establish Theorem 25, we need to determine the rank of all differentials in the spectral sequence. As morphisms between pure Hodge structures of different weights are necessarily trivial, one remains with five differentials to investigate, all of the form d_{1}^{p,q}: E_{1}^{p,q} → E_{1}^{p+1,q}. We will denote them by

\begin{matrix}δ_{0} : E_{1}^{0,0} ⟶ E_{1}^{1,0}, & δ′_{0} : E_{1}^{4,0} ⟶ E_{1}^{5,0}, \\ δ_{1} : E_{1}^{1,1} ⟶ E_{1}^{2,1}, & δ_{2} : E_{1}^{2,2} ⟶ E_{1}^{3,2}, & δ_{3} : E_{1}^{3,3} ⟶ E_{1}^{4,3}. \end{matrix}

Lemma 27. All the differentials δ_{0}, δ′_{0}, δ_{1}, δ_{2} and δ_{3} have rank 1.

Proof. Since β_{4}^{perf} is connected, one has H^{0}(β_{4}^{perf}; ℚ) = ℚ. This implies that δ_{0} has rank 1. Next, we consider the differential δ′_{0} : E_{1}^{4,0} ≅ ℚ ⟶ E_{1}^{5,0} ≅ ℚ. From the description of the strata given in Table 14, we have E_{1}^{4,0} = H_{c}^{4}(Z_{C_{321}}; ℚ) and E_{1}^{5,0} = H_{c}^{5}(Z_{C_{5}}; ℚ) for the cones

\begin{matrix}C_{5} = ⟨x_{1}^{2}, x_{2}^{2}.(x_{1} − x_{4})^{2}, (x_{2} − x_{3})^{2}, (x_{3} − x_{4})^{2}⟩, \\ C_{321} = ⟨x_{1}^{2}, x_{2}^{2}, x_{4}^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{3})^{2}, (x_{3} − x_{4})^{2}⟩. \end{matrix}

The cone C_{5} is contained in C_{321}, hence Z_{C_{321}} is contained in the closure of Z_{C_{5}}. Furthermore, the rank of δ′_{0} must coincide with the rank of the differential η_{0} : H_{c}^{4}(Z_{C_{321}}; ℚ) → H_{c}^{5}(Z_{C_{5}}; ℚ) of the Gysin long exact sequence associated with the inclusion of Z_{C_{321}} in the partial compactification Z_{C_{5}} ∪ Z_{C_{321}} of Z_{C_{5}}. If one considers the stabilizers, one observes that G_{C_{321}} is a subgroup of G_{C_{5}}. Therefore, one can view η_{0} as a map from the cohomology of (ℂ^{∗})^{4} to the cohomology of (ℂ^{∗})^{5} in the following way:

[figure: H_{c}^{4}(Z_{C_{321}}; ℚ); η_{0}; H_{c}^{4}((ℂ^{∗})^{4}; ℚ)^{G_{C_{321}}}; η_{0}; H_{c}^{5}(Z_{C_{5}}; ℚ); H_{c}^{5}((ℂ^{∗})^{5}; ℚ)^{G_{C_{5}}}; H_{c}^{5}((ℂ^{∗})^{5}; ℚ)^{G_{C_{321}}} ,]

where we used the fact that the G_{C_{321}}-invariant part of H_{c}^{5}((ℂ^{∗})^{5}; ℚ) coincides with the G_{C_{5}}-invariant part. This new interpretation relates the map η_{0} to the differential

(7.2)  H_{c}^{4}((ℂ^{∗})^{4}; ℚ) ≅ ℚ ⟶ H_{c}^{5}((ℂ^{∗})^{5}; ℚ) ≅ ℚ

of the Gysin exact sequence of an inclusion (ℂ^{∗})^{4} ↪ ℂ × (ℂ^{∗})^{4}, with complement isomorphic to (ℂ^{∗})^{5}. In particular, since H_{c}^{k}(ℂ × (ℂ^{∗})^{4}; ℚ) vanishes for k ≤ 5, the differential (7.2) is an isomorphism, and the same holds for η_{0}. Let us consider the differential

δ_{1} : E_{1}^{1,1} ≅ H_{c}^{2}(Z_{K_{5}−1}; ℚ) ⊕ H_{c}^{2}(Z_{K_{3,3}}; ℚ) → E_{1}^{2,1} ≅ H_{c}^{3}(Z_{K_{5}−1−1}; ℚ).

Note that both Z_{K_{5}−1} and Z_{K_{3,3}} are contained in the closure of Z_{K_{5}−1−1} ⊂ β_{4}^{perf}. We choose to investigate the inclusion i_{3,3} of Z_{K_{3,3}} in the partial compactification Z_{K_{3,3}} ∪ Z_{K_{5}−1−1} of Z_{K_{5}−1−1}. Then the rank of δ_{1} cannot be smaller than the rank of the differential

η_{1} : H_{c}^{2}(Z_{K_{3,3}}; ℚ) ⟶ H_{c}^{3}(Z_{K_{5}−1−1}; ℚ)

in the Gysin long exact sequence associated with i_{3,3}, even though there is no canonical isomorphism between the kernel of η_{1} and that of δ_{1}. In Vallentin’s notation, the cone K_{3,3} is given by

\begin{aligned}K_{3,3} = & ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{3}, (x_{1} − x_{3})^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{3})^{2}, (x_{2} − x_{4})^{2}, \\ & (x_{1} + x_{2} − x_{3} − x_{4})^{2}⟩. \end{aligned}

In particular, its subcone

K_{5} − 1 − 1b = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}, (x_{1} − x_{3})^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{3})^{2}, (x_{2} − x_{4})^{2}⟩

belongs to the same GL(4, ℤ)-orbit as K_{5} − 1 − 1, so that Z_{K_{5}−1−1b} ⊂ β_{4}^{perf} coincides with Z_{K_{5}−1−1}. The stabilizer G_{K_{5}−1−1b} of the cone K_{5} − 1 − 1b is generated by −Id_{ℤ^{4}} and by the two automorphisms

\{ \begin{matrix}x_{1} ↦ x_{3} \\ x_{2} ↦ x_{4} \\ x_{3} ↦ x_{2} \\ x_{4} ↦ x_{1} \end{matrix} and \{ \begin{matrix}x_{1} ↦ x_{1} \\ x_{2} ↦ x_{2} \\ x_{3} ↦ x_{4} \\ x_{4} ↦ x_{3} \end{matrix} .

In particular, one can check that the group G_{K_{5}−1−1b} is contained in the stabilizer G_{K_{3,3}} of the cone K_{3,3}. Analogously to the case of η_{0}, we can reduce the study of η_{1} to the study of the long exact sequence of an inclusion ℂ^{∗} ↪ ℂ × ℂ^{∗} with complement isomorphic to (ℂ^{∗})^{2}, by exploiting the diagram

[figure: H_{c}^{2}(Z_{K_{3,3}}; ℚ); η_{1}; H_{c}^{2}(ℂ^{∗}; ℚ)^{G_{K_{3,3}}}; H_{c}^{2}(ℂ^{∗}; ℚ)^{G_{K_{5}−1−1b}}; η_{1}; H_{c}^{3}(Z_{K_{5}−1−1b}; ℚ); H_{c}^{3}((ℂ^{∗})^{2}; ℚ)^{G_{K_{5}−1−1b}}.]

Then the claim follows from the fact that the differential H_{c}^{2}(ℂ^{∗}; ℚ) ≅ ℚ(−1) → H_{c}^{3}((ℂ^{∗})^{2}; ℚ) in the Gysin long exact sequence associated with the inclusion ℂ^{∗} ↪ ℂ × ℂ^{∗} has rank 1. The proof for δ_{2} and δ_{3} is completely analogous to that for δ_{1}. In the case of δ_{2} one considers the inclusion of the 2-dimensional stratum Z_{K_{5}−2} in the 3-dimensional stratum Z_{K_{5}−2−1}, given by the inclusion of the cone

K_{5} − 2 − 1b = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{3})^{2}, (x_{2} − x_{4})^{2}, (x_{3} − x_{4})^{2}⟩,

which lies in the same GL(4, ℤ)-orbit as K_{5} − 2 − 1, in

K_{5} − 2 = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{3})^{2}, (x_{2} − x_{4})^{2}, (x_{3} − x_{4})^{2}⟩.

In this case, the stabilizers of K_{5} − 2 and of K_{5} − 2 − 1b coincide as subgroups of GL(4, ℤ). In the case of δ_{3}, one considers the inclusion of the 3-dimensional stratum Z_{C_{2221}} in the 4-dimensional stratum Z_{C_{222}}, given by the inclusion of the cone

C_{222} = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{4})^{2}, (x_{3} − x_{4})^{2}⟩

in

C_{2221} = ⟨x_{1}^{2}, x_{2}^{2}, x_{3}^{2}, x_{4}^{2}, (x_{1} − x_{4})^{2}, (x_{2} − x_{4})^{2}, (x_{3} − x_{4})^{2}⟩.

Again, the stabilizers of C_{222} and C_{2221} coincide. □

7.4. Cones containing e. We shall now prove Theorem 26.

Proof of (1)⇒(2) in Theorem 26. Assume that the cohomology with compact support of the exceptional divisor E of the blow-up A_{4}^{Vor} → A_{4}^{perf} is as stated in (1). The Gysin long exact sequence associated with the closed inclusion E ⊂ β_{4} is as follows:

(7.3) ⋯ → H_{c}^{k−1}(E; ℚ) ⟶^{d_{k}} H_{c}^{k}(β_{4} \ E; ℚ) → H_{c}^{k}(β_{4}; ℚ) → H_{c}^{k}(E; ℚ) → ⋯

Similarly the Gysin sequence of the pair {P_{sing}} ⊂ β_{4}^{perf} reads

⋯ → H_{c}^{k−1}(P_{sing}; ℚ) → H_{c}^{k}(β_{4}^{perf} \ {P_{sing}}; ℚ) →
→ H_{c}^{k}(β_{4}^{perf}; ℚ) → H_{c}^{k}(P_{sing}; ℚ) → ⋯

Since A_{4}^{Vor} → A_{4}^{perf} is an isomorphism outside E, the complement β_{4} \ E is isomorphic to β_{4}^{perf} \{P_{sing}}. By Theorem 25 the odd cohomology with compact support of β_{4}^{perf} vanishes and hence H_{c}^{k}(β_{4}^{perf} \{P_{sing}}; ℚ) = H_{c}^{k}(β_{4} \E; ℚ) = 0 for odd k ≥ 3. Moreover, H_{c}^{1}(β_{4}^{perf} \{P_{sing}}; ℚ) = H_{c}^{1}(β_{4} \E; ℚ) = 0 since P_{sing} is a point and β_{4}^{perf} is compact (which implies that cohomology with compact support and ordinary cohomology coincide). Furthermore by the description of H_{c}^{•}(E; ℚ) from (1) we know that all odd cohomology of E vanishes. This ensures that all differentials d_{k}, k ≥ 1 are zero. This implies that the Betti numbers b_{k} of β_{4} with k ≥ 1 are as stated in Theorem 26. Also the description of the Hodge structures follows from Theorem 25 and from part (1) in view of the long exact sequence (7.3). Finally, the fact that H_{c}^{0}(β_{4}; ℚ) is one-dimensional follow from the connectedness of β_{4}. To complete the proof, recall that β_{4} is compact, so that cohomology and cohomology with compact support agree. □

Lemma 28. For every k, the cohomology group H^{k}(E; ℚ) carries a pure Hodge structure of weight k.

Proof. To prove the claim, we consider the second Voronoi compactification A_{4}^{Vor}(n) of the moduli space of principally polarized abelian fourfolds with a level-n structure (n ≥ 3). Recall that A_{4}^{Vor}(n) is a smooth projective scheme and that the map π(n): A_{4}^{Vor}(n) → A_{4}^{Vor} is a finite group quotient. The preimage π(n)^{−1}(E) of E is the union of finitely many irreducible components, all of which are smooth and pairwise disjoint. This follows from the toric description, since these components are themselves toric varieties given by the star Star(⟨e⟩) in the lattice Sym^{2}(ℤ^{4})/ℤe. In particular, this implies that the Hodge structures on the cohomology groups of π(n)^{−1}(E) are pure of weight equal to the degree. As π(n) is finite, the pull-back map

π(n)|_{π(n)^{−1}(E)}^{∗}: H^{k}(E; ℚ) → H^{k}(π(n)^{−1}(E); ℚ)

is injective. This implies that each cohomology group H^{k}(E; ℚ) is a Hodge substructure of H^{k}(π(n)^{−1}(E); ℚ), thus yielding the claim. □

Proof of (1) in Theorem 26. In view of Lemma 28, determining the cohomology of E is equivalent to computing its Hodge Euler characteristics, i.e.

e_{Hdg}(E) = ∑_{k∈ℤ}(−1)^{k}[H_{c}^{k}(E; ℚ)],

where [·] denotes the class in the Grothendieck group K_{0}(HS_{ℚ}) of Hodge structures. Hodge Euler characteristics are additive, so we are going to work with Table 16. GL(4, ℤ)-orbits of cones of dimension ≥ 6 containing e

σ dim σ e_{Hdg}(Z_{σ})
111+ 10 1
111− 10 1
211+ 9 L − 1
211− 9 L
311+ 8 L^{2} − L
311− 8 L^{2}
22′1 8 L^{2} − L
221+ 8 L^{2}
221− 8 L^{2} + L
411 7 L^{3} − L^{2}
321+ 7 L^{3} − L^{2} + L − 1

σ dim σ e_{Hdg}(Z_{σ})
321− 7 L^{3} − 2L^{2} + L
222′ 7 L^{3} − L^{2}
22′2′′ 7 L^{3}
222+ 7 L^{3}
222− 7 L^{3}
421 6 L^{4} − L^{3} + L^{2} − L
331+ 6 L^{4} + 1
331− 6 L^{4} − L^{3} − L + 1
322+ 6 L^{4} − L^{3}
322− 6 L^{4} − L^{3}
322′ 6 L^{4} − 2L^{3} + 2L^{2} − 2L + 1

Table 17. GL(4, ℤ)-orbits of cones of dimension ≤ 5 containing e

Σ  dim Σ e_{Hdg}(Z_{σ})
422′ 5 L^{5} + L^{3} − L^{2} + L
332− 5  L^{5} − L^{4} + L^{3} − 3L^{2} + 2L
431  5  L^{5} − L^{4} + L^{3} − L^{2} + L − 1
422  5  L^{5} − L^{4}
332+ 5  L^{5} − 2L^{4} + L^{3} − L^{2} + 2L − 1
432 4 L^{6} − 2L^{5} + 2L^{4} − 4L^{3} + 5L^{2} − 2L
333− 4  L^{6} + 2L^{2}
441 4 L^{6} + L^{2}
333+ 4  L^{6} − L^{5} − L^{3} + 2L^{2} − L
433  3  L^{7} − L^{6} + L^{5} − L^{4} + 4L^{3} − 4L^{2}
442  3  L^{7} + 2L^{3} − L^{2}
443  2  L^{8} + 2L^{4} − 3L^{3}
444  1  L^{9} − L^{4}
TOT.  L^{9} + L^{8} + 2L^{7} + 3L^{6} + 3L^{5} + 3L^{4} + 3L^{3} + 2L^{2} + L + 1

a locally closed stratification of E and add up the Hodge Euler characteristics to get the result. The toroidal construction of A_{4}^{Vor} yields that E is the union of toric strata Z_{σ} for all cones σ belonging to the second Voronoi decomposition but not to the perfect cone decomposition. Note that for such σ the variety Z_{σ} automatically maps to A_{0} under the map A_{4}^{Vor} → A_{4}^{Sat}. Furthermore, up to the action of GL(4, ℤ), one can assume that these cones contain the extremal ray ⟨e⟩ defined in (7.1) as extremal ray.

Since cones that lie in the same GL(4, ℤ)-orbit give the same variety Z_{σ}, we have to work with a list of representatives of all GL(4, ℤ)-orbits of cones fulfilling our conditions. Such a list is given in [Val, §4.4.5]. As in the proof of Theorem 14, we compute for each cone σ in Vallentin’s list the generators of its stabilizer G_{σ} in GL(4, ℤ), as well as their action on H^{1}((ℂ^{∗})^{10−dim σ}; ℚ). Then we use the computer algebra program Singular [GPS] to calculate all positive Betti numbers of the quotient Z_{σ} = (ℂ^{∗})^{10−dim σ}/G_{σ}. The results are given in Tables 16 and 17, where we list all cones and the Hodge Euler characteristics of the corresponding strata of E. As already explained, the Hodge Euler characteristic of E is the sum for the Euler characteristics of all strata Z_{σ} and is computed at the bottom of Table 17. In view of Lemma 28, and recalling that L is the notation of the weight 2 Tate Hodge structure ℚ(−1) in the Grothendieck group of rational Hodge structures, we can conclude that the Betti numbers of E agree with those given in the statement of Theorem 26. □

Remark 29. Note that the Betti numbers of E satisfy Poincaré duality. Indeed, this must be the case as E is smooth up to finite group action.

Appendix A. Cohomology of A_{3} with coefficients in symplectic local systems

In this section, we recollect the information on the cohomology of local systems on A_{2} and A_{3} that we used in the course of the paper. Let us recall that the cohomology of local systems of odd weight on A_{g} vanishes because it is killed by the abelian involution. Therefore, we only need to deal with local systems of even rank. The cohomology of A_{2} and A_{3} with constant coefficients is known. The moduli space A_{2} is the disjoint union of the moduli space M_{2} of genus 2 curves and the locus Sym^{2} A_{1} of products. Since it is known that the rational cohomology of both these spaces vanishes in positive degree, we have

Lemma 30. The only non-trivial rational cohomology groups with compact support of A_{2} are H_{c}^{4}(A_{2}; ℚ) = ℚ(−2) and H_{c}^{2}(A_{2}; ℚ) = ℚ(−1).

The rational cohomology of A_{3} was computed by Hain ([Ha]). We state below his result in terms of cohomology with compact support.

Theorem 31 (Hain). The non-trivial Betti numbers with compact support of A_{3} are

\begin{matrix}i & 12 & 10 & 8 & 6 \\ b_{i} & 1 & 1 & 1 & 2 \end{matrix}

Furthermore, all cohomology groups are algebraic with the exception of H_{c}^{6}(A_{3}; ℚ), which is an extension of ℚ(−3) by ℚ.

We deduce the results we need on non-trivial symplectic local systems with weight ≤ 2 from results on moduli spaces of curves ([BT],[T3]). Note that the result for 𝕍_{1,1} was already proven in [HT, Lemma 3.1].

Lemma 32. The cohomology groups with compact support of the weight 2 symplectic local systems on M_{2} are as follows: the cohomology of 𝕍_{2,0} vanishes in all degrees, whereas the only non-zero cohomology group with compact support of 𝕍_{1,1} is H_{c}^{3}(M_{2}; 𝕍_{1,1}) = ℚ.

Lemma 33. The rational cohomology of M_{3} with coefficients in 𝕍_{1,0,0} and 𝕍_{2,0,0} is 0 in all degrees. The only non-trivial cohomology group with compact support of M_{3} with coefficients in 𝕍_{1,1,0} is H_{c}^{9}(M_{3}; 𝕍_{1,1,0}) = ℚ(−5).

Proof of Lemma 33. Following the approach of [G2], we use the forgetful maps p_{1} : M_{3,1} → M_{3} and p_{2} : M_{3,2} → M_{3} to obtain information. Note that p_{1} is the universal curve over M_{3} and that the fibre of p_{2} is the configuration space of 2 distinct points on a genus 3 curve. According to [BT, Cor. 1], there is an isomorphism H^{•}(M_{3,1}; ℚ) ≅ H^{•}(M_{3}; ℚ) ⊗ H^{•}(ℙ^{1}; ℚ) as vector spaces with mixed Hodge structures. If we compare this with the Leray spectral sequence in cohomology associated with p_{1}, we get that the cohomology of M_{3} with coefficients in 𝕍_{1,0,0} must vanish. Next, we analyze the Leray spectral sequence in cohomology associated with p_{2}. Taking the 𝔖_{2}-action into account, the cohomology of the fibre of p_{2} induces the following local systems on M_{3}:

local system:

local system:

deg. invariant part

alternating part

0  ℚ  0
1  𝕍_{1,0,0}  𝕍_{1,0,0}
2  ℚ(−1) ⊕ 𝕍_{1,1,0} ℚ(−1) ⊕ 𝕍_{2,0,0}
3 0 𝕍_{1,0,0}(−1)

This implies that the cohomology of M_{3} with coefficients in 𝕍_{2,0,0} (respectively, in 𝕍_{1,1,0}) is strictly related to the 𝔖_{2}-alternating (resp. 𝔖_{2}-invariant) part of the cohomology of M_{3,2}. The rational cohomology of M_{3,2} is described with its mixed Hodge structures and the action of the symmetric group in [T3, Thm 1.1]. By comparing this with the E_{2}-term of the Leray spectral sequence associated with p_{2}, one obtains that the cohomology of 𝕍_{2,0,0} vanishes and that the only non-trivial cohomology group of 𝕍_{1,1,0} is H^{3}(M_{3}; 𝕍_{1,1,0}) = ℚ(−3). Then the claim follows from Poincaré duality. □

Proof of Lemma 32. The proof is analogous to that of Lemma 33. In this case, one needs to compare the Leray spectral sequence associated with p_{2} : M_{2,2} → M_{2} with the cohomology of M_{2,2} computed in [T2, II,2.2]. Note that in this case the cohomology of 𝕍_{1,0} vanishes because it is killed by the hyperelliptic involution on the universal curve over M_{2}. □

Next, we compute the cohomology of the weight 2 local systems on A_{2} and A_{3} we are interested in, by using Gysin long exact sequences in cohomology with compact support and the stratification A_{2} = τ_{2}(M_{2}) ⊔ Sym^{2} A_{1} of A_{2}, respectively, the stratification A_{3} = τ_{3}(M_{3}) ⊔ τ_{2}(M_{2}) × A_{1} ⊔ Sym^{3} A_{1} of A_{3}.

The result on the cohomology with compact support of 𝕍_{1,1} was already proved in [HT, Lemma 3.1].

Lemma 34. The only non-trivial cohomology groups of A_{2} with coefficients in a local system of weight 2 are H_{c}^{3}(A_{2}; 𝕍_{1,1}) = ℚ and H_{c}^{3}(A_{2}; 𝕍_{2,0}) = ℚ(−1).

Proof. Using branching formulae as in [BvdG, §§7–8], one proves that the restriction of 𝕍_{2,0} to Sym^{2} A_{1} ⊂ A_{2} coincides with the symmetrization of 𝕍_{2}×𝕍_{0} on A_{1} × A_{1}. Its cohomology with compact support is then ℚ(−1) in degree 3 and trivial in all other degrees by e.g. [G1, Thm. 5.3]. Analogously, one shows that the cohomology of Sym^{2} A_{1} with coefficients in the restriction of the local system 𝕍_{1,1} is trivial. Then the claim follows from the Gysin long exact sequence associated with the inclusion Sym^{2} A_{1} ⊂ A_{2}. □

Lemma 35. The cohomology with compact support of A_{3} in the local system 𝕍_{1,1,0} is non-trivial only in degree 5 and possibly in degrees 8 and 9 and is given in these degrees by H_{c}^{5}(A_{3}; 𝕍_{1,1,0}) = ℚ(−1) and H_{c}^{8}(A_{3}; 𝕍_{1,1,0}) ≅ H_{c}^{9}(A_{3}; 𝕍_{1,1,0}) = ℚ(−4)^{⊕ϵ} with ϵ ∈ {0, 1}.

Proof. Branching formulae yield that the cohomology with compact support of the restriction of 𝕍_{1,1,0} to τ_{2}(M_{2}) × A_{1} is equal to ℚ(−5) (coming from the local system 𝕍_{1,1} ⊗ 𝕍_{0} on M_{2} × A_{1}) in degree 8, to ℚ(−1) in degree 5 (coming from the local system 𝕍_{0} ⊗ 𝕍_{0}(−1)) and is trivial in all other degrees. Moreover, the restriction of 𝕍_{1,1,0} to Sym^{3} A_{1} is trivial, as is easy to prove if one looks at the cohomology of the restriction of the universal abelian variety over A_{3} to Sym^{3} A_{1}. It remains to consider the Gysin long exact sequence associated with the closed inclusion A_{3}^{red} ⊂ A_{3}. The only differential which can possibly be non-trivial is

ℚ(−5) = H_{c}^{8}(A_{3}^{red}; 𝕍_{1,1,0}) ⟶ H_{c}^{9}(M_{3}; 𝕍_{1,1,0}) = ℚ(−5).

From this the claim follows. □ In the investigation of the cohomology with compact support of the locus β_{2}^{0} of semi-abelic varieties of torus rank 2 we also need to consider the cohomology with compact support of the weight 4 local system 𝕍_{2,2} on A_{2}. In the forthcoming preprint [T4], we will show that the cohomology of 𝕍_{2,2} vanishes in all degrees. For our application, however, we do not need such a complete result. The following lemma suffices:

Lemma 36. The cohomology with compact support of A_{2} with coefficients in the local system 𝕍_{2,2} is 0 in all degrees different from 3, 4. Furthermore, for every weight k there is an isomorphism

Gr_{k}^{W} (H_{c}^{3}(A_{2}; 𝕍_{2,2})) ≅ Gr_{k}^{W} (H_{c}^{4}(A_{2}; 𝕍_{2,2}))

between the graded pieces of the weight filtration.

Proof. First, we prove that the result holds in the Grothendieck group of rational Hodge structures. This requires to prove that the Euler characteristic of H_{c}^{•}(A_{2}; 𝕍_{2,2}) in the Grothendieck group of rational Hodge structures vanishes. By branching formulae, the cohomology with compact support of the restriction of 𝕍_{2,2} to Sym^{2} A_{1} is equal to the cohomology of the local system 𝕍_{0} ⊗ 𝕍_{0}(−2), which is equal to ℚ(−4) in degree 4 and trivial otherwise. On the other hand, the Euler characteristic of H_{c}^{•}(M_{2}; 𝕍_{2,2}) was proved in [Ber, Theorem 11.6] to be equal to −[ℚ(−4)]. Then the additivity of Euler characteristics ensures that the Euler characteristic of 𝕍_{2,2} on M_{2} vanishes. This means that the Euler characteristic of each graded piece of the weight filtration on H_{c}^{•}(A_{2}; 𝕍_{2,2}) is 0. More generally, the fact that M_{2} and Sym^{2} A_{1} are affine of dimension 3 and 2 respectively, combined with the Gysin long exact sequence associated to Sym^{2} A_{1} ↪ A_{2} implies that the cohomology of A_{2} with values in any local system is trivial in degree greater than 3. Thus, by Poincaré duality, the cohomology with compact support of A_{2} can be non-trivial only in degree larger than or equal to 3. Furthermore, for non-trivial irreducible local systems H^{0} (and hence H_{c}^{6}) vanishes, whereas H^{1} (and hence H_{c}^{5}) is always zero by the Raghunathan rigidity theorem [R]. This means that the cohomology with compact support of 𝕍_{2,2} on A_{2} can be non-zero only in degrees 3 and 4. The cohomology groups in these degrees are then isomorphic when passing to the associated graded pieces of the weight filtration as a consequence of the vanishing of the Euler characteristic in the Grothendieck group of Hodge structures. □

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Klaus Hulek Leibniz Universität Hannover Institut für Algebraische Geometrie Welfengarten 1 D-30167 Hannover Germany hulek@math.uni-hannover.de Orsola Tommasi Leibniz Universität Hannover Institut für Algebraische Geometrie Welfengarten 1 D-30167 Hannover Germany tommasi@math.uni-hannover.de