j. differential geometry 64 (2003) 171-246

TODA AND KDV

DAVID GIESEKER

Abstract

The main object of this paper is to produce a deformation of the KdV hierarchy of partial differential equations. We construct this deformation by taking a certain limit of the Toda hierarchy. This construction also provides a deformation of the Virasoro algebra.

1. Introduction

Our aim in this paper is to produce a deformation of the KdV hierarchy whose existence was conjectured in [5]. To describe KdV algebraically following Gelfand and Dickey [2], let

R_{0} = C[w^{(0)}, w^{(1)}, …]

be a polynomial ring in infinitely many variables. Introduce a C derivation ∂ on R_{0} by

∂w^{(k)} = w^{(k+1)}.

An element of R_{0} is intended to represent an abstract differential operator in one variable. If f is a C^{∞} function on R, then define

P (f) = P (f, \frac{df}{dx}, …) ,

i.e., substitute f for w^{(0)}, \frac{df}{dx} for w^{(1)}, etc.

I wish to thank Pantelis Damianou, Ed Frenkel, and Ali Kisisel to helpful conversations. This work was partially supported by the NSF.

Received 10/10/2000.

To describe translationally invariant PDE’s algebraically, consider C derivations D of R_{0} which commute with ∂. The set of such D is naturally just R_{0} under the correspondence D → D(w^{(0)}). So R_{0} inherits the structure of a Lie algebra, since the commutator of two derivations is a derivation. Let

K_{1} = w^{(3)} + w^{(1)}w^{(0)},

called the KdV element of R_{0}. One of the main results of KdV theory is that K_{1} lies in a large abelian subalgebra of R_{0}. In fact, there is a sequence of elements K_{n} of R_{0} so that K_{n+1} is not in the linear span generated by the lower K_{n}, …, K_{1} and all the K_{k} commute. These K_{k} are called the KdV hierarchy.

The main object of this paper will be to produce interesting deformations of the KdV hierarchy. That is, we seek to produce mutually commuting L_{k} ∈ R_{0}[[ϵ]] which become the K_{k} when we set ϵ = 0, where R_{0}[[ϵ]] is the formal power series ring in ϵ.

While the ring R_{0} captures much of the algebraic structure of KdV, sometimes one considers solutions or representations of KdV. Suppose that M is an analytic manifold and let P_{0} = ∂, P_{1}, … be derivations of R_{0} commuting with ∂. Let χ_{0}, χ_{1}, …, be vector fields on M and let f be a function on M. We say that f, χ_{0}, χ_{1}, …, form a representation of P_{0}, P_{1}, …, if

χ_{k}(f) = P_{k}(f, χ(f), χ^{2}(f), …),

where we regard the vector field χ_{k} as a derivation on functions on M and χ = χ_{0}.

The main example of representations of the K_{k} is the following: Let X be a hyperelliptic curve and let Q be a Weirstrass point. Let

ϑ : H^{1}(O_{X}) → C

be the theta function. Then we can find translationally invariant vector fields, χ_{0}, χ_{1}, …, so that if

f = χ_{0}^{2}(log ϑ),

then f, χ_{0}, …, form a representation of the KdV hierarchy.

Our aim is to develop a difference version of KdV hierarchy to obtain a deformation of the KdV hierarchy. Here the idea is basically to discretize a differential equation. The heuristic motivation for these discretizations in given in [5]. To be rigorous we need a method to describe difference equations. We consider the ring

S_{1} = C[…, X_{−1}, X_{0}, X_{1}, …; …, Y_{−1}, Y_{0}, Y_{1}, …]

and let

S_{2} = C[…, a_{−1}, a_{0}, a_{1}, …; …, b_{−1}, b_{0}, b_{1}, …].

Let T : S_{2} → S_{2} be the C algebra homomorphism defined by T (a_{n}) = a_{n+1} and T (b_{n}) = b_{n+1}. Now given P, Q ∈ S_{1}, we can define a derivation D_{P,Q} : S_{2} → S_{2} by

D_{P,Q}(a_{n}) = P (…, a_{n−1}, â_{n}, a_{n+1}, …; …, b_{n−1}, b̂_{n}, b_{n+1}, …)

and

D_{P,Q}(b_{n}) = Q(…, a_{n−1}, â_{n}, a_{n+1}, …; …, b_{n−1}, b̂_{n}, b_{n+1}, …),

where the ˆ indicates that a_{n} should be substituted for X_{0} and b_{n} should be substituted for Y_{0}. This construction gives all the derivations of S_{2} commuting with T and so introduces a Lie algebra structure on S_{1} ⊕S_{1}. The interesting example is the Toda equations:

T_{1} = (P_{1}, Q_{1}) = (Y_{−1} − Y_{0}, Y_{0}(X_{0} − X_{1})).

The main theorem here due to Toda, Flaschka and many others is that T_{1} lies in an unexpectedly large Abelian sub-algebra of the Lie algebra S_{1} ⊕ S_{1}. In fact, there is a whole sequence of mutually commuting T_{k} ∈ S_{1} ⊕ S_{1}.

We can also describe solutions of the Toda hierarchy using algebraic geometry following van Moerbeke. Let N be a positive integer. Let C be the space of all complex valued functions on Z. Let T : C → C be translation by N, T (f)(n) = f(n + N). Let C_{N} be the set of translation invariant functions: T (f) = f. Given A and B in C_{N}, we define

L_{(A,B)} : C → C

by the formula

L_{(A,B)}(ψ)(n) = ψ(n + 1) + A(n)ψ(n) + B(n)ψ(n − 1).

Thus L_{(A,B)}(ψ) is a second order linear difference operator. By definition of C_{N}, the operators L_{(A,B)} and T commute, so we can reasonably look for common eigenfunctions of these two operators. If you think of L_{(A,B)} as a discrete analogue of a Schrödinger operator, this amounts to finding the energy levels with a given quasi-momentum of a particle traveling through a periodic potential, a problem frequently encountered in solid state physics [1]. We then define the Bloch spectrum B_{(A,B)} of L_{(A,B)} to be the set of (λ, α) ∈ C×C^{∗} so that there is a nonzero function ψ with L_{(A,B)}(ψ) = λψ and T (ψ) = αψ. By projecting to the λ axis, it is easy to see that B_{(A,B)} is a hyperelliptic curve, possibly singular. Indeed, there are in general two values of α associated to any fixed λ. B_{(A,B)} can be compactified to a curve B_{(A,B)} by adding two points P and Q over λ = ∞. It turns out that the divisor N(P − Q) is linearly equivalent to zero. It also turns out that B_{(A,B)} does not determine (A, B). There are interesting ways of moving (A, B) so that B_{(A,B)} remains fixed. Such a deformation of (A, B) keeping the Bloch spectrum fixed is called an isospectral deformation. The set of all (A′, B′) isospectral to (A, B) turns out to be isomorphic to the Jacobian of B_{(A,B)} in a birational sense for generic A and B. In particular, any linear flow on the Jacobian becomes a non-linear flow on C_{N} × C_{N}, which in turn is a linear combination of Toda flows. For example, there is a linear flow on the Jacobian so that if (A_{t}, B_{t}) indicates the flow of (A_{0}, B_{0}) after time t, then

\frac{dA_{t}(k)}{dt} = B_{t}(k − 1) − B_{t}(k)

and

\frac{dB_{t}(k)}{dt} = B_{t}(k)(A_{t}(k) − A_{t}(k + 1)).

Conversely, given a hyperelliptic curve C of genus g and two points P and Q on C and a suitably generic line bundle L of degree g so that N(P − Q) is linearly equivalent to zero, we can define A_{L} and B_{L} in C_{N} so that linear flows on the Jacobian become Toda flows on C_{N} × C_{N}. (See [7] for instance.)

[5] developed a method for taking the limit of Toda equations. Our purpose in this paper is to develop an algebraic framework for these limits and to prove the main conjecture of [5]. Here is a formalism to allow us to make sense of taking a limit of Toda equations. Let

R = C[v^{(0)}, v^{(1)}, …; w^{(0)}, w^{(1)}, …].

Again we introduce a derivation ∂ by ∂v^{(k)} = v^{(k+1)} and by considering R_{0} as a subring of R. We think of elements P ∈ R as being differential expressions in two functions f and g:

P (f, g) = P (f, \frac{df}{dx}, …; g, \frac{dg}{dx}, …) .

For k ∈ Z, let E_{k} : R[[ϵ]] → R[[ϵ]] be defined by

E_{k} = exp(kϵ∂)

as a formal power series in ϵ. Now suppose we have (P_{1}, P_{2}) ∈ S_{1} ⊕ S_{1}. We can then define a derivation D′_{P_{1},P_{2}} : R[[ϵ]] → R[[ϵ]] commuting with ∂ and continuous in ϵ topology by

(1)  D′_{P_{1},P_{2}}(v^{(0)}) = P_{1}(…, E_{−1}(v^{(0)}), Ê_{0}(v^{(0)}), E_{1}(v^{(0)}), …;
…, E_{−1}(w^{(0)}), Ê_{0}(w^{(0)}), E_{1}(w^{(0)}), …)

and

(2)  D′_{P_{1},P_{2}}(w^{(0)}) = P_{2}(…, E_{−1}(v^{(0)}), Ê_{0}(v^{(0)}), E_{1}(v^{(0)}), …;
…, E_{−1}(w^{(0)}), Ê_{0}(w^{(0)}), E_{1}(w^{(0)}), …),

where the ˆ is again a place holder. Define a C algebra endomorphism of R[[ϵ]] commuting with ∂ by

Φ(v^{(0)}) = −2 + ϵ^{2}v^{(0)}

and

Φ(w^{(0)}) = 1 + ϵ^{2}w^{(0)}.

Φ does not have an inverse on R[[ϵ]], but does have one on R((ϵ)), the ring of Laurent series in ϵ which contain only finitely many negative powers of ϵ and finally define

D_{P_{1},P_{2}} = ΦD′_{P_{1},P_{2}}Φ^{−1}.

In particular, it turns out that we get a series of mutually commuting derivations

D_{T_{k}} (3)

of R[[ϵ]] coming from the Toda hierarchy.

Suppose that P ∈ R_{0}[[ϵ]] and consider the element W = v^{(0)} − P and let I_{P} be the closure of the ideal of R[[ϵ]] generated by W, ∂W, ∂^{2}W, etc. Notice that R[[ϵ]]/I_{P} is naturally isomorphic to R_{0}[[ϵ]]. The following is one of our main results:

Theorem 1.0.1. There is a P so that I_{P} is invariant under all the D_{T_{k}}. Thus the D_{T_{k}} induce derivations D_{T_{k}} of R_{0}[[ϵ]]. Suitable linear combinations of the D_{T_{k}} over C((ϵ)) are a deformation of the KdV hierarchy.

We construct this P recursively in powers of ϵ. Suppose we have found a P which works to order ϵ^{n} and we write P′ = P +ϵ^{n}P_{1}. For D_{T_{1}} to preserve I_{P′} to order ϵ^{n+1} it turns out that there is an element Q_{n} computed in terms of the original P so that ∂P_{1} = Q_{n}. But it is not at all obvious why Q_{n} should be the total derivative of anything. Thus at each stage of constructing P, we meet a highly nontrivial obstruction. We will show that there are lots of functions g so that

∫_{z}^{z+1} Q_{n}(g) = 0 (4)

for generic z ∈ C. If (4) is true for generic enough g, then Q_{n} is a derivative.

We define a generalized idea of representations. Suppose that we have an element P ∈ R. Let M be an analytic manifold and let χ be an analytic vector field on M and let f and g be two meromorphic functions on M. We define P_{χ}(f, g) to be P (f, χf, χ^{2}f, …, g, χg, χ^{2}g, …). Next, suppose we have a function h on M. For any given N ∈ Z^{+}, we can extend this definition to P ∈ R[[ϵ]] by

P_{χ,N}(∑_{n=0}^{∞} P_{n}ϵ^{n})(f, g) = ∑_{n=0}^{N} h^{n}P_{n,χ}(f, g).

If h_{1} and h_{2} are two functions on M, we say

h_{1} ≡ h_{2} mod h^{N}

if (h_{1} − h_{2})/h^{N} is analytic at all the points P where both h is equal to zero at P and both h_{1} and h_{2} are analytic at P. We will use a similar terminology for vector fields on M.

Suppose we are given derivations D_{1}, …, D_{n} of R[[ϵ]] and we are given vector fields χ, χ_{1}, …, χ_{n} on a manifold M. Suppose we are also given a function h on M which is killed by χ and all the χ_{j}. We further suppose that the f and g do not have poles along the set {h = 0}.

Definition 1.0.2. In the above situation, we say that (f, g, h; χ, χ_{1}, …, χ_{n}) form a representation of D_{1}, …, D_{n} if

χ_{i}(P (f, g)) ≡ D_{i,χ,M}(P)(f, g) mod h^{N}

for any P ∈ R[[ϵ]] and any positive integer N and M sufficiently large depending on N. We also assume that these equations are true with the convention that D_{0} = ∂ and that χ_{0} = χ.

Definition 1.0.3. We say that D_{i} is slow under the above representation if D_{i}(P) is not in the ideal (ϵ) ⊂ R[[ϵ]] for some P ∈ R[[ϵ]], but χ_{i} vanishes on the set h = 0.

We can construct representations using algebraic geometry. Let T be the disk in C and let π : X → T be a smooth proper family of curves of genus n. We will suppose there are sections P : T → X, Q : T → X, R : T → X so that for each t ∈ T, we have P (t) + Q(t) is a divisor on X_{t} = π^{−1}(t) linearly equivalent to the divisor 2R(t). We will assume that P (0) = Q(0) = R(0). Choosing a homology basis of H_{1}(X , Z), we can identify C^{g} × T locally with the relative Jacobian of the family X → T. Suppose γ ∈ H_{1}(X , Z) is given. We will suppose there is function h on T so that

∫_{Q(t)}^{P (t)} ω = h(t) ∫_{γ} ω (5)

for all holomorphic one forms on X_{t} = π^{−1}(t). Geometrically, (5) says that under the Abel-Jacobi map which sends a piece of the curve to C^{g}, the secant line from Q to P passes through γ. This is meaningful as long as P (t), Q(t) are all close together. Under our identification, a point (v, t) ∈ C^{g} × T gives a line bundle L on X_{t}. When h(t) = 1/N for N an integer, we can introduce the functions A_{L} and B_{L} as being defined at points (v, t) with h(t) = 1/N. It is easy to see that there are meromorphic functions f and g which coincide with A and B, and these f, g and h and certain linear flows produce a representation of the Toda D_{T_{k}}. Further, by studying the geometry of the situation, we can show that this representation is slow for D_{T_{2}} + 2D_{T_{1}}. This turns out to mean that f can be computed asymptotically up to an additive constant from g. Lemma 2.6.1 contains the crucial step. It says that if f is computed in terms of g up to order h^{n} and if the representation is slow, then we can compute f in terms of g up to an additive constant modulo h^{n+1}. This computation is just that there is a P ∈ R_{0}[[ϵ]] so that f = P (g)+C and this P is the desired P of Theorem 1.0.1. This is a strange relation, since for any particular N, there’s no relation between A and B. The subtlety here is that the geometric genus of the Bloch spectrum for a arbitrary A and B of periodicity N grows with N, but the geometric genus of the Bloch spectrum associated to the A_{L} and B_{L} above remains g, although the arithmetic genus does grow as we let h(t) = 1/N. In fact, X_{t} is the normalization of the Bloch spectrum, but the Bloch spectrum has many nodes which are resolved by the normalization. Further, the line bundle L on the normalization of the Bloch spectrum becomes a torsion free sheaf on the Bloch spectrum, which is not locally free at new nodes.

The basic problem turns out to be to construct lots of such representations. We look at families of curves inside P^{1} × P^{1} of bidegree (n + 1, 2) with affine coordinates x and y. Intuitively, the condition (5) imposes g − 1 conditions, so there should be lots of curves satisfying the condition (5). We investigate curves near the following curve C_{0} defined by

0 = (y^{2} − x)(x − 1) (x − \frac{1}{2^{2}} ) (x − \frac{1}{3^{2}} ) ⋯ (x − \frac{1}{n^{2}} ) .

Our object roughly is to show that the subset of curves satisfying (5) is smooth of codimension g−1 near C_{0}. Further, we let L_{0} be a line bundle on C_{0} of degree n which has degree one on all the vertical components {0 = (x − 1/k^{2})} of C_{0} and degree zero on the component {y^{2} − x = 0}. Then we can explicitly calculate the functions f and g we are interested in when we deform the pair (C_{0}, L_{0}) in certain directions. For instance, when we deform one of the nodes of C_{0} away, but still have the curves satisfying (5). This gives enough information to produce generic enough g′s.

There are several technical problems in establishing our results. One is finding a suitable definition of generic. Another is that f is only determined up to a constant by g. This problem is overcome by a monodromy argument Lemma 3.7.4.

The sequence of commuting derivations D_{T_{k}} can be put in the context of Poisson brackets, so that we can consider algebraically the setup of Hamiltonian completely integrable systems with conserved quantities in involution with respect to a Poisson bracket and the associated flows from the conserved quantities. I learned about this type of construction from papers of E. Frenkel. Let

R̂ = C[…, â_{−1}, â_{0}, â_{1}, …, b̂_{−1}, b̂_{0}, b̂_{1}, …].

We say a monomial in the â_{k} and b̂_{k} has weight r if the sum of the subscripts of the â_{k} and b̂_{l} sum to r. So the monomial â_{1}â_{2}b̂_{−3} has weight 0. Let I_{k} ⊂ R̂ be the C span of all the elements of weight k. Let M_{N} be the ideal of R generated by

â_{N}, â_{N+1}, …, â_{−N}, â_{−N−1}, …; …, b̂_{N}, b̂_{N+1}, …, b̂_{−N}, b̂_{−N−1}.

Let Î_{k} be the completion of I_{k} with respect to subspaces I_{k} ∩ M_{N} as N → ∞. Then

F = ⨁_{k} Î_{k}

is called the Fourier ring. F is naturally a graded ring. We can construct a series of maps f_{n} : R[[ϵ]] → F[[ϵ]] so that f_{n}(v^{(0)}) = â_{n} and f_{n}(w^{(0)}) = b̂_{n} and the f_{n} behave like Fourier coefficients, e.g.,

f_{n}(HK) = ∑_{k+l=n} f_{k}(H)f_{l}(K).

One can form an analogous ring F_{0} from the ring

R̂_{0} = C[…, b̂_{−1}, b̂_{0}, b̂_{1}, …].

One can then show that the Toda derivations on R[[ϵ]] induce derivations on F[[ϵ]] which are compatible with the f_{n}. Now each Toda lattice can be put in a Poisson framework and we can make a formal version of these Poisson brackets to obtain a Poisson bracket on F[[ϵ]]. Further, the Toda flows on F[[ϵ]] come from conserved quantities in F[[ϵ]]. Let Î_{P} ⊂ F[[ϵ]] be the closure of the ideal generated by all the Fourier coefficients f_{n}(v^{(0)} − P) for n ≠ 0 and a certain Casimir. Then we can find an induced Dirac bracket on F[[ϵ]]/Î_{P} ≃ F_{0}[[ϵ]] so that the Toda derivations come by bracketing with conserved quantities. Further, we can find β̂_{k} ∈ F[[ϵ]]/(Î_{P}) for k ∈ Z so that modulo ϵ the β_{k} generate F_{0} topologically and satisfy the defining relations of the Virasoro algebra modulo ϵ.

A similar construction of the deformation of KdV discovered by Frenkel and Reshetikhin [4] in terms of difference equations has been made by Frenkel [3]. I believe the techniques of this paper will produce many such deformations of KdV hierarchy as well as deformations of W-algebras. [6] contains some very interesting work on deformations of KdV inspired by [5], which was produced independently of this paper.

2. Differential algebra

2.1

Let R be the ring of polynomials with complex coefficients with generators v^{(i)} and w^{(j)} where i and j run over the nonnegative integers,

R = C[v^{(0)}, w^{(0)}, v^{(1)}, w^{(1)}, …]

We introduce a C derivation ∂ by the formulas

∂v^{(i)} = v^{(i+1)}

and

∂w^{(i)} = w^{(i+1)}.

Then ∂ on any polynomial in R is defined by the Leibnitz rule. We have a subring R_{0} ⊂ R defined to be the ring generated by the w^{(n)}.

This ring R is considered to be the ring of translation invariant differential operators in two functions f(x) and g(x). An element of R can be regarded as such a differential operator by making the substitutions

v^{(n)} = \frac{∂^{n}f(x)}{∂x^{n}}

and

w^{(n)} = \frac{∂^{n}g(x)}{∂x^{n}}

so that ∂ just becomes \frac{∂}{∂x.} If P ∈ R and f(x) and g(x) are C^{∞} functions of x, then we define

P (f, g)(x)

to be the result of making the above substitution. So for example, if P = v^{(1)}w^{(2)}, then

P (f, g)(x) = \frac{∂f(x)}{∂x} \frac{∂^{2}g(x)}{∂x^{2}}.

If f and g depend on a auxiliary variable t, then we write P (f, g)(x, t). The ring R can be used to study systems of equations:

\frac{∂f(x, t)}{∂t} = P (f, g)(x, t)

\frac{∂g(x, t)}{∂t} = Q(f, g)(x, t),

where P and Q are elements of R. We can encode the pair (P, Q) by defining a derivation D_{(P,Q)}.

Definition 2.1.1. D_{(P,Q)} is the derivation of R commuting with the derivation ∂ with the additional properties

D_{(P,Q)}(v^{(0)}) = P

and

D_{(P,Q)}(w^{(0)}) = Q.

Any derivation of R commuting with ∂ is of this form.

2.2

We will mostly be concerned with the ring R[[ϵ]]. The elements of this ring are formal power series in ϵ so that the coefficients of ϵ^{n} are just elements of R. We extend ∂ to be a continuous derivation of R[[ϵ]] by taking ∂ϵ = 0. We next introduce an important series of maps E_{k} : R[[ϵ]] → R[[ϵ]] by the formulas

E_{k}(P) = P + kϵ∂P + \frac{k^{2}ϵ^{2}∂^{2}P}{2!} + \frac{k^{3}ϵ^{3}∂^{3}P}{3!} + · · · .

Formally, we can write

E_{k} = exp(kϵ∂).

We have that

E_{k}E_{j} = E_{k+j}.

Note that E_{k}(v^{(0)}) is just the Taylor series for f(x + kϵ)

E_{k}(v^{(0)})(f(x), g(x)) = f(x) + \frac{kϵ∂f(x)}{∂x} + ⋯,

when we make the substitution of f(x) for v^{(0)} described above. Note that if D is a continuous derivation of R[[ϵ]] commuting with ∂ and D(ϵ) = 0, then D is uniquely specified by D(v^{(0)}) and D(w^{(0)}). Conversely, given F and G in R[[ϵ]], we can find a continuous derivation D commuting with ∂ and with D(ϵ) = 0. Let us call such a derivation a tame derivation.

We will use the ring R[[ϵ]] to describe the asymptotic behavior of difference equations.

Definition 2.2.1. S_{1} is the ring of polynomials in the variables

…, X_{−1}, X_{0}, X_{1}, X_{2}, …

and the variables

…, Y_{−1}, Y_{0}, Y_{1}, Y_{2}, …

.

In our context, a difference equation will be given by two polynomials P_{1} and P_{2} in the X_{i} and Y_{j}. In order to facilitate substitution, we will write

P_{1}(…, a, b̂, c, …; …, α, β̂, γ, …)

to mean the result of substituting a for X_{−1}, b for X_{0}, c for X_{1} and also substituting α for Y_{−1}, β for Y_{0}, etc. That is the ˆ is just to indicate the variable to be substituted for X_{0} or Y_{0}. Let S_{2} be the polynomial ring over C with variables …, a_{−1}, a_{0}, a_{1}, … and …, b_{−1}, b_{0}, b_{1}, …. (Of course, R, S_{1} and S_{2} are all the same polynomial ring on a denumerable number of variables, but it is convenient to have different names the variables.) Given P_{1} and P_{2}, we can define a derivation D_{P_{1},P_{2}} of S_{2} by

D_{P_{1},P_{2}}(a_{n}) = P_{1}(…, a_{n−1}, â_{n}, a_{n+1}, …; …, b_{n−1}, b̂_{n}, b_{n+1}, …)

and

D_{P_{1},P_{2}}(b_{n}) = P_{2}(…, a_{n−1}, â_{n}, a_{n+1}, …; …, b_{n−1}, b̂_{n}, b_{n+1}, …).

Let T be the automorphism of S_{2} defined by T (a_{n}) = a_{n+1} and T (b_{n}) = b_{n+1}. Then D_{P_{1},P_{2}} is translation invariant in the sense that D_{P_{1},P_{2}} commutes with T. Conversely, any derivation of S_{2} commuting with T is of the form D_{P_{1},P_{2}}. Since the commutator of derivations is a derivation, this allows us to define the commutator of (P_{1}, P_{2}) with (Q_{1}, Q_{2}) by

D_{[(P_{1},P_{2}),(Q_{1},Q_{2})]} = [D_{P_{1},P_{2}}, D_{Q_{1},Q_{2}}].

We will now define a continuous derivation D_{P_{1},P_{2}} of R[[ϵ]] by

(6)  D_{P_{1},P_{2}}(v^{(0)}) = P_{1}(…, E_{−1}(v^{(0)}), Ê_{0}(v^{(0)}), E_{1}(v^{(0)}), …;
…, E_{−1}(w^{(0)}), Ê_{0}(w^{(0)}), E_{1}(w^{(0)}), …)

and

(7)  D_{P_{1},P_{2}}(w^{(0)}) = P_{2}(…, E_{−1}(v^{(0)}), Ê_{0}(v^{(0)}), E_{1}(v^{(0)}), …;
…, E_{−1}(w^{(0)}), Ê_{0}(w^{(0)}), E_{1}(w^{(0)}), …).

We then define

D_{P_{1},P_{2}}(v^{(n)}) = ∂^{n}D_{P_{1},P_{2}}(v^{(0)})

and

D_{P_{1},P_{2}}(w^{(n)}) = ∂^{n}D_{P_{1},P_{2}}(w^{(0)})

and extend by the Leibnitz rule. Note that the commutator [D_{P_{1},P_{2}}, ∂] vanishes on the generators v^{(n)} and w^{(n)}, so D_{P_{1},P_{2}} commutes with ∂. It is an exercise in the chain rule that

[D_{(P_{1},P_{2})}, D_{(Q_{1},Q_{2})}] = D_{[(P_{1},P_{2}),(Q_{1},Q_{2})]}.

2.3

Suppose that we have an element P ∈ R. Let M be an analytic manifold and let χ be an analytic vector field on M and let f and g be two meromorphic functions on M. We define P_{χ}(f, g) to be P (f, g, χf, χg, χ^{2}f, χ^{2}g, …). Next, suppose we have a function h on M. For any given positive integer N, we can extend this definition to P ∈ R[[ϵ]] by

P_{χ,N} (∑_{n=0}^{∞} P_{n}ϵ^{n}) (f, g) = ∑_{n=0}^{N} h^{n}P_{n,χ}(f, g).

If h_{1} and h_{2} are two functions on M, we say

h_{1} ≡ h_{2} mod h^{N}

if (h_{1} − h_{2})/h^{N} is analytic on an open dense set of the set {h = 0}. We will use a similar terminology for vector fields on M.

Suppose we are given tame derivations D_{1}, …, D_{n} of R[[ϵ]] and we are given vector fields χ, χ_{1}, …, χ_{n}. Suppose we are also given a function h on M which is killed by χ and all the χ_{j}. We further suppose that the f and g do not have poles along the set {h = 0}.

Definition 2.3.1. In the above situation, we say that ρ = (f, g, h; χ, χ_{1}, …, χ_{n}) is a representation of D_{1}, …, D_{n} if

χ_{i}(P (f, g)) ≡ D_{i,χ,M}(P)(f, g) mod h^{N}

for any P ∈ R[[ϵ]] and any positive integer N and M sufficiently large depending on N. We also assume that these equations are true with the convention that D_{0} = ∂ and that χ_{0} = χ. We define ρ(P) = P (f, g). We will only use ρ(P) in congruences modulo h^{N}, so in the context of congruence, the formal power series in h makes sense. Analogously, suppose that D_{1}, …, D_{n} ∈ R_{0}[[ϵ]]. ρ = (g, h; χ, χ_{1}, …, χ_{n}) is a representation of D_{1}, …, D_{n} if

χ_{i}(P (g)) ≡ D_{i,χ,M}(P)(g) mod h^{N}

for any P ∈ R_{0}[[ϵ]] and any positive integer N and M sufficiently large depending on N.

In the Definition, it suffices to check the cases P = v^{(0)} and P = w^{(0)} to check the equality of the definition for all P, since both sides are derivations.

Definition 2.3.2. Suppose D = ∑ a_{i}D_{i}is a linear combinations of the D_{i}. We say that D is slow under ρ if there is a P ∈ R[[ϵ]] so that D(P) is not in the ideal (ϵ) ⊂ R[[ϵ]], but ∑ a_{i}χ_{i} vanishes on the set h = 0.

2.4

We will be constructing representations be in the following context: Let V be an analytic manifold and let M = V ×C^{g}. Let π be the projection of M onto V. h will be the pullback of some function on V via π. σ = σ_{0} and σ_{1}, …, σ_{k} will denote sections of π, σ_{k} : V → M. Now any section τ of π induces vertical vector field D_{τ} on M by

D_{τ} (f)(x) = lim_{p→0} \frac{f(x + pτ(x)) − f(x)}{p}.

We can define representations of R[[ϵ]] when we have C^{∞} functions f and g which satisfy difference equations.

Proposition 2.4.1. Suppose that f and g satisfy the following equations:

D_{σ_{n}}(f)(x)=P_{1,n}(…, f(x − h(x)σ(π(x))), f̂(x), f(x + h(x)σ(π(x))), …; (8)
…, g(x − h(x)σ(π(x))), ĝ(x), g(x + h(x)σ(π(x))), …)

D_{σ_{n}}(g)(x)=P_{2,n}(…, f(x − h(x)σ(π(x))), f̂(x), f(x + h(x)σ(π(x))), …; (9)
…, g(x − h(x)σ(π(x))), ĝ(x), g(x + h(x)σ(π(x))), …),

where P_{l,k} are in S_{1} and the ˆ is the place indicator. We let D_{k} be the element D_{P_{1,k},P_{2,k}}, a tame derivation of R[[ϵ]] defined above and let χ_{k} = D_{σ_{k}} and χ be D_{σ}. Then (f, g, h; χ, χ_{1}, …, χ_{n}) form a representation of D_{1}, …, D_{n}.

Proof. All we need to do is to check that

χ_{i}(f) ≡ D_{i,χ,N}(v^{(0)})(f, g) mod h^{N}

and

χ_{i}(g) ≡ D_{i,χ,N}(w^{(0)})(f, g) mod h^{N}.

In fact,

D_{i,χ,N}(v^{(0)})(f, g) (10)

≡ P_{1,n}(…, f(x − h(x)σ(π(x))), f̂(x), f(x + h(x)σ(π(x))), …;

…, g(x − h(x)σ(π(x))), ĝ(x), g(x + h(x)σ(π(x))), …) mod h^{N}.

This in turn follows from

f(x + ph(x)σ(π(x))) ≡ E_{p}(v^{(0)})_{χ,N}(f, g) mod h^{N},

which in turn is just Taylor’s theorem. q.e.d.

2.5

We will frequently use this construction when

f = −2 + h^{2}f_{1}

and

g = 1 + h^{2}g_{1},

where f_{1} and g_{1} are meromorphic functions of M which do not have polar divisors containing {h = 0}. To this end, define a C algebra endomorphism of R[[ϵ]] commuting with ∂ by

Φ(v^{(0)}) = −2 + ϵ^{2}v^{(0)}

and

Φ(w^{(0)}) = 1 + ϵ^{2}w^{(0)}.

Φ does not have an inverse on R[[ϵ]], but does have one on R((ϵ)), the ring of Laurent series in ϵ which contain only finitely many negative powers of ϵ.

Definition 2.5.1. Suppose D is a tame derivation of R[[ϵ]]. We define a new derivation D_{Φ} of R((ϵ)) by

D_{Φ} = ΦDΦ^{−1}.

In situations we will be considering D_{Φ} will turn out to be a tame derivation of R[[ϵ]].

186 d. gieseker

Lemma 2.5.2. Suppose f = −2 + h^{2}f_{1} and g = 1 + h^{2}g_{1}. If

ρ = (f, g, h; χ, χ_{1}, …, χ_{n})

is a representation of D_{1}, …, D_{n} then ρ_{Φ} = (f_{1}, g_{1}, h; χ, χ_{1}, …, χ_{n}) is a representation of D_{1,Φ}, …, D_{n,Φ}.

Proof.

(Φ(P))(f_{1}, g_{1}) = P (f, g)

for any P ∈ R[[ϵ]]. So

(11)  χ_{i}((Φ(P))(f_{1}, g_{1})) = χ_{i}(P (f, g))
≡ D_{i}(P)(f, g) mod h^{N}
≡ Φ(D_{i}(P))(f_{1}, g_{1}) mod h^{N}
≡ (D_{Φ,i})(Φ(P))(f_{1}, g_{1}) mod h^{N}.

Given Q ∈ R[[ϵ]], we let P = Φ^{−1}(Q) and then we have

χ_{i}(Q)(f_{1}, g_{1})) ≡ (D_{Φ,i})(Q)(f_{1}, g_{1}) mod h^{N}

so we have a representation. q.e.d.

Next we work out a simple example of these definitions. We take

P_{1} = Y_{−1} − Y_{0} ∈ S_{1}

and

P_{2} = Y_{0}(X_{0} − X_{1}).

These are the Toda equations. Then define D_{1} by

(12)  D_{1}(v^{(0)}) = w^{(0)} − w^{(1)}ϵ + w^{(2)}ϵ^{2}/2! + ⋯ − w^{(0)}
= −w^{(1)}ϵ + w^{(2)}ϵ^{2}/2! + ⋯

and

D_{1}(w^{(0)}) = w^{(0)}(−v^{(1)} − v^{(2)}ϵ/2! + ⋯).

Then

D_{1,χ,N}(v^{(0)})(f, g) = −hχ(g) + h^{2}χ^{2}(g)/2! + ⋯

and

D_{1,χ,N}(w^{(0)})(f, g) = g(−hχ(f) − h^{2}χ^{2}(f)/2! + ⋯).

So if (f, g, h; χ_{1}) is a representation of D_{1}, then we will have

χ_{1}(f) = −hχ(g) + h^{2}χ^{2}(g)/2! + ⋯

and

χ_{2}(g) = g(−hχ(f) − h^{2}χ^{2}(f)/2! + ⋯),

where these equations are taken to be true near {h = 0} modulo high powers of h, so the expansions are considered to be asymptotic.

2.6

Recall that R_{0} ⊂ R is the C algebra generated by the w^{(n)}. Next, we consider P ∈ R_{0}[[ϵ]] and assume that

P = w^{(0)} + ϵP_{1}.

Assume we have a representation (f, g, h; χ, χ_{1}) of D_{1}. Assume that

D_{1}(v^{(0)}) = v^{(1)} − w^{(1)} + ϵQ_{1}

and that

D_{1}(w^{(0)}) = w^{(1)} − v^{(1)} + ϵQ_{2}.

Assume that this representation is slow for D_{1}, so that χ_{1} vanishes on h = 0. Let D be the derivation defined by

D(v^{(0)}) = v^{(1)} − w^{(1)}

and

D(w^{(0)}) = w^{(1)} − v^{(1)}.

Then

D_{1} = D + ϵE.

Let W = (v^{(0)} − P) and let I_{P} be the closure of the ideal generated by W, ∂W, ∂^{2}W, …. If Q ∈ R[[ϵ]], let Q be the image of Q in R[[ϵ]]/I_{P} . Then there is the natural map

σ_{1} : R_{0}[[ϵ]] → R[[ϵ]]/I_{P} .

Note that σ_{1} is an isomorphism. We let σ(Q) be σ_{1}^{−1}(Q). Thus σ(Q) is just the result of replacing any occurrence of v^{(0)} in Q by P, any occurrence of v^{(1)} by ∂P, etc.

Lemma 2.6.1. Given P and D_{1} as above, there is an H ∈ R_{0}[[ϵ]] so that if

ρ(v^{(0)} − P) ≡ 0 mod h^{n}.

Then

χ(ρ(v^{(0)} − P)) ≡ ρ(H) mod h^{n+1}.

Note that H does not depend on ρ.

Proof. We let s = ρ(v^{(0)} − P). Notice that if Q ∈ R, then

ρ(Q) ≡ ρ(σ(Q)) mod h^{n}, (13)

since P = σ(v^{(0)}) and

(14)  ρ(v^{(0)}) ≡ ρ(P) mod h^{n}
≡ ρ(σ(v^{(0)})) mod h^{n}.

Let x = χ_{1}(s). Then

x = ρ(D_{1}(v^{(0)} − P)) (15)

= ρ(D_{1}(v^{(0)} − w^{(0)} − ϵP_{1}))

= ρ(D(v^{(0)} − w^{(0)} − ϵP_{1}) + ϵE(v^{(0)} − w^{(0)} − ϵP_{1}))

= ρ(2∂(v^{(0)} − w^{(0)}) + ϵ(ρ(−D(P_{1}) + E(v^{(0)} − w^{(0)} − ϵ(P_{1}))

= 2χ(ρ(v^{(0)}) + ρ(−w^{(0)})) + ϵ(ρ(−D(P_{1}) + E(v^{(0)} − w^{(0)} − ϵP_{1}))

= 2χ(s) + 2χρ(P) + ρ(−2w^{(1)}) + ϵ(ρ(−D(P_{1}) + E(v^{(0)} − w^{(0)} − ϵP_{1})) = 2χ(s) + ρ(2∂P − 2w^{(1)}) + ϵ(ρ(−D(P_{1}) + E(v^{(0)} − w^{(0)} − ϵP_{1})).

Noting that ρ(σ(−D(P_{1})+E(v^{(0)}−w^{(0)}−ϵP_{1}))) is congruent to ρ(−D(P_{1}) +E(v^{(0)} − w^{(0)} − ϵP_{1})) modulo h^{n} from (13), we can continue

(16) x = 2χ(s) + ρ(σ(2∂P − 2w^{(1)} + ϵ(−D(P_{1}) + E(v^{(0)} − w^{(0)} − ϵP_{1})) mod h^{n+1}.

But

x ≡ 0 mod h^{n+1}

since D_{1} is slow for ρ so we obtain the conclusion of the lemma. q.e.d.

Definition 2.6.2. Let D_{i}(v^{(0)}) = P_{i} and D_{i}(w^{(0)}) = Q_{i}. We say D_{1}, …, D_{n} are A-nice if

P_{i} (v^{(0)} + K, v^{(1)}, …; w^{(0)}, w^{(1)}, …)

= ∑_{j≤i} \binom{i}{j}P_{j} (v^{(0)}, v^{(1)}, …; w^{(0)}, w^{(1)}, …) (AK)^{j}

and

Q_{i} (v^{(0)} + K, v^{(1)}, …; w^{(0)}, w^{(1)}, …)

= ∑_{j≤i} \binom{i}{j}Q_{j} (v^{(0)}, v^{(1)}, …; w^{(0)}, w^{(1)}, …) (AK)^{j}.

Definition 2.6.3. Let D_{i}(v^{(0)}) = P_{i} and D_{i}(w^{(0)}) = Q_{i}. We say the ρ has weights r_{1}, …, r_{n} if

P_{i} (uv^{(0)}, uv^{(1)}, …; u^{2}w^{(0)}, u^{2}w^{(1)}, …)
= u^{r_{i}}P_{i} (v^{(0)}, v^{(1)}, …; w^{(0)}, w^{(1)}, …) .

and

Q_{i} (uv^{(0)}, uv^{(1)}, …; u^{2}w^{(0)}, u^{2}w^{(1)}, …)
= u^{r_{i}+1}Q_{i} (v^{(0)}, v^{(1)}, …; w^{(0)}, w^{(1)}, …) .

Remark 2.6.4. Note that if the D_{i} are 1-nice, then D_{i,Φ} are ϵ^{2}-nice.

Definition 2.6.5. Suppose that ρ = (f, g, h, χ; χ_{1}, …, χ_{n}) is a ϵ^{2}- nice representation and that s is a meromorphic function on M with χ(s) = 0 and χ_{i}(s) = 0. Let

χ′_{i} = ∑_{j≤i} \binom{i}{j}ϵ^{2j}χ_{j}s^{j}

Then ρ′ = (f + s, g, h, χ; χ′_{1}, …, χ′_{n}) is a representation of D_{1}, …, D_{n} and we call ρ′ the translation of ρ by s.

Definition 2.6.6. Suppose ρ = (f, g, h, χ; χ_{1}, …, χ_{n}) has weights r_{1}, …, r_{n}. We define an extended representation ρ′ = (f′, g′, h′, χ′, χ′_{1}, …) on M × C by first defining a function u : M × C → C by

u(m, z) = 1 + h(m)^{2}z

and

f′(m, z) = u(m, z)f(m)

and

g′(m, z) = u(m, z)^{2}g(m)

and χ′_{i} is u^{1−r_{i}} times the natural pullback of χ_{i} and h′ and χ′ are the natural pullbacks of h and χ to M × C.

Remark 2.6.7. This definition works fine without the particular choice of u = 1 + h^{2}z we have made, but the next remark does not.

Remark 2.6.8. In the situation of Definition 2.6.5, suppose that

f = −2 + h^{2}f_{1}

and

g = 1 + h^{2}g_{1}.

Then

ρ′_{Φ}(w^{(0)})(m, z) ≡ ρ_{Φ}(w^{(0)})(m) + 2z mod h.

Definition 2.6.9. If W ∈ R_{0}[[ϵ]], we say g = ρ(w^{(0)}) satisfies the equation W nontrivially mod h^{n} if

ρ(W) ≡ 0 mod h^{n}

and g does not vanish at a generic point of {h = 0} and W ∉ (ϵ^{n}).

Remark 2.6.10. Note that in the above definition, we can find W_{1} ∈ R_{0}[[ϵ]] so that W_{1} = ϵ^{k}W and W_{1} ∉ (ϵ). Then k < n, so ρ(W_{1})(g) is zero when restricted to {h = 0}.

2.7

Let D ∈ R_{0}. We introduce the variational derivative of D

δD = ∑_{k=0}^{∞}(−1)^{k}∂^{k} (\frac{∂D}{∂w_{k}} ) .

This operator has the property that

[\frac{d}{dϵ} ∫_{0}^{2π} D(f + ϵg)]_{ϵ=0} = ∫_{0}^{2π} δD(f)g, (17)

if f and g are periodic. δ(D)(f) can be somewhat more intuitively defined by Equation (17). Evaluating

∫_{0}^{2π} D(f + ϵg)

will yield the integral of a differential polynomial in f and g. To take the limit at ϵ → 0, we can throw away all the non-linear terms in g. Further, we can eliminate any occurences of derivatives of g by integration by parts. The resulting differential polynomial in f is δ(D)(f).

Lemma 2.7.1. Suppose we have a function f(t_{1}, t_{2}, …, t_{n}; x) which is periodic with period 1 in x. Suppose that for generic z, we have

∫_{z}^{z+1} D(f)(x)dx = 0.

Then

∫_{z}^{z+1} δ(D)(f)\frac{∂f}{∂t_{i}} = 0,

for all i.

Proof. We assume that i = 1 for convenience. Fixing t_{1}, t_{2}, …, we let

g(x, ϵ) = \frac{f(t_{1} + ϵ, t_{2}, …, t_{n}) − f(t_{1}, t_{2}, …, t_{n})}{ϵ}

Note that g(x, ϵ) is holomorphic function of x, even when ϵ = 0. In fact,

g(x, 0) = \frac{∂f}{∂x_{1}} .

So

(18)  ∫_{z}^{z+1} D(f + ϵg) = ∫_{z}^{z+1} D(f)(t_{1} + ϵ, t_{2}, …, t_{n}; x)dx
= 0.

Taking the derivative of both sides of the equation with respect to ϵ and setting ϵ = 0, we obtain

∫_{z}^{z+1} δ(D)(f)\frac{∂f}{∂t_{1}} = 0.

q.e.d.

Lemma 2.7.2. Let D ∈ R_{0} be of order m, i.e., the highest derivative occuring is of order m. Suppose that

\frac{∂f}{∂t_{k}} (0, 0, …, 0; x) = a_{k} + b_{k} exp(2πikx),

with b_{k} ≠ 0. Then D(f) is not identically zero for k ≤ m.

Proof. Fix a point x_{0}. We define a map ϕ_{x_{0}} : C^{m} → C^{m} by

ϕ_{x_{0}} ( t_{1} ; t_{2} ; t_{3} ; ⋮ ; t_{m}) = (f(t_{1}, t_{2}, …, t_{m}; x_{0}) ; \frac{∂f(t_{1},t_{2},...,t_{m};x_{0})}{∂x} ; \frac{∂^{2}f(t_{1},t_{2},...,t_{m};x_{0})}{∂^{2}x} ; ⋮ ; \frac{∂^{m}f(t_{1},t_{2},...,t_{m};x_{0})}{∂^{m}x}) .

We can compute the Jacobian matrix dϕ_{x_{0}} of ϕ_{x_{0}} at (0, 0, …, 0).

dϕ_{x_{0}}(0, 0, 0, …, 0)

= (\begin{matrix}a_{1} + b_{1}e^{2πix_{0}} & a_{2} + b_{2}e^{4πix_{0}} & … & a_{m} + b_{m}e^{2mπix_{0}} \\ 2πie^{2πix_{0}} & 4πib_{2}e^{4πix_{0}} & … & 2πimb_{m}e^{2mπix_{0}} \\ … & … & ... & … \\ (2πi)^{m}e^{2πix_{0}} & b_{2}(4πi)^{m}e^{4πix_{0}} & … & b_{m}(2πim)^{m}e^{2mπix_{0}} \end{matrix} ) .

We claim that det(dϕ_{x_{0}}(0, 0, 0, …, 0)) ≠ 0 for generic values of x_{0}. It suffices to show that det(W_{x_{0}}) ≠ 0, where

W_{x_{0}} = (\begin{matrix}\frac{a_{1}}{b_{1}} e^{−2πix_{0}} + 1 & \frac{a_{2}}{b_{2}} e^{−4πix_{0}} + 1 & … & \frac{a_{m}}{b_{m}} e^{−2πimx_{0}} + 1 \\ 1 & 2 & … & m \\ 1 & 4 & … & m^{2} \\ … & … & ... & … \\ 1 & 2^{m} & … & m^{m} \end{matrix}) .

In particular if det(W_{x_{0}}) = 0, then we would have

det (\begin{matrix}\frac{a_{1}}{b_{1}} z + 1 & \frac{a_{2}}{b_{2}} z^{2} + 1 & … & \frac{a_{m}}{b_{m}} z^{m} + 1 \\ 1 & 2 & … & m \\ 1 & 4 & … & m^{2} \\ … & … & ... & … \\ 1 & 2^{m} & … & m^{m} \end{matrix}) = 0

for any value of z. In particular, taking z = 0, we would have that a Vandermond determinant was zero.

So the image of ϕ_{x_{0}} would contain an open set. On the other hand, the equation D(f) = 0 says that the numbers

f(t_{1}, t_{2}, …, t_{m}; x_{0}), \frac{∂f(t_{1},t_{2},...,t_{m};x_{0})}{∂x}, \frac{∂^{2}f(t_{1},t_{2},...,t_{m};x_{0})}{∂^{2}x}, …, \frac{∂^{m}f(t_{1},t_{2},...,t_{m};x_{0})}{∂^{m}x}

satisfy a nontrivial algebraic equation H = 0 independent of t_{1}, t_{2}, …, t_{m}. But {H = 0} cannot contain an open set. q.e.d.

2.8

We wish to study representations in the following somewhat degenerate context: We have a sequence of functions g_{r} on M_{r} = V_{r} × C and let π_{r} be the projection of M_{r} → V_{r}. We can write g_{r}(v, z), where v ∈ V_{r} and z ∈ C. We will assume that g_{r} are periodic with period one with respect to the second variable, e.g.,

g_{r}(v, z) = g_{r}(v, z + 1).

Let D_{r} be the vector field \frac{∂}{∂z} which is tangent to the fibers of π_{r}. We can define P (g_{r}) for all P ∈ R_{0} so that (w^{(0)})(g_{r}) = g_{r} and

D_{r}(P (g_{r})) = ∂(P)(g_{r}).

We also assume there are maps k_{r} : V_{r} → V_{r+1} so that the pullback of g_{r+1} is g_{r} via k_{r} × id. We also assume we are given points Q_{r} ∈ V_{r} compatible with the maps π_{r}.

Definition 2.8.1. We say the family {g_{r}} is generic if first for any D ∈ R_{0}, there is an r so that D(g_{r}) is not identically zero. Second suppose that we are given a D ∈ R_{0}. Each point v of V_{r} yields a function of x by

F_{v,r}(x) = g_{r}(v, x).

Suppose that

∫_{z}^{z+1} D(F_{v,r})(x) dx = 0

for generic z,v and all r. Then there is an E ∈ R_{0} so that D = ∂E.

Next we give a criterion for the family {g_{r}} to be generic. We further assume that for any positive integer n, there is an integer r so that for any integer k, −n ≤ k ≤ n there are maps ϕ_{k,r} of the unit disk D ⊂ C to V_{r} with the following properties: Let G_{r,k} be the pullback of g_{r} to D × C via ϕ_{k,r} × id. Let E be the vector field on D × C defined by

(EG)(t, z) = \frac{∂G(t, z)}{∂t}.

We suppose that

(EG_{r,k})(0, z) = a_{k,r} + b_{k,r} exp(2πikz), (19)

where a_{k,r}, b_{k,r} ∈ C and b_{k,r} ≠ 0 and a_{r,0} = 0. Further, we suppose that ϕ_{k,r}(0) = Q_{r}.

ϕ_{k,r} induces a map ψ_{k,r} from the tangent space T_{0} of the disk D at zero to the tangent space T_{Q_{r}} of V_{r}. Note that the vectors

W_{k,r} = ψ_{k,r} (\frac{∂}{∂z} ) ,

are all independent, since otherwise we would get a linear dependence relation between the functions a_{k,r} + b_{k,r} exp(2πikz). We can introduce a coordinate system t_{1}, t_{2} … t_{p} on V_{r} so that Q_{r} is the origin of the coordinate system and the span of the functions

\frac{∂g_{r}}{∂t_{k}} (0, 0, 0, …, 0; x)

contains the span of the functions exp(2πikz) for k from −m to m. This is because the linear span of the functions exp(2πikz) is the same as the linear span of the functions a_{k,r} + b_{k,r} exp(2πikz).

Lemma 2.8.2. Suppose that for each r, we have that there is a periodic meromorphic function w_{r} on V_{r} × C so that D_{r}(w_{r}) = D(g_{r}). Then under the assumption of (19), there is a E ∈ R_{0} so that D = ∂E.

Proof. If δ(D) = 0, then we can find E so that ∂E = D + C, where C is constant. But our assumption implies that

∫_{z}^{z+1} D(g_{r}) = 0,

for generic z, where we integrate along a straight line for z to z + 1 in C. Since E(g_{r}) is periodic, we have

∫_{z}^{z+1} ∂E(g_{r}) = 0.

So we would have C = 0. So we may assume that δ(D) ≠ 0. Note that δ(D) ∉ C. Consequently, using Lemma 2.7.2 we can find a map γ : D′ → V_{m} so that

(δ(D)(g_{r}))(γ(t), x) = t^{n}G(t, x),

where G(0, x) is not identically zero as a function of x and D′ is the disk. Now G(v, x + 1) = G(v, x). Thus for generic x there is an integer k so that

∫_{x}^{x+1} G(0, z) exp(2πikz)dz ≠ 0.

Choose a p so that exp(2πikz) is in the span of the functions

\frac{∂g_{p}}{∂t_{l}} (0, 0, 0, …, 0; x).

On the other hand, we have that

0 = ∫_{z}^{z+1} δ(D)(g_{p})\frac{∂g_{p}}{∂t_{l}}

from Lemma 2.7.1 Hence we can find a function R so that

0 = ∫_{z}^{z+1} δ(D)(g_{p})(γ(t), x)R(t, x) dx

and

R(Q_{p}, x) = exp(2πikz)

by taking R to be a linear combination of the functions \frac{∂g_{p}}{∂t_{l}} . In particular, we obtain

(20)  0 = ∫_{z}^{z+1} δ(D)(g_{p})(γ(t), x))R(t, x)
= t^{n} ∫_{z}^{z+1} G(t, x)R(t, x).

So

0 = ∫_{z}^{z+1} G(t, x)R(t, x).

Evaluating at t = 0 yields a contradiction. q.e.d.

Lemma 2.8.3. Suppose that D_{1} … D_{n} ∈ R_{0}. and that the family {g_{r}} is generic. Then we can find an r so that if D is a nontrivial linear combination of the D_{k} and D(g_{r}) = 0, then D = 0.

Proof. We can assume that the D_{i} are linearly independent. Let W_{r} ⊂ C^{n} be the set of (a_{1}, …, a_{n}) so that

∑_{k} a_{k}D_{k}(g_{r}) = 0.

W_{r} is a sequence of linear subspaces which decrease with r. Further, by the first property of generic {g_{r}}, any element of C^{n} is eventually not in some W_{r}. q.e.d.

2.9

We will next study the following situation: Let N_{r} = V_{r} × C^{r}, where C^{r} has a basis δ_{1}, …, δ_{r} and let π_{r} be the projection from N_{r} to V_{r}. We suppose there are inclusions k_{r} : V_{r} → V_{r+1}. For simplicity, we will identify the image of V_{r} under k_{r} with V_{r}. Suppose there are functions h_{r} on V_{r} so that the restriction of h_{r+1} is h_{r}. Let Q_{r} be a series of points in V_{r} with Q_{r} = Q_{r+1} under the identification and assume h_{r}(Q_{r}) = 0. Then we can send ψ_{r} : π_{r+1}^{−1}(V_{r}) → N_{r} by

ψ_{r}(v_{r}, c_{1}, …, c_{r+1}) = (v_{r}, c_{1}, …, c_{r}).

Suppose that f_{r} and g_{r} are functions on N_{r} and that ψ_{r}^{∗}(f_{r}) is the restriction of f_{r+1} and ψ_{r}^{∗}(g_{r}) is the restriction of g_{r+1}. We will assume that f_{r} and g_{r} are generically defined on the zero section on N_{r} → V_{r}.

Suppose that D_{1}, …, D_{n} are tame derivations of R[[ϵ]] and that χ_{0,r}, χ_{1,r}, …, χ_{n,r} are vertical vector fields on N_{r} which form representations of D_{1}, …, D_{n} and so that χ_{0,r} represents ∂. We will also assume that χ_{0,r} is differentiation in the direction δ_{1} +2δ_{r} +⋯+rδ_{r}. Note that f_{r+1} and g_{r+1} are constant when restricted to the fibers of ψ_{r} and further we assume that if F is a function on N_{r},

χ_{k,r+1}(ψ_{r}^{∗}(F)) = ψ_{r}^{∗}(χ_{k,r}(F))

for k ≤ r.

Definition 2.9.1. We say that a sequence of representations as above are a compatible sequence.

Let W_{r} ⊂ N_{r} be defined by h_{r} = 0. We can generate a certain class of functions on W_{r} in the following way: Take any P ∈ R[ϵ]. Then P (f, g) is a meromorphic function on N_{r}. Then we find the minimal power p of h_{r} so that P (f, g)h_{r}^{p} is generically defined on W_{r}. We can then restrict P (f, g)h_{r}^{p} to W_{r} to obtain a meromorphic function on W_{r}. Let C be the set of all functions on W_{r} arising from such P.

Definition 2.9.2. The sequence of compatible representations is general if the following holds: Suppose f is in the class C. Suppose is that

f_{1}(v_{r}, z) = f(v_{r}, z, 2z, …, rz)

is constant as a function of z and v_{r} ∈ V_{r} is generic in V_{r}. Then f is constant on the fibers of the projection W_{r} → V_{r}.

Now define functions f′_{r} and g′_{r} on V_{r} × C by the formulas:

f′_{r}(v_{r}, z) = f_{r}(v_{r}, z, 2z, …, rz).
g′_{r}(v_{r}, z) = g_{r}(v_{r}, z, 2z, …, rz).

Notice that the restriction of g′_{r+1} to V_{r} ×C is g′_{r}. Let G_{r} be the function g′_{r} restricted to h_{r} = 0. We will assume that the (G_{r}, Q_{r}) form a generic sequence. We also assume that all functions in the class C are periodic when translated by δ_{1} + 2δ_{2} + rδ_{r} and that the compatible sequence of representations is general.

Lemma 2.9.3. Given an integer n, suppose there is a P ∈ R_{0}[ϵ] so that:

1. f_{r} ≡ P (g_{r}) mod h_{r}^{n}.

2. D_{1}(v^{(0)}) ≡ v^{(1)} − w^{(1)} mod ϵ.

3. D_{1}(w^{(0)}) ≡ w^{(1)} − v^{(1)} mod ϵ.

4. χ_{1,r} vanishes on {h_{r} = 0}.

5. χ_{0,r}, … form a general representation of D_{1}, ….

6. This representation is slow for D_{1}.

Then we can find a Q ∈ R_{0}[ϵ] congruent to P mod ϵ^{n} so that

\frac{f_{r} − Q(g_{r})}{h_{r}^{n}}

is constant on the fibers of the projection W_{r} → V_{r}.

Proof. Using Lemma 2.6.1, we can find a Q_{1} ∈ R_{0}[ϵ] so that

χ(f_{r}) ≡ Q_{1}(g_{r}) mod h_{r}^{n+1}.

In particular, we can restrict and get

D_{r}(f′_{r}) = Q_{1}(g′_{r}).

Notice that

(∂P − Q_{1})(g′_{r}) ≡ 0 mod h_{r}^{n}.

Since the g′_{r} are generic, we must have

∂P ≡ Q_{1} mod ϵ^{n}.

If not, we can suppose that

∂P ≡ Q_{1} mod ϵ^{k}

for some maximal k < n. Then let

E = \frac{∂P − Q}{ϵ^{k}}

and let F ∈ R_{0} be the constant term of E as a power series in ϵ. Then E(g′_{r}) vanishes on the set h_{r} = 0. Thus F = 0, contradicting the maximality of k.

So we can write

Q_{1} ≡ ∂P + ϵ^{n}Q_{2} mod ϵ^{n+1}

for some Q_{2} ∈ R_{0}. In particular,

h_{r}^{n}Q_{2}(g′_{r}) ≡ \frac{d}{dz} (f′_{r}(v_{r}, z) − P (g_{r})(v_{r}, z)) mod h_{r}^{n+1}.

Thus Q_{2}(G_{r}) is the derivative of a periodic function. Since the family G_{r} is general, we must have Q_{2} = ∂Q_{3} and hence we may set Q = P +ϵ^{n}Q_{3}. Then (f′_{r} − Q(g′_{r}))(v_{r}, z, 2z, …, rz) is constant as a function of z. So

\frac{f_{r} − Q(g_{r})}{h^{n}}

is constant on the fibers of the projection W_{r} → V_{r}. q.e.d.

Proposition 2.9.4. Under the hypotheses of Lemma 2.9.3, we can find functions F_{r} and vector fields χ′_{1}, …, χ′_{n} so that F_{r}, g_{r}, h_{r}, χ, χ′_{1}, …, χ′_{n} form a representation of D_{1}, …, D_{n} and a Q ∈ R_{0}[[ϵ]] so that

F_{r} ≡ Q(G_{r}) mod h_{r}^{n+1}

and

Q ≡ P mod ϵ^{n}.

Further, χ′_{1,r} vanishes on {h_{r} = 0}.

Proof. Let s_{n,r} be the restriction of f_{r} − Q(g_{r}) to the zero section of N_{r} over V_{r}. We can think of s_{n,r} as a function on V_{r} and hence we can think of s_{n,r} as a function on N_{r}. We define F_{r} = f_{r} − s_{n,r}. We can modify the χ_{k,r} to χ′_{k,r} by translation by −s_{n,r} This new family of representations remains compatible. χ′_{1,r} still vanishes on {h_{r} = 0}. For

χ_{1} ≡ χ′_{1} mod h_{r},

since the difference χ_{1} − χ′_{1} is divisible by s_{n,r} Note that we start out with f_{r} ≡ g_{r} mod h_{r}. q.e.d.

Theorem 2.9.5. Suppose there is a generic compatible family of representations with:

1. G_{r}, Q_{r} general.

2. χ_{1,r} vanishes on {h_{r} = 0}.

3. D_{1}(v^{(0)}) ≡ v^{(1)} − w^{(1)} mod ϵ.

4. D_{1}(w^{(0)}) ≡ w^{(1)} − v^{(1)} mod ϵ.

Then there is a Q ∈ R_{0}[[ϵ]] so that

D_{i}(v^{(0)} − Q) ∈ I_{Q}

for all i.

Proof. Suppose we have constructed a Q_{n} so that

f_{r} ≡ Q_{n}(g_{r}) mod h^{n}

for a given n and all r. By the Proposition 2.9.4, we can find

Q_{n+1} ≡ Q_{n} mod ϵ^{n}

200 d. gieseker

and modify f_{r} so that

f_{r} ≡ Q_{n+1}(g_{r}) mod h_{r}^{n+1}.

Notice that for any P ∈ R[[ϵ]], we have that

P (f_{r}, g_{r}) ≡ σ_{Q_{n+1}}(P)(g_{r}) mod h_{r}^{n+1}.

Since σ_{Q_{n+1}} is a ring homomorphism commuting with ∂, it suffices to check the equation for P = v^{(0)} and P = w^{(0)}. Now

(21)  0 ≡ χ_{i,r}(f_{r} − Q_{n+1}(g_{r}))
≡ D_{i}(v^{(0)} − Q_{n+1})(f_{r}, g_{r})
≡ σ_{Q_{n+1}}(D_{i}(v^{(0)} − Q_{n+1}))(g_{r}) mod h_{r}^{n+1}.

We let Q be the limit of the Q_{n}. Then

σ_{Q}(D_{i}(v^{(0)} − Q))(g_{r}) ≡ 0 mod h_{r}^{n}

for all n. Since the g_{r} are generic, we have

σ_{Q}(D_{i}(v^{(0)} − Q)) = 0,

i.e., D_{i}(v^{(0)} − Q) ∈ I_{Q}. q.e.d.

Definition 2.9.6. Suppose that

D_{i}(v^{(0)} − Q) ∈ I_{Q}.

Then we define derivations

D_{i} : R_{0}[[ϵ]] → R_{0}[[ϵ]]

by

D_{i}(P) = σ(\overline{D_{i}(P)})

where σ is the inverse of the natural map

σ_{1} : R_{0}[[ϵ]] → R[[ϵ]]/I_{Q}

and T is the image of T ∈ R[[ϵ]] in R[[ϵ]]/I_{Q}.

Theorem 2.9.7. Suppose that ρ = (f, g, h; χ_{1}, …, χ_{n}) is a representation of D_{1}, …, D_{n} and that

f ≡ Q(g) mod h^{n}

for all n. Then using the notation of Definition 2.9.6, (g, h; χ_{1}, …, χ_{n}) is a representation of D_{1}, …, D_{n}.

Proof.

(22)  χ_{i}(g) ≡ D_{i}(w^{(0)})(f, g)) mod h^{n}
≡ D_{i}(w^{(0)})(Q(g), g) mod h^{n}
≡ σ(D_{i}(w^{(0)})(g) mod h^{n}
≡ D_{i}(w^{(0)})(g) mod h^{n}.

q.e.d.

Definition 2.9.8. If D ∈ R_{0}[[ϵ]] so that

D = D_{0} + D_{1}ϵ + D_{2}ϵ^{2} + ⋯,

we define

D^{[0]} = D_{0}.

Similarly, we let χ_{i}^{[0]} be the restriction of χ_{i} to the set {h = 0}.

Lemma 2.9.9. Suppose that D_{1}, …, D_{n} ∈R_{0}. Suppose that E_{1}, …, E_{n} ∈ R_{0}. Suppose that (g, 0; χ_{1}, …, χ_{n}) is a representation of D_{1}, …, D_{n} and (g, 0; χ′_{1}, …, χ′_{n}) is a representation of E_{1}, …, E_{n}. Assume further that χ_{1}, …, χ_{n} and χ′_{1}, …, χ′_{n} span the same n dimensional vector space. Further, assume that if g satisfies any nontrivial linear combination D of the D_{i} and E_{i}, then D = 0. Then linear span of the D_{i} is the same as the linear span of the E_{i}.

Proof. We can assume that the χ′_{i} = χ_{i}. Then (D_{i} − E_{i})(g) = 0, so D_{i} = E_{i}. q.e.d.

Corollary 2.9.10. Suppose that D_{1} ∈ R_{0}. Suppose that (g, 0; χ_{1}) is a representation of D_{1}. Further, assume that D_{1}(g) = 0 implies that D_{1} = 0. Then if χ_{1}(g) = 0, then D_{1} = 0.

3. Constructing deformations

3.1

Let T be a smooth analytic manifold and let π : X → T be a proper flat family of curves of arithmetic genus n. Let t_{0} ∈ T be a fixed point. We will usually be interested in the behavior of the family and related objects near t_{0}. We will suppose there are sections P : T → X, Q : T → X, R : T → X so that for each t ∈ T, we have P (t) + Q(t) is a divisor on X_{t} = π^{−1}(t) is linearly equivalent to the divisor 2R(t). We will also assume that P (t), Q(t) and R(t) are all smooth points of X_{t}.

Let δ : T × S^{1} → X be a map defined over T so that π is smooth at every point of the image of δ and let ω be a section of π_{∗}(ω_{X /T}). (ω_{X /T} is the sheaf of relative dualizing differentials and S^{1} is the unit circle.) Then δ(t) is a cycle on X_{t} and we can form

I_{δ}(t)(ω_{t}) = ∫_{δ(t)} ω_{t}

where I_{δ} is a section of the dual of π_{∗}(ω_{X /T}). So we can consider I_{δ} as a section of R^{1}π_{∗}(O_{X}). We will suppose that we can find δ_{1}, …, δ_{n} so that I_{δ_{1}}, …, I_{δ_{n}} are a basis of R^{1}π_{∗}(O_{X}). Let v_{k} be sections of π_{∗}(ω_{X /T}) which are dual to the I_{δ_{k}}.

We will assume that there is an open set U ⊂ X so that z : U → D × T is an isomorphism of D × T as T spaces, where D is the unit disk. We assume that the images of the sections P, Q and R are all contained in U. For simplicity, we will assume that the image of P is the zero section and let

h = z(R).

We will assume that

dz = v_{1}.

Further, we will assume that we can find anti-derivatives h_{k} of the v_{k} defined on U and that the functions

\frac{∫_{Q}^{P} v_{k}}{∫_{Q}^{P} v_{1}} = k.

This condition insures that the secant line between Q and P on the local curve

t → (h_{1}(t), h_{2}(t), …, h_{n}(t))

passes through (1, 2, …, n) when h_{1}(t) ≠ 0 and insures that the tangent line to P passes through (1, 2, …, n) when h_{1}(t) = 0. The existence of such families is not obvious at this point, but we will exhibit such families in §4.

3.2

Let

T_{1} = T × C^{n}

and let

X_{1} = X × C^{n} = X ×_{T} T_{1}.

We will define a relative line bundle L on X_{1} over T_{1}. Let π_{1} : X_{1} → T_{1} be the projection. We have the coordinate functions z_{1}, …, z_{n} on C^{n}. We will again denote the pullback of z_{i} to T_{1} by z_{i}. On the other hand, we will denote the pullback of I_{δ_{i}} to a section of R^{1}π_{1,∗}(O_{X_{1}}) by I_{δ_{i}}. Thus we can consider I = ∑ z_{k}I_{δ_{k}} as a section of R^{1}π_{1,∗}(O_{X_{1}}). Then there is a relative line bundle L on X_{1} so that L corresponds to the same section of R^{1}π_{1,∗}(O_{X_{1}}^{∗}) as exp(2πiI). We also assume that we have a line bundle M of relative degree n on X_{1} which is the pullback of a line bundle on X → T. Let N = M ⊗ L.

We will assume that P (T) and Q(T), which are divisors on X , meet transversally. Consider Z = π(P (T) ∩ Q(T)), where π is the projection of X → T. We will assume Z is a divisor in T and that the map from P (T) ∩ Q(T) → Z is an isomorphism. Let Z_{1} be the inverse image of Z in T_{1}. We assume that R(T) ∩ P (T) and R(T) ∩ Q(T) are both equal to P (T) ∩ Q(T). We will assume that Z_{1} is defined by an equation h = 0, where h is the pullback of a function on T.

We will also assume that we have chosen a nonconstant section λ of O_{X} (P (T) + Q(T) − 2R(T)). Let’s fix a point t ∈ T_{1}. Then on X_{1,t}, we have the line bundles N_{k,t} = N_{t}(k(P (t) − Q(t))).

Definition 3.2.1. Suppose P (t) ≠ Q(t). Then t is N-good if each of the line bundles N_{k,t}(−P (t)), is non-special for |k| < N. We then define sections s_{k} ∈ H^{0}(N_{k,t}) for |k| < N so that

\frac{s_{k+1}}{λs_{k}} (P (t)) = 1.

Remark 3.2.2. Given our choice of λ, these s_{k} are determined up to an nonzero multiplicative constant independent of k.

We will assume we have made a special choice to λ to be denoted λ_{0} so that zλ_{0}(P) = 1. Let’s fix N and let T_{2} ⊂ T_{1} be the set of N-good points.

Theorem 3.2.3. There are functions A_{k,t,λ} and B_{k,t,λ} defined on T_{2} for |k| < N − 1 so that

λs_{k} = s_{k+1} + A_{k,t,λ}s_{k} + B_{k,t,λ}s_{k−1}.

The function B is never zero on T.

Proof. The dimension of H^{0}(N_{p}(P (p) + Q(p))) = 3. So there must be a linear dependence relations

Cλs_{k} + Ds_{k+1} + As_{k} + Bs_{k−1} = 0,

where A, B, C and D are all in C. Now we have chosen our normalizations of the s_{m} and of λ so that

\frac{s_{k+1}}{λs_{k}} = 1

at P (p). We must then have C = −D. Further, we cannot have C = D = 0, since this would lead to a dependence relation between s_{k} and s_{k−1}, which have different order poles at P (p). So we can completely normalize by taking C = −1 and D = 1 and finally writing

λs_{k} = s_{k+1} + As_{k} + Bs_{k−1}.

q.e.d.

If u is a nowhere zero function on T, then

A_{k,t,uλ} = uA_{k,t,λ}

and

B_{k,t,uλ} = u^{2}A_{k,t,λ}.

Further, we have

A_{k,t,C+λ} = A_{k,t,λ} + C,

while B is unchanged by adding C to λ for any function C on T. We can now normalize the B_{k} in the following way: Let

B_{k,t} = \frac{B_{k,t,λ}}{B_{k,0,λ}} .

3.3

We next discuss theta functions following [8]. Let C = X_{t} and denote P_{t} by P, etc. We assume that C is nonsingular. We regard H^{1}(C, Z) as a subgroup of H^{0}(C, Ω)^{∗}. Let C_{1}^{∗} be the set of all complex numbers of absolute value one. Choose a map

α : H^{1}(C, Z) → C_{1}^{∗}

so that

\frac{α(u_{1} + u_{2})}{α(u_{1})α(u_{2})} = e^{iπ⟨u_{1},u_{2}⟩}.

There is a unique hermitian form H on H^{0}(C, Ω)^{∗} so that

ℑ H(x, y) = ⟨x, y⟩.

Let ϑ defined on H^{0}(C, Ω)^{∗} be the function satisfying the functional equation

ϑ(z + u) = α(u)e^{πH(z,u)+πH(u,u)/2}ϑ(z)

for z ∈ H^{0}(C, Ω)^{∗} and u ∈ H^{1}(C, Z). There is a map γ : U ∩ C → H^{0}(C, Ω)^{∗} defined by

γ(q) = ∫_{P}^{q} ω,

where the path from P to q is chosen to lie in U ∩ C. Thus we can regard γ(q) ∈ H^{1}(C, O). Given any non-special line bundle L = O_{C}(D) of degree n on C with D effective, here is a constant K_{L} ∈ H^{1}(O_{C}) so that the zeros of the function p → ϑ(γ(p) + K_{L}) are just D counting multiplicity. Further,

exp(2πiK_{L}) = L.

Also

K_{L(q−P)} = K_{L} − γ(q)

modulo periods. Let K_{0} be a constant corresponding to the line bundle M and choose a line bundle M_{1} = O(D_{1}) so that P is a point of multiplicity one of D_{1} and all the other points of D_{1} are outside U. Select K_{1} corresponding to M_{1}. So ϑ(γ(z) + K_{1}) vanishes exactly once at P and at no other point of U.

Using the theta function, we can write down an expression for a function λ_{0} initially valid in U.

λ_{0}(z) = α\frac{ϑ(γ(z) − γ(R) + K_{1})^{2}}{ϑ(γ(z) + K_{1})ϑ(γ(z) + K_{1} − γ(Q))},

where

α = \frac{∇(ϑ)(K_{1}) · γ′(P)ϑ(K_{1} − γ(Q))}{ϑ(K_{1} − γ(R))^{2}}.

This is a well-defined meromorphic function on C, since P +Q is linearly equivalent to 2R so by Abel’s theorem, γ(P)+γ(Q) = 2γ(R). Using the periodicity properties of ϑ,

\frac{ϑ(Z − γ(R) + K_{1})^{2}}{ϑ(Z + K_{1})ϑ(Z + K_{1} − γ(Q))}

is periodic in Z ∈ H^{1}(O).

Next, we develop a formula for

u = \frac{λ_{0}s_{0}}{s_{1}} .

Attached to the point t, there is a line bundle N_{0,t} and let L be the projection on t to C^{n}. The zeros of ϑ(γ(z) + L + γ(Q)) match the zeros of s_{1} and the zeros of ϑ(γ(z) + L) match the zeros of s_{0}. On the other hand,

\frac{ϑ(γ(z) + K_{1} − γ(Q))}{ϑ(γ(z) + K_{1})}

has a simple pole at P and a simple zero at Q. So s_{1}/s_{0} is a multiple of the rational function

\frac{ϑ(γ(z) + K_{1} − γ(Q))ϑ(γ(z) + L + γ(Q))}{ϑ(γ(z) + K_{1})ϑ(γ(z) + L)}.

Consequently, u is a multiple of

\frac{ϑ(γ(z) + K_{1} − γ(R))^{2}ϑ(γ(z) + L)}{ϑ(γ(z) + K_{1} − γ(Q))^{2}ϑ(γ(z) + L + γ(Q))}.

Using that u(P) = 1 and γ(P) = 0 we obtain that

u = \frac{ϑ(γ(z) + K_{1} + γ(R))^{2}ϑ(γ(z) + L)ϑ(K_{1} − γ(Q))^{2}ϑ(L + γ(Q))}{ϑ(γ(z) + K_{1} − γ(Q))^{2}ϑ(γ(z) + L + γ(Q))ϑ(K_{1} − γ(R))^{2}ϑ(L)}.

Let a denote the constant term of the Laurent series for λ − s_{1}/s_{0} developed around z = 0. We get a = \frac{d}{dz} (u) evaluated at z = P. Since u = 1 at z = 0, we obtain

(22) a = −2\frac{∇(ϑ)(K_{1} − γ(Q)) · γ′(P)}{ϑ(K_{1} − γ(Q))} − \frac{∇(ϑ)(L + γ(Q)) · γ′(P)}{ϑ(L + γ(Q))}

+ 2\frac{∇(ϑ)(K_{1} + γ(R)) · γ′(P)}{ϑ(K_{1} + γ(R))} + \frac{∇(ϑ)(L) · γ′(P)}{ϑ(L)}.

For future reference, we give a formula for the constant term C_{0} of the Laurent expansion of s_{1}/s_{0}, namely

(23)  C_{0} = \frac{∇(ϑ)(K_{1} − γ(Q)) · γ′(P)}{ϑ(K_{1} − γ(Q))} + \frac{∇(ϑ)(L + γ(Q)) · γ′(P)}{ϑ(L + γ(Q))}

− \frac{∇(ϑ)(L) · γ′(P)}{ϑ(L)} − \frac{W}{∇(ϑ)(L) · γ′(P)}

where

W = \frac{1}{2} \frac{d^{2}}{dz^{2}} (ϑ(γ(z)))

evaluated at z = P.

We will also be interested in evaluating

v = \frac{λ_{0}s_{0}}{s_{−1}}

at Q. We obtain that

\frac{s_{−1}}{s_{0}} = F \frac{ϑ(γ(z) + K_{1})ϑ(γ(z) + L − γ(Q))}{ϑ(γ(z) + K_{1} − γ(Q))ϑ(γ(z) + L)}

with

F = \frac{ϑ(K_{1} − γ(Q))ϑ(L)}{∇(ϑ)(K_{1}) · γ′(P)ϑ(L − γ(Q))}.

So

\frac{λ_{0}s_{0}}{s_{−1}} = F_{1} \frac{ϑ(γ(z) + L)ϑ(γ(z) − γ(R) + K_{1})^{2}}{ϑ(γ(z) + K_{1})^{2}ϑ(γ(z) + L − γ(Q))},

where

F_{1} = \frac{(∇(ϑ)(K_{1}) · γ′(P))^{2}ϑ(L − γ(Q))}{ϑ(−γ(R) + K_{1})^{2}ϑ(L)}.

So evaluating at Q, we get

(24)  b = \frac{λ_{0}s_{0}}{s_{−1}} (Q)

= F_{1} \frac{ϑ(L + γ(Q))ϑ(γ(Q) − γ(R) + K_{1})^{2}}{ϑ(γ(Q) + K_{1})^{2}ϑ(L)}

= (\frac{ϑ(γ(Q) − γ(R) + K_{1})∇(ϑ)(K_{1}) · γ′(P)}{ϑ(−γ(R) + K_{1})ϑ(γ(Q) + K_{1})})^{2}

· \frac{ϑ(γ(Q) + L)ϑ(−γ(Q) + L)}{ϑ(L)^{2}}

= (\frac{ϑ(γ(R) + K_{1})∇(ϑ)(K_{1}) · γ′(P)}{ϑ(−γ(R) + K_{1})ϑ(2γ(R) + K_{1})})^{2}

· \frac{ϑ(2γ(R) + L)ϑ(−2γ(R) + L)}{ϑ(L)^{2}}

bearing in mind that 2γ(R) = γ(Q).

We have defined a and b in Equations (22) and (24). a and b depend on t.

Proposition 3.3.1 For our choice of λ_{0}, we have

a = A_{0,t,λ_{0}}

and

b = B_{0,t,λ_{0}}

near t_{0}.

3.4

We will now make a canonical choice of λ near the set Z. Specifically, let

U = \sqrt{B_{0,(x,0),λ_{0}}}.

The expression for b makes it clear that such a meromorphic function exists, since

lim_{γ(R)→0} \frac{ϑ(L)^{2}}{ϑ(2γ(R) + L)ϑ(−2γ(R) + L)} = 1

for any L, in particular for L = 0. In particular, we set

U = (\frac{ϑ(γ(R) + K_{1})∇(ϑ)(K_{1}) · γ′(P)}{ϑ(−γ(R) + K_{1})ϑ(2γ(R) + K_{1})}) \sqrt{\frac{ϑ(2γ(R))ϑ(−2γ(R))}{ϑ(0)^{2}}}

where we choose the branch of the square root which is near 1. When R is near P, then

ϑ(γ(R) + K_{1}) ≈ γ(R) · ∇(ϑ)(K_{1})

and

ϑ(−γ(R) + K_{1}) ≈ −γ(R) · ∇(ϑ)(K_{1})

and

ϑ(2γ(R) + K_{1}) ≈ 2γ(R) · ∇(ϑ)(K_{1})

Toda and KdV 209

so

U ≈ −\frac{∇(ϑ(K_{1}) · γ′(P)}{∇(ϑ)(K_{1}) · γ(Q)}.

On the other hand, we have that

ϑ(K_{1} − γ(Q)) ≈ −∇(ϑ(K_{1})) · γ(Q),

so the constant term of s_{1}/s_{0} is approximately

−\frac{∇(ϑ)(K_{1}) · γ′(P)}{∇(ϑ)(K_{1}) · γ(Q)}

from the expression for C_{0}. So the constant term of U^{−1}s_{1}/s_{0} is just 1. Now choose C, a function on T, so that

U^{−1}A_{0,(x,0),λ_{0}} + C = −2.

Proposition 3.4.1. We can canonically choose

λ_{can} = U^{−1}λ_{0} + C

so that:

1. 2 + A_{k,t,λ_{can}} is divisible by h^{2}.

2. −1 + B_{k,t,λ_{can}} is divisible by h^{2}.

3. 2 + A_{k,0,λ_{can}} = 0.

4. −1 + B_{k,0,λ_{can}} = 0.

5. We have

λ = α_{−1} \frac{h}{z} + α_{0} + α_{1} \frac{z}{h} + ⋯,

where α_{−1}(t) vanishes on Z and α_{0} becomes −1 on Z. Of course, the canonical choice depends on the line bundle M, but is otherwise completely determined near Z by properties three, four and five.

Proof. Note that

V (z, h) = λ_{0} \frac{z(z − Q)}{(z − R)^{2}}

does not have zeros or poles when h ≠ 0 and h and z are near zero. Now near h = z = 0, V (z, h) is a power of h times a unit. We have

λ_{can} = \frac{U^{−1}s_{n+1}}{s_{n}} + A_{can} + B_{can} \frac{Us_{n−1}}{s_{n}}

Hence the constant term of λ_{can} as a Laurent series in z is the constant term of

\frac{U^{−1}s_{n+1}}{s_{n}}

plus the constant term of A_{can}. So modulo h, the constant term of λ_{can} is −1. The Laurent series for λ_{0} is

\frac{1}{z} + \frac{J}{h} + · · · .

So hV (c, h) is a unit. But the Laurent series of hV (z, h) is a power series in z/h. So λ_{can} is a power series in \frac{z}{h}. q.e.d.

Definition 3.4.2.

A_{k,t} = A_{k,t,λ_{can}}
f = A_{0,t}
g = B_{0,t}.

f and g are functions on T × C^{n}.

Lemma 3.4.3. f and g are defined near any 2-good point.

Let χ be differentiation in the direction (1, 2, …, n).

Proposition 3.4.4.

g(t, L) = 1 + 4h(t)^{2}(χ^{2}(log(ϑ)(L) − χ^{2}(log(ϑ)(0)))) + higher order terms in h.

Proof. We define

g_{1}(L) = \frac{ϑ(L)^{2}}{ϑ(L + γ(Q))(ϑ(L − γ(Q))}

and then

g = \frac{g_{1}(L)}{g_{1}(0)}.

On the other hand,

γ(Q) = 2h(1, 2, …, n)

and we know that γ′′(Q/2) = 0. q.e.d.

Let’s examine the function G(t, L) which is the analytic continuation of

\frac{g(t, L) − 1}{h(t)^{2}}.

We can introduce a series of vectors v_{k} ∈ C^{n} by using the identification of C^{n} with

H^{0}(ω_{X_{1,t}})^{∗}

via integration over the δ_{i} by

v_{k}(ω) = Res_{P (t)} \frac{ω}{z^{k}} .

Let K_{1}, K_{2}, … be the KdV differential operators. Let χ_{k} be directional derivative in the directions v_{k}.

Proposition 3.4.5. We have

λ = α_{−1} \frac{h}{z} + α_{0} + α_{1} \frac{z}{h} + · · · .

We can find β_{k,l} which are universal polynomials in the α_{k} so that

∑_{l≤k} β_{k,l}Res_{P (t)}λ^{l}ω ≡ h^{2k−1}v_{2k+1}(ω) mod h^{2k}.

Further,

β_{k,k} = n_{k}α_{−1}^{n_{k}},

where n_{k} ∈ Q and n_{l} ≠ 0.

Proof. First note that we can find γ_{k,l}, universal polynomials in the α_{k}, so that

∑_{l≤k} γ_{k,l}Res_{P (t)}λ^{l}ω ≡ α_{−1}^{k} \frac{h^{k}}{z^{k}} .

Indeed,

λ = α_{−1} \frac{h}{z} + ⋯,

λ^{2} = α_{−1}^{2} \frac{h^{2}}{z^{2}} + 2α_{−1}α_{0} \frac{h}{z} + ⋯,

λ^{3} = α_{−1}^{3} \frac{h^{3}}{z^{3}} + 3α_{0}α_{−1}^{2} \frac{h^{2}}{z^{2}} + (3α_{−1}α_{0}^{2} + 3α_{−1}^{2}α_{1})\frac{h}{z} + · · · .

So we can express

μ_{k} = α_{−1}^{k} \frac{h^{k}}{z^{k}}

in terms of λ, …, λ^{k} with the coefficients universal polynomials in the α_{k}’s.

Note that z −h vanishes on R(t) and that z −h is the anti-derivative of a holomorphic differential. Since the curve is hyperelliptic and R(t) is a Weirstrass point, z − h is an odd function under the involution of the hyperelliptic curve. However, any differential is even under the involution. Thus any differential can be expanded around P (t) as a power series in the even powers of z −h times dz. So if ω is a differential, we can write

(25)  ω = (a_{0} + a_{2}(z − h)^{2} + a_{4}(z − h)^{4} + ⋯)dz
= f((z − h)^{2})dz.

So

Res_{P (t)}μ_{1}ω = α_{−1}hf(h^{2}).
Res_{P (t)}μ_{2}ω = 2h^{3}α_{−1}^{2}f′(h^{2}).
Res_{P (t)}μ_{3}ω = h^{3}α_{−1}^{3}(2f′(h^{2}) + 4h^{2}f′′(h^{2})).
Res_{P (t)}μ_{4}ω = h^{4}α_{−1}^{4}(2hf′′(h^{2}) + 8hf′′(h^{2}) + 8h^{3}f′′′(h^{2})).

Continuing in this way, we see that we can express

h^{2k−1}f^{(k)}(h^{2})

as a linear combination of

Res_{P (t)}μ_{l}ω

for l ≤ k. On the other hand,

f^{(k)}(h^{2}) ≡ v_{2k+1}(ω) mod h.

q.e.d.

Definition 3.4.6. Suppose D_{1}, …, D_{n} ∈R_{0}[[ϵ]]. We assume D_{1}, …, D_{n} are linearly independent over C[[ϵ]]. Let M be the C[[ϵ]] module generated by the D_{k}. Then we can find an increasing sequence of integers m_{1}, …, m_{n} and E_{1}, …, E_{n} ∈ R_{0}[[ϵ]] so that the E_{k}^{[0]} are all linearly independent over C and so that ϵ^{m_{k}}E_{k} are a basis of M. m_{1}, …, m_{n} are uniquely determined and called the characteristic numbers of D_{1}, …, D_{n}. The ϵ^{m_{1}}E_{1}, …, ϵ^{m_{n}}E_{n} are called a normalized basis for M. Note that E_{1}^{[0]}, …, E_{n}^{[0]} are linearly independent over C.

Corollary 3.4.7. Suppose that T is one dimensional and {h = 0} is just a point x and that h is a parameter near x. Let g be the function on C^{n} defined by

g(L) = G(x, L).

In the context of the Definition 3.4.6 suppose that (G, h; χ, χ_{1}, …, χ_{n}) is a representation of D_{1}, …, D_{n} and g does not satisfy any nonzero linear combination of K_{1}, …, K_{n}, E_{1}^{[0]}, …, E_{n}^{[0]}. Further assume that

v_{2j+1}(g) ≡ ∑_{i≤j} β_{i,j}K_{i}(g) mod h

with β_{j,j} ≠ 0. Then the characteristic numbers of D_{1}, …, D_{n} are 1, 3, …, 2n − 1. Further, the span of E_{1}^{[0]}, …, E_{n}^{[0]} is the same of the span of K_{1}, …, K_{n}.

Proof. There are polynomials P_{i,j} ∈ C[ϵ] so that

∑_{i=1}^{j} P_{i,j}(h)χ_{i} ≡ h^{2j−1}v_{2j+1} mod h^{2j}

with P_{j,j}(0) ≠ 0. Then let

E′_{j} = ∑_{i=1}^{j} P_{i,j}(ϵ)D_{i}.

Let ϵ^{r_{j}}F_{j} be the leading term of E′_{j} as a power series in ϵ. Then F_{j} is a linear combination of E_{1}^{[0]}, …, E_{n}^{[0]}. If r_{j} < 2j − 1, then

F_{j}(g) = 0.

So we would have to have F_{j} = 0. We conclude that r_{j} ≥ 2j − 1. On the other hand,

v_{2j+1}(g) ≡ ∑_{i≤j} β_{i,j}K_{i}(g) mod h

with β_{j,j} ≠ 0. So r_{j} ≥ 2j − 1 and so r_{j} = 2j − 1. Let

E_{j} = ϵ^{−2j+1}E′_{j}.

Then

E_{j}^{[0]} = ∑_{i≤j} β_{i,j}K_{i}.

Since the K_{i} are independent, so are the E_{j}^{[0]}. q.e.d.

3.5

For future reference, we will now calculate the function B in a very special situation. Let T be a point and let X be a curve of arithmetic genus one with one node. We can find a normalization map π : P^{1} → X so that π(0) = π(∞) is the node. Let p and s be points of P^{1}, q = 1/p and r = 1. We set P = π(p), Q = π(q), S = π(s) and R = π(r). Let δ_{1} be the image of a circle traversed counterclockwise around 0 ∈ P^{1}. We let M = O_{X} (S). Then the function B will just depend on a number α ∈ C and on p and s. The normalized differential is just

ω_{1} = \frac{1}{2πi} \frac{dz}{z} .

Explicitly, let

L_{α} = O_{X} (π(x) − S).

Then we should have

\frac{1}{2πi} ∫_{S}^{x} \frac{dz}{z} = α.

So

x = Se^{2πiα}.

We use the usual parameter z on X coming from the parameter z on P^{1} to normalize our expressions for λ and the s_{k}. We can now write down these expressions using a degenerate theta function

ϑ_{k}(z) = 1 − \frac{z}{k} .

So we get

λ(z) = \frac{ϑ_{1}(z)^{2}K_{1}}{ϑ_{p}(z)ϑ_{q}(z)},

where K_{1} is chosen so that the first Laurent coefficient of λ at p is one. On the other hand,

\frac{s_{0,α}}{s_{−1,α}} (z) = \frac{ϑ_{p}(z)ϑ_{x/p^{2}}(z)K_{2}}{ϑ_{q}(z)ϑ_{x}(z)}

and define

r(z, t) = \frac{ϑ_{t/p^{2}}(z)}{ϑ_{t}(z)}.

Lemma 3.5.1.

(26)  B_{α} = \frac{r(q, x)r(p, s)}{r(p, x)r(q, s)}

= \frac{ϑ_{x/p^{2}}(q)ϑ_{s}(q)ϑ_{x}(p)ϑ_{s/p^{2}}(p)}{ϑ_{x/p^{2}}(p)ϑ_{s}(p)ϑ_{x}(q)ϑ_{s/p^{2}}(q)}

= \frac{(1 − p/x)(1 − 1/ps)(1 − p/x)(1 − p^{3}/s)}{(1 − p^{3}/x)(1 − p/s)(1 − 1/xp)(1 − p/s)}.

3.6

We will now consider how to use the family X_{1} → T_{1} to define representations. For each positive integer k, we define a map

π_{k} : π_{∗}(ω_{X_{1}/T_{1}}) → O_{T_{1}}

by

ϕ_{k}(ω) = Res_{P} (λ^{k}ω),

where Res_{P (t)}(λ^{k}ω) indicates the residue at P (t) of the meromorphic section of the dualizing sheaf of X_{1,t} = C obtained by multiplying ω by λ^{k}. Notice that the ϕ_{k} all vanish on Z, since f_{−1} vanishes on Z. We define

χ_{k} = ϕ_{k}.

Lemma 3.6.1.

ψ = \frac{χ_{2} + 2χ_{1}}{ϵ}

vanishes on Z.

Proof. This follows from Proposition 3.4.1 (5). Indeed, using the fact that α_{0} ≡ −1 mod h, we see that

Res_{P (t)}(λ_{can}^{2} + 2λ_{can})ω

vanishes to order h^{2}. So χ_{2} + 2χ_{1} vanishes on Z. q.e.d.

We will initially assume C is smooth. We will also assume that

∫_{Q}^{P} v_{k} = hk.

We can now associate an infinite tridiagonal matrix C_{p} by defining its ij^{th} entry C_{p,i,j} as

C_{p,i,j} = \begin{cases}1, & for i = j + 1 \\ A_{i,p} & for i = j \\ B_{i,p} & for i + 1 = j \\ 0 & otherwise. \end{cases}

Note that C_{p}^{n} is a well-defined infinite matrix. If we have an infinite matrix E, we define a matrix E^{+} by

E_{i,j}^{+} = \begin{cases}E_{i,j} & for i < j \\ 0 & otherwise. \end{cases}

We have identified H^{1}(O_{C}) with C^{n} using the I_{k}. So we may think of the ϕ_{k} as sections of the map T_{1} = T × C^{n} → T. So we can define vertical vector fields χ_{1}, χ_{2}, … on T_{1} corresponding to differentiating the direction ϕ_{1}, ϕ_{2}, …. We define χ to be differentiation in the direction (1, 2, …, n).

Theorem 3.6.2.

χ_{k}(C_{p}) = [(C_{p}^{k})^{+}, C_{p}],

where [E, F] indicates the commutator of E and F.

Proof. Assume that h ≠ 0 and that h = \frac{N}{M} , with N and M integers. Then Mγ(Q) ∈ Z^{n} and consequently there is a function α having a pole of order M at P and a zero of order M at Q. We normalize α so that α/λ^{M} has value one at P. Then we have

s_{k+M} = αs_{k}

and consequently, A_{k} and B_{k} are periodic. Theorem 3.6.2 is just Theorem 4 of [7]. On the other hand, the set of points with h rational is dense and conclusion of this theorem holds generally. q.e.d.

We can find T_{k} ∈ S_{1} ⊕ S_{1} so that D_{T_{k}}(A_{l}) is the l^{th} diagonal entry of [(C_{p}^{k})^{+}, C_{p}] and D_{T_{k}}(B_{l}) is the l^{th} off diagonal entry of [(C_{p}^{k})^{+}, C_{p}].

Definition 3.6.3. D_{k} = D_{T_{k}}.

For t ∈ T, let σ : T → T × C^{n} be defined by

σ(t) = (t, 1, 2, …, n).

Then using Definition 3.4.2

A_{k,x} = f(x + kh(x)σ(π(x))

and

B_{k,x} = g(x + kh(x)σ(π(x)).

Theorem 3.6.4.

(f, g, χ, χ_{1}, …, χ_{n})

form a representation of

D_{1}, …, D_{n}.

Further, both f and g are periodic under translation by (1, 2, …, n). Further,

f_{1} = \frac{f + 2}{h^{2}}

and

g_{1} = \frac{g − 1}{h^{2}}

are meromorphic functions which are holomorphic at any point t with h(t) = 0 provided that the N_{t} is non-special on the curve X_{1,t}. Further, the D_{1}, …, D_{n} are 1-nice.

Proof. We have checked the last statement when the curve X_{1,t} is smooth. But if N_{t} is non-special, then all the nearby line bundles on all the nearby curves are non-special. Hence f_{1} and g_{1} are analytic at points w near t provided that h(w) ≠ 0. Further, at points w so that h(w) = 0, the two functions are holomorphic provided that X_{1,t} is smooth. Hence both functions are holomorphic on a neighborhood of t, except perhaps for a subset of codimension ≥ 2. So by Hartog’s theorem, these functions are holomorphic at t. q.e.d.

Let ψ = χ_{2} + 2χ_{1}. We can define P_{1} and P_{2} in S_{2} so that

D_{P_{1},P_{2}}(A_{k}) = A_{k−1,p}B_{k−1,p} + A_{k,p}B_{k−1,p} + 2 B_{k−1,p} − A_{k,p}B_{k,p} − A_{k+1,p}B_{k,p} − 2 B_{k,p},

218 d. gieseker

and

D_{P_{1},P_{2}}(B_{k}) = B_{k,p}(B_{k−1,p} + A_{k,p}^{2} + 2 A_{k,p}

− A_{k+1,p}^{2} − 2 A_{k+1,p} − B_{k+1,p}).

Thus we have the tame derivation D = D_{P_{1},P_{2}} of R[[ϵ]].

Then we can use Theorem 3.6.2 to calculate:

ψ(A_{k,p}) = D_{P_{1},P_{2}}(A_{k}),

ψ(B_{k,p}) = D_{P_{1},P_{2}}(B_{k}).

Proposition 3.6.5. (f, g, h, χ, ψ) form a representation of D_{P_{1},P_{2}}. Further, ψ is slow for this representation.

Let’s calculate the first terms of D_{Φ} as a series in ϵ. We get

(27) D(v^{(0)}) = E_{−1}(v^{(0)})E_{−1}(w^{(0)}) + (v^{(0)})E_{−1}(w^{(0)})
+ 2E_{−1}(w^{(0)}) − (v^{(0)})w^{(0)} − E_{1}(v^{(0)})w^{(0)} − 2w^{(0)}.

We have

D_{Φ}(v^{(0)}) = Φ(D(v^{(0)}).

So

(28) D_{Φ}(ϵ^{2}v^{(0)}) = (E_{−1}(−2 + ϵ^{2}v^{(0)})E_{−1}(1 + ϵ^{2}w^{(0)})
+ (−2 + ϵ^{2}v^{(0)})E_{−1}(1 + ϵ^{2}w^{(0)})
+ 2E_{−1}(1 + ϵ^{2}w^{(0)}) − (−2 + ϵ^{2}v^{(0)})(1 + ϵ^{2}w^{(0)})
− E_{1}(−2 + ϵ^{2}v^{(0)})(1 + ϵ^{2}w^{(0)}) − 2(1 + ϵ^{2}w^{(0)}).

Evaluating we get

D_{Φ}(v^{(0)}) = ϵ(v^{(1)} − w^{(1)}) + higher order terms in ϵ.

Similarly,

D_{Φ}(w^{(0)}) = ϵ(−v^{(1)} + w^{(1)}) + higher order terms inϵ. Definition 3.6.6. D_{k} = D_{k,Φ}.

Theorem 3.6.7.

(f_{1}, g_{1}, χ, χ_{1}, …, χ_{n})

form a representation of

D_{1}, …, D_{n}.

Further, both f_{1} and g_{1} are periodic under translation by (1, 2, …, n). Further, set

D′_{1} = \frac{D_{2} + 2D_{1}}{ϵ}

and

χ′_{1} = \frac{χ_{2} + 2χ_{1}}{h} .

Then the representation

(f_{1}, g_{1}, χ, χ′_{1})

is slow for D′_{1}.

3.7

We will be applying this construction of slow representations to prove Theorem 1.0.1. We need to have a criterion for checking that such representations are general.

Definition 3.7.1. Let S be a complex manifold and let π : X → S be a family of stable curves parameterized by S. Let s_{0} ∈ S be a point and suppose that N_{1}, N_{2}, …, N_{p} are the nodes of X_{x_{0}}. Then we can find functions f_{k} defined near s_{0} so that the deformation of the node N_{k} is locally isomorphic to xy = f_{k}. We say the nodes are independent at s_{0} if the differentials of the f_{k} are independent at x_{0}.

Remark 3.7.2. Suppose the g_{1}, …, g_{r} are functions on S. Suppose that the g_{1}, …, g_{r} have independent differentials when restricted to some smooth S′ on which all the f_{k} from the above definition vanish. Let S′′ be submanifold defined by the vanishing of all the g_{i}. Then the family X ×_{S} S′′ has independent nodes.

We will consider the following situation: Let M be a connected complex manifold and let Λ be a sheaf of abelian groups locally isomorphic to Z^{2n}. We will assume that Λ has a symplectic form ⟨ , ⟩. Let p be a point of M. There is the usual monodromy representation ρ of π_{1}(M, p) on the stalk Λ_{p}. Suppose that there exist δ_{1}, δ_{2}, …, δ_{n} in Λ_{p} so that the endomorphisms T_{i} of Λ_{p} defined by

T_{i}(γ) = γ + ⟨γ, δ_{i}⟩δ_{i}

are all in the image of ρ. We also assume that ⟨δ_{i}, δ_{j}⟩ = 0 for all i and j. Let

δ = δ_{1} + 2δ_{2} + ⋯ + nδ_{n}.

Let π : V → M be a vector bundle of rank n. Here V is a physical bundle, i.e., V is a complex manifold and for q ∈ M, the fibers of π, π^{−1}(q) = V_{q}, are given the structure of complex vector spaces. Let V be the sheaf of analytic sections of V, so that V is locally free of rank n on M. Suppose that Λ is a subsheaf of V so that each λ ∈ Λ_{p} can be considered a local section of V defined around p. In particular, by evaluating at p, we get a map μ_{p} from Λ_{p} → V_{p}. We assume that the image of μ_{p} is a lattice in V_{p}. We assume that the images of the δ_{i} give a complex basis of V_{p} for each p.

Let f be a meromorphic function on V. In particular, we can look at f_{p}, the restriction of f to V_{p}. We will assume that f_{p} is invariant under translation by elements of Λ_{p} for all p.

Definition 3.7.3. Let W be a complex subbundle of V. We say that W is good if f is constant on all the cosets of W_{p} ⊂ V_{p} and the elements of Λ_{p} ∩ W_{p} span W as a complex vector space.

Let U ⊂ M be an connected open set and let λ be a section of Λ over U. The set of p ∈ U so that λ(p) ∈ W_{p} is either all of U or is defined by nontrivial analytic conditions and so is nowhere dense. Consequently, we may find a set of the second category T ⊂ M so that if p ∈ T and λ(p) ∈ W_{p}, then λ(q) ∈ W_{q} for all q ∈ U. We call such a point very general.

If W is good, then the sheaf Λ_{W} = Λ∩W ⊂ Λ is a locally isomorphic to Z^{k} for some k and Λ_{W} ⊗ O_{M} = W.

Lemma 3.7.4. Suppose that the subbundle of V generated by δ is good. Then f_{p} is constant.

Proof. It suffices to prove the assertion for a very general point p. Let W be a good subbundle. First note that the monodromy representation ρ leaves L_{p} = Λ_{p} ∩ W_{p} invariant, since p is very general.

Let Q_{p} be a maximal complex subspace of V_{p} so that f_{p} is constant on the cosets of Q_{p}. Note that if f_{p} is constant on the cosets of Q_{1} and Q_{2}, then there f_{p} is constant on the cosets of Q_{1} + Q_{2}, so such a maximal Q_{p} exists. First, suppose that L = Q_{p} ∩ Λ_{p} is not a lattice in Q_{p}. Let q : V_{p} → V_{p}/Λ_{p} be the quotient map and consider the closure X of q(Q_{p}) in the torus V_{p}/Λ_{p}. Note that f can be considered a function on V_{p}/Λ_{p} which is constant on the cosets of q(Q_{p}) and hence on the cosets of X. But X is a closed subgroup of V_{p}/Λ_{p} and hence there is a real subspace Q′_{p} of V_{p} so that q(Q′_{p}) = X and f_{p} is constant on the cosets of Q′_{p}. Further, Q′_{p} ∩ Λ_{p} is a lattice in Q′_{p}. f_{p} is meromorphic, so f_{p} is constant on the complex subspace Q′′_{p} spanned by the vectors in Q′_{p}. Thus Q′′_{p} = Q_{p}, so L_{p} = Q_{p} ∩ Λ_{p} is a lattice in Q_{p}.

For any point p, there is a simply connected neighborhood U ⊂ M of p and a set T ⊂ U of the second category in U so that Q_{q} ∩Λ_{q} = Q_{r} ∩Λ_{r} for all q and r in T, since the set of subgroups of Z^{2n} is denumerable. Here we have identified Λ_{r} with Λ_{q}, since they are both identified with the global sections of Λ over U. We can then find a subsheaf W of V so that W_{q} = Q_{q} for all q ∈ T. Our sheaf W has been constructed in a neighborhood of a arbitrary point p, but these sheaves constructed at different points coincide on the overlaps, since their fibers coincide over sets of the second category. Consequently, we have a sheaf W so that the fiber of W at q is Q_{q} for a dense set of q. Let Λ_{W} = W ∩ Λ. Let W ⊂ V be the physical bundle associated with W. Then f is constant on the cosets of W_{q} for a dense set of q. Consequently, W_{p} ⊆ Q_{p} for all p with equality for a set of the second category T.

Assume that p ∈ T. Then monodromy operates on W_{p} = Q_{p}. Let U_{p} be the real span of the δ_{i} . The δ_{i} form a complex basis of V_{p}, so the real dimension of Q_{q} ∩U_{p} is less than or equal to the complex dimension of Q_{p}. Also note that we can find a symplectic form ⟨ , ⟩ on V_{p} as a real vector space extending the form on Λ. Observe that if v ∈ Q_{p} and ⟨v, δ_{k}⟩ ≠ 0, then δ_{k} ∈ Q_{p}, since monodromy acts. Consider the map T : Q_{p} → Q_{p} ∩ U_{p} defined by

T (v) = ∑_{k}⟨v, δ_{k}⟩δ_{k}.

The observation shows that T does map to U_{p}. Since U_{p} is maximal isotropic, the kernel of T is contained in Q_{p} ∩ U_{p}. So the real dimension of Q_{p} is less than or equal to twice the real dimension of Q_{p} ∩ U_{p} and the map T is onto. The dimension of the image of T is the number of k so that ⟨v, δ_{k}⟩ ≠ 0 for some v ∈ Q_{p}. Hence Q_{p} ∩ U_{p} has a real basis consisting of some subset of the δ_{k}. But δ ∈ Q_{p} ∩ U_{p} and δ is linear combination of the δ_{k} so that all the δ_{k} appear nontrivially in δ. So Q_{p} = V_{p} and f_{p} is constant. q.e.d.

4. Explicit construction of curves

4.1

Our aim is to construct a family of representations which are generic and general and then to apply Theorem 2.9.5. Theorem 3.6.7 already checks that our families will satisfy conditions (2), (3) and (4) of Theorem 2.9.5, except that we want to use the following for the definition of D_{1} in Theorem 2.9.5:

D_{1} = \frac{D_{2} + 2D_{1}}{ϵ} .

To construct the n^{th} representations, we will be considering families of curves in P^{1}×P^{1} over C, which are generically double sheeted coverings of the second factor. We will suppress the n in the notation until we are ready to put all the representations into a family. Let

T_{0} = H^{0}(P^{1} × P^{1}, O(n + 1, 2)).

So an element of T_{0} is a polynomial in the variable X_{0}, X_{1}, Y_{0}, Y_{1}, which is homogeneous of degree n + 1 in X_{0}, X_{1} and homogeneous of degree 2 in Y_{0}, Y_{1}. Usually, we will use affine coordinates to describe the elements of T_{0} we will be considering , where we set X_{0} = 1, X_{1} = x, Y_{0} = 1 and Y_{1} = y. One can easily pass from the affine coordinates to the homogenous coordinates. So the elements of T_{0} of interest to us can be described by

a(x)y^{2} + b(x)y + c(x).

Let T_{1} ⊂ P^{1} × P^{1} × T_{0} be the universal curve and let π_{3} : T_{1} → T_{0} and π_{1} : T_{1} → P^{1} be the obvious projections.

There is an birational involution ι of T_{1} over T_{0} given by

ι(x, y) = (x, −\frac{b(x)}{a(x)} − y) .

Note that ι is only defined for those points (x, y) with a(x) ≠ 0. Consider the map Λ from C^{n} to T_{0}:

Λ(α_{1}, α_{2}, …, α_{n}) = (y^{2} − x)(x − α_{1}) … (x − α_{n}) = P_{α_{1},...,α_{n}}.

Let L_{0} be the image of Λ in T. Let

C(α_{1}, α_{2}, …, α_{n})

be the curve

π_{0}^{−1}(Λ(α_{1}, α_{2}, …, α_{n})).

The α_{k} form a partial system of local coordinates around any point Λ(α_{1}, α_{2}, …, α_{n}) on T_{0} as long as the α_{k} are distinct and so L_{0} is a submanifold of T_{0} at those points.

Let P_{0} denote Λ (1, \frac{1}{2^{2}} , \frac{1}{3^{2}} , …, \frac{1}{n^{2}} ) ∈ T_{0}. We will be investigating curves in a neighborhood of

C_{0} = C (1, \frac{1}{2^{2}} , \frac{1}{3^{2}} , …, \frac{1}{n^{2}} ) .

For P near P_{0}, T_{1} → T_{0} forms a family of semi-stable curves parameterized by a neighborhood U ⊂ T_{0} of P_{0}. On C_{0}, we have 2n nodes N_{k} = (k^{2}, k) for k between −n and −1 and between 1 and n.

Lemma 4.1.1. The nodes of the family T_{1} → T_{0} are independent near P_{0}. (See Definition 3.7.4 for the meaning of independent.)

Proof. By replacing U by a smaller neighborhood of P_{0}, we can find f_{k} defined on U so that the deformation of the node N_{k} is locally given by

(x − \frac{1}{k^{2}} ) (y^{2} − x) + f_{k}(P)

for P ∈ T_{0} near P_{0}. The f_{k} have independent differentials at P_{0}. Indeed, it suffices to show that for any k, we can construct a map ψ_{k} : D → T_{0}, where D is the unit disk, so that ψ_{k}^{∗}(f_{p}) vanishes identically if p ≠ k, but vanishes to exactly order one at 0 ∈ D if p = k.

W (x, y, t)

= ((y^{2} − x) (x − \frac{1}{k^{2}} ) + (t^{2} + \frac{2t}{k}) (y + \frac{1}{k} )^{2}) ∏_{p^{2}≠k^{2}} (x − \frac{1}{p^{2}} ) .

Note that

\frac{∂W}{∂t} (\frac{1}{k^{2}} , \frac{1}{k} , 0) ≠ 0

so that the total space family of curves over D defined by W = 0 is smooth at (\frac{1}{k^{2}} , \frac{1}{k}, 0), so that ψ_{k}^{∗}(f_{k}) vanishes exactly once at t = 0. On the other hand, (\frac{1}{k^{2}} , −\frac{1}{k}) continues to be a node of the curve W (x, y, t) = 0 so ψ_{−k}^{∗}(f_{k}) = 0 and two distinct nodes continue to lie over x = p^{2} for p^{2} ≠ k^{2}. So ψ_{p}^{∗}(f_{k}) = 0. q.e.d.

4.2

For k > 0, choose small circles β_{k} oriented counterclockwise around the points \frac{1}{k^{2}} and let δ_{k,0} be the lift of β_{k} to C_{0} which is near to the point (\frac{1}{k^{2}} , \frac{1}{k}) so that π_{1} ∘ δ_{k,0} = β_{k}. For some neighborhood U ⊂ T_{0} of P_{0}, we can find a map δ_{k} : S^{1} × U → π_{3}^{−1}(U) defined over U which restricts to δ_{k,P_{0}}, where π_{3} : T_{1} → T_{0} is the projection. Let ω_{T_{1}/T_{0}} be the sheaf of relative dualizing differentials. Let δ_{k,Q} be the cycle δ_{k}(S^{1}, Q) on T_{1,Q} for Q ∈ T_{0}. By possibly shrinking U, a section w of ω_{T_{1}/T_{0}} over V ⊂ U can be integrated fiberwise over the cycle δ_{k,Q}. Thus we obtain a maps over V,

∫_{δ_{k}} : π_{3,∗}(ω_{T_{1}/T_{0}}) → O_{T_{0}},

i.e.,

(∫_{δ_{k}} ω)_{Q} = ∫_{δ_{k}(S^{1},Q)} ω_{Q}.

Thus we get a map

Ψ : π_{3,∗}(ω_{T_{1}/T_{0}}) → ⨁_{k} O_{T_{0}}

as the direct sum of the ∫_{δ_{k}}.

Note that we can compute the dualizing differentials on D = C(α_{1}, …, α_{n}). Namely, let w_{k} be the differential

\frac{\sqrt{α}_{k}dy}{πi(y^{2} − α_{k})}.

Then w_{k} extends to a section of ω_{D} and

∫_{δ_{p}} w_{k} = δ_{p,k},

where δ_{p,k} indicates the Kronecker delta function. So by shrinking U again to a neighborhood of P_{0}, we can assume that Ψ is an isomorphism. We have established:

Lemma 4.2.1. We can find local sections v_{p} of π_{3,∗}(ω_{T_{1}/T_{0}}) so that

∫_{δ_{k}} v_{p} = δ_{k,p}.

A point t = (E, p) of T_{1} consists of an equation E(x, y) ∈ T_{0} for a curve C ⊂ P^{1} × P^{1} and a point p ∈ C. Let U_{1} be a small neighborhood of (P_{0}, (0, 0)). We introduce functions h_{p} of z ∈ U_{1} by the formula

h_{p}(z) = ∫_{ι(z)}^{z} v_{p},

where we define this when the projection of z to P^{1} × P^{1} near zero and P = π_{3}(z) is near P_{0}. Here we must specify a path γ from ι(z) to y. First we ask that π_{3}(γ) be a point and that the projection of γ to P^{1} × P^{1} lie near (0, 0). Notice that on C_{0}, the point (0, 0) is fixed under ι. So as long as we stay near to (0, 0) and C_{0}, it makes sense to ask that the path from stays near (0, 0). With these assumptions, h_{p} is well-defined. The functions h_{k} all vanish on the ramification locus R of the map π_{1} × π_{3} : U → P^{1} × T_{0}, when the h_{k} are defined, for R is just defined by ι(z) = y. Also π_{3} : R → T_{0} is a local isomorphism at the points we are considering.

4.3

We can compute these h_{k} on the curves C(α_{1}, …, α_{n}). The projection of our path γ lies on the curve C(α_{1}, …, α_{n}). Let the projection of z to the second factor of P^{1} × P^{1} be y_{0}.

(29)  h_{k}(z) = ∫_{−y_{0}}^{y_{0}} \frac{\sqrt{α}_{k}dy}{πi(y^{2} − α_{k})}

= \frac{\sqrt{α}_{k}}{πi} log (\frac{α_{k} − y_{0}}{α_{k} + y_{0}} )

≈ \frac{2y_{0}}{πi\sqrt{α}_{k}} .

where ≈ indicates approximately when y_{0} is close to zero.

This means that each of the h_{k} = 0 defines R ∩ π_{3}^{−1}(L_{0}) as a subscheme of π_{3}^{−1}(L_{0}) in a neighborhood of P_{0}, since y = 0 vanishes to order one R ∩ π_{3}^{−1}(L_{0}) ⊂ π_{3}^{−1}(L_{0}). Since all the h_{k} vanish on R in a neighborhood of (0, 0, y), we see that all the h_{k} vanish to order one along R ∩ π_{3}^{−1}(L_{0}).

Let

H_{k} = \frac{h_{k}}{h_{1}}

for k from 2 to n. R ∩ π_{3}^{−1}(L_{0}) can be identified with L_{0} locally via π_{3}, so we can use the coordinates α_{1}, …, α_{n} as coordinates on R∩π_{3}^{−1}(L_{0}). Then when restricted to R ∩ π_{3}^{−1}(L_{0}), the (H_{k})_{|R∩π_{3}^{−1}(L_{0})} just become

(H_{k})_{|R∩π_{3}^{−1}(L_{0})} =\sqrt{\frac{α_{k}}{α_{1}}} .

Note that the equations h_{1}, H_{2}, …, H_{n} all have independent differentials at (0, 0, P_{0}), since the H_{k} have independent differentials when restricted to R ∩ π_{3}^{−1}(L_{0}).

Lemma 4.3.1. We have

\frac{∂}{∂α_{k}} H_{k} ≠ 0

near (0, 0), P_{0}.

4.4

Let

T = {t ∈ T_{1}|H_{k}(t) = k}

and let

X = T ×_{T_{0}} T_{1}.

The map π : X → T has a canonical section P : T → X defined in the following way: A point t of T consists of an equation E(x, y) ∈ T_{0} for a curve C ⊂ P^{1} × P^{1} and a point p ∈ C. π^{−1}(t) is canonically identified with C. So we define P (t) = p. We have another section Q : T → X defined by Q(t) = ι(p). On the other hand, the ramification locus R ⊂ U is locally isomorphic to T_{0}. So we can find a section R : U′ → X so that ι(R(t)) = R(t), where U′ is a neighborhood of (0, 0, P_{0}). By shrinking U, we can assume that the images of P (t), Q(t) and R(t) in P^{1} ×P^{1} are all near (0, 0). By localizing on T, we can assume that there is a section s of O(2R(t)) which is not constant when restricted to any fiber of π. By possibly further restricting T we can a map z of a neighborhood of R to T × D over T so that

s = \frac{1}{z^{2}} .

Using Lemma 3.7.4, we see:

Lemma 4.4.1. T is smooth near (P_{0}, (0, 0)) = t_{0}. π : X → T is a family of semi-stable curves . X_{t_{0}} has 2n nodes and these nodes are independent. The I_{δ_{k}} for k > 0 form a basis of the R^{1}π_{∗}(O) locally. Let T_{2} be the subset of T defined by h = 0 and let X_{1} = π^{−1}(T_{2}). Let V be the physical bundle associated to R^{1}π_{∗}(O_{X_{1}}) Let f be a meromorphic function V defined on a neighborhood of the inverse image of t_{0}. Then if f is constant on the fibers of the line bundle generated by

δ = δ_{1} + δ_{2} + ⋯ + δ_{n},

then f is constant on the fibers of V.

At this point, we will make a choice of a line bundle M on T_{1}. Note that

T_{1} ⊂ T_{0} × P^{1} × P^{1}.

A point of T_{1} consists of an equation E ∈ T_{0} and a point (x, y) with E(x, y) = 0. So we can map ϕ : T_{1} → P^{1} by ϕ(E, x, y) = y. On the other hand, locally around P_{0}, we can find a map γ : U_{0} → T_{1} so that γ(U_{0}) ⊂ R. Thus γ(U_{0}) is a divisor on the inverse image U_{1} of U in T_{1}. We let

M = ϕ^{∗}(O_{P^{1}}(1)) ⊗ O_{U_{1}}(−γ(U_{0})).

We denote the pullback of M to X by M again. We now have a function B(z, α_{1}, α_{2}, …, α_{n}) defined locally on T_{1} −T_{2}. Thus M restricted to the curve {E = 0} = C is just

ϕ^{∗}(O_{P^{1}}(1)) ⊗ O_{C}(−R)

on the curve C(α_{1}, α_{2}, …, α_{n}). This bundle has degree one on all the vertical components of C(α_{1}, α_{2}, …, α_{n}), but degree zero on the curve {y^{2} = x}.

4.5

Let b be a nonzero integer between −n and n. Let

W_{t}(x, y) = ((y^{2} − x)(x − b^{2}) + (t^{2} − 2bt)(y − b)^{2})).

Note that the point (b^{2}, b) is a node the curve W_{t} = 0 for all t. Fixing t, let (x_{1}, y_{1}) be a generic point of W_{t} = 0. Let

S = ∏_{k≠b}(x − g_{k}).

Let Z = (W_{t}S, (x_{1}, y_{1})) ∈ T_{1}. Our aim is to evaluate

B(Z, α_{1}, α_{2}, …, α_{n}).

We can find a map of ψ_{1,t} : P^{1} → P^{1} by

ψ_{1,t}(w) = −\frac{4 wt^{2} − 8 wt b − b^{2} + 2 b^{2}w − b^{2}w^{2}}{(w + 1)^{2}}

and ψ_{2,t} : P^{1} → P^{1} by

ψ_{2,t}(w) = −\frac{4 wt^{2} − 8 wt b + bt − 2 b^{2} − bw^{2}t + 2 b^{2}w}{(wt − t + 2 b) (w + 1)} .

Let ψ_{t} = (ψ_{1,t}, ψ_{2,t}) : P^{1} → P^{1} × P^{1}. Then ψ_{t} maps P^{1} to the curve W_{t} = 0 and is in fact the normalization of this curve for t generic. Further, we have ψ_{1,t}(0) = ψ_{1,t}(∞) = b^{2} and ψ_{2,t}(0) = ψ_{2,t}(∞) = b, so 0 and ∞ map to the node of W_{t} = 0. Further, ψ_{1,t} is ramified at 1 and in fact ψ_{1,t}(w) = ψ_{1,t}(1/w). Now ψ_{2,t}^{−1}(∞) = {−1, 1 − 2b/t} = {P_{1}, P_{2}}. Let C ⊂ X_{p} be the curve with equation W_{t} = 0.

Let L be any line bundle on X_{z} which has degree zero on all the irreducible components of X_{z}.

Lemma 4.5.1. The natural restriction map ϕ : H^{0}(X_{z}, L ⊗ M_{z}) → H^{0}(C, L ⊗ M_{z} ⊗ O_{C}) is an isomorphism.

Proof. Note that L⊗M_{z}⊗O_{C} has degree one on a curve of arithmetic genus one, while L ⊗ M_{z} has degree n on a curve of arithmetic genus n. Hence, we need only show that ϕ is injective. A section s in the kernel of ϕ is a section of L ⊗ M_{z} which vanishes on the curve C. The other components of X_{z} are all fibers of the projection of P^{1} × P^{1} onto the first factor. As such, the degree of M_{z} on these components is one. But these components meet C in two points. So the restriction of s to these components is a section of a bundle of degree one which vanishes at two points. Hence the section vanishes on all the vertical components, and so s = 0. q.e.d.

For v_{0} ∈ P^{1}, let ψ_{t}(v_{0}) = (x_{1}, y_{1}). We will develop conditions on the g_{k} so that Z ∈ T. In fact, we will write g_{k} as a function of v_{0} and t. We can write

w_{k} = \frac{1}{2πi} (\frac{1}{z − g_{k}} − \frac{1}{z − 1/g_{k}} ) dz

for k ≠ 0, while

w_{0} = \frac{1}{2πi} \frac{dz}{z} .

We will consider g_{k} close to

\frac{b − k}{b + k} .

The w_{k} are the pullbacks of the canonical differentials on curve in P^{1} × P^{1} corresponding to Z. Then we have

(30)  h_{k} = ∫_{1/v_{0}}^{v_{0}} w_{k}

= \frac{1}{2πi}( log(v_{0} − g_{k}) − log(1/v_{0} − g_{k})

− log(v_{0} − 1/g_{k}) + log(1/v_{0} − 1/g_{k}))

= \frac{1}{πi} log (\frac{v_{0} − g_{k}}{1 − g_{k}v_{0}} ) .

Note that we have chosen the usual branch of the log so that h_{k} vanishes when v_{0} = 1. We have

(31)  h_{b} = ∫_{1/v_{0}}^{v_{0}} w_{b}

= \frac{1}{πi} log(v_{0}).

Next we choose g_{1}, …, g_{b−1}, g_{b+1}, … so that

H_{k} = k.

We do this by first choosing g_{1} as a function of v_{0} and t to make h_{b} = bh_{1}. Indeed, we can just take

g_{1} = v_{0} \frac{1 − v_{0}^{b−1}}{1 − v_{0}^{b+1}} .

Note that g_{1} is analytic even when v_{0} = 1 and in fact

g_{1} = \frac{b − 1}{b + 1},

when v_{0} = 1. We can find similar formulas for g_{k} in terms of v_{0} for the rest of the k which are not b. Let

R(t, v_{0})(x, y) = W_{t}(x, y) ∏_{j≠b}(x − ψ_{1,t}(g_{k}(v_{0}, t))).

Recall that

h = \frac{1}{b}h_{b} = \frac{1}{biπ} log(v_{0})

so that v_{0} = exp(biπh). Then we can construct a map

Φ_{b} : C × C → T

by

Φ_{b}(h, t) = (R(t, v_{0}), (x_{1}, y_{1})).

Note that Φ_{b}(0, 0) = (P_{0}, (0, 0)). Then Φ_{b}(h, t) ∈ T. We will use Lemma 3.5.1 to compute

B(Φ_{b}(h, t), α_{1}, …, α_{n}).

Let s = 2b/t − 1. Note that the pullback of M to curve C = {W_{t} = 0} is O_{C}(ψ_{t}(S)), where S = π(s). Let x = se^{2πiα_{b}}. Then

B(Φ_{b}(h, t), α_{1}, …, α_{n}) = \begin{matrix}(−x + v_{0}) & 2 & (v_{0}s − 1) & (−s + v_{0}^{3}) \\ (v_{0}x − 1) & (−s + v_{0}) & 2 & (−x + v_{0}^{3}) \end{matrix}.

Let

H = (\frac{−1 + B(Φ_{b}(h, t), α_{1}, …, α_{n})}{h^{2}})_{h=0} .

Then we can compute

H = 4 \frac{s (s^{2}β^{2} − β + 1 − s^{2}β) π^{2}}{(sβ − 1)^{2} (s − 1)^{2}} ,

where

β = \frac{x}{s} = exp(2iπα_{b}).

So we get:

Lemma 4.5.2.

(\frac{dH}{dt})_{t=0} = \frac{2π^{2}(β − 1)b}{β}.

4.6

Recall that the family π : X → T depends on the integer n. (The curves have bidegree (2, n + 1).) Let’s rename T as V_{n} and Φ_{b} as Φ_{b,n} and Φ_{b}(0, 0) = Q_{n}. We also define

ϕ_{b,n}(t) = Φ_{b,n}(0, t).

We also denote N_{n} = V_{n} × C^{n}. We claim that we can fit the N_{n} into a compatible family. Our first task is to construct a map k_{n} : V_{n} → V_{n+1}. Let

T_{0,n} = H^{0}(P^{1} × P^{1}, O(n + 1, 2)).

We map u_{n} : T_{0,n} × C → T_{0,n+1} by

u_{n}(P, β)(x, y) = P(x, y)(x − β).

Here we will only deal with β near 1/(n + 1)^{2} so the curve x − β = 0 will meet the curve P (x, y) = 0 transversally. Now we see that the fiber C_{n}(P) of T_{1,n} → T_{0,n} over P is naturally a subcurve of the fiber of C_{n+1}(u_{n}(P)), where T_{1,n} is the universal curve over T_{0,n}. Thus we have natural maps

k_{n} : T_{1,n} → T_{1,n+1}.

Further, the normalized differentials v_{k,n+1} on the curve C_{n+1}(P) restrict to the normalized differentials v_{k,n} for k = 1, …, n. So the functions

h_{k,n}(P, β) = ∫_{Q}^{P} v_{k,n+1}

on the curve C_{k_{n}(P,β)} are independent of β and in fact

h_{k,n}(P, β) = ∫_{Q}^{P} v_{k,n},

where the latter integral is taken on the curve P(x, y) = 0. On the other hand,

\frac{∂}{∂β} \frac{h_{n+1,n+1}}{h_{1,n+1}} ≠ 0

when β is near

\frac{1}{(n + 1)^{2}}

by Lemma 4.5.2.

Now

V_{n} ⊂ T_{1,n}

is the set of (P, P) with P ∈ C_{n}(P) so that

∫_{ι(P)}^{P} v_{k,n} = k ∫_{ι(P)}^{P} v_{1,n} = kh_{1,n}.

The functions k_{n} map V_{n} → V_{n+1}. Hence we have functions f_{n} and g_{n} on V_{n} × C^{n} which form a representation of D_{1}, …, D_{n}, …. Further,

f_{n}(v, z_{1}, z_{1}, …, z_{n}) = f_{n+1}(k_{n}(v), z_{1}, …, z_{n+1}).

We let V′_{n}, f′_{n} be the extended representations. (Definition 2.6.5.) Let g′_{n} be the meromorphic functions defined on (V′_{n}×C)∩{h_{1,n}} = 0

g′_{n}(v, z) = g′_{1,n}(v, z, 2z, …, nz).

Lemma 4.6.1. The family {g′_{n}} is generic.

Proof. Let b be an nonzero integer between −n and n. Then Lemma 4.5.2 shows that there are maps ϕ_{b,n} from the disk D × C to V_{n} so that

(\frac{∂ϕ_{b,n}^{∗}(g_{1,n})}{∂t}) (0, z) = \frac{4π^{2}}{b} (1 − exp(−2πibz)).

We can then consider V_{n} as a subset of V′_{n} = V_{n} × C by sending v to (v, 0) and hence we can consider ϕ_{b,n} as a map to V′_{n} × C.When b = 0, we let

ϕ_{0,n}(s, z) = ((Q_{r}, s), z).

Then

(\frac{∂ϕ_{0,n}^{∗}(g′_{1,n})}{∂s}) (0, z) = 2.

q.e.d.

Theorem 4.6.2. There is a Q ∈ R_{0}[[ϵ]] so that

D_{i}(v^{(0)} − Q) ∈ I_{Q}.

Proof. We have constructed a representation satisfying the hypotheses of Theorem 2.9.5. q.e.d.

Theorem 4.6.3. The characteristic numbers of D_{1}, …, D_{n} are 1, 3, …, 2n − 1. Further, the span of E_{1}^{[0]}, …, E_{n}^{[0]} is the same of the span of K_{1}, …, K_{n}, where the E_{k}^{[0]} are the leading terms of a normalized basis of the C[[ϵ]] module M generated by the D_{k}. (Definition 3.4.6.)

Proof. Using Lemma 2.8.3 and Lemma 4.6.1, we can construct a representation of D_{1}, …, D_{n} satisfying the hypotheses of Corollary 3.4.7.

q.e.d.

5. Poisson structures

5.1

Let

R̂_{0} = C[…, â_{−1}, â_{0}, â_{1}, …, b̂_{−1}, b̂_{0}, b̂_{1}, …]

and let

R̂ = R_{0}[ζ, ζ^{−1}].

We say a monomial in the â_{k} and b̂_{l} has weight r if the sum of the subscripts of the â_{k} and b̂_{l} sum to r. So the monomial â_{1}â_{2}b̂_{−3} has weight 0, as does ζ. Let I_{k} ⊂ R be the C span of all the elements of weight k. Let M_{N} be the ideal of R_{0} generated by

â_{N}, â_{N+1}, …, â_{−N}, â_{−N−1}, …, b̂_{N}, b̂_{N+1}, …, b̂_{−N}, b̂_{−N−1},

i.e., a monomial is in M_{N} if it involves â_{k} or b̂_{k} with |k| ≥ N. We also let M_{N} denote the induced ideal in R̂. Let Î_{k} be the completion of I_{k} with respect to subspaces I_{k} ∩ M_{N} as N → ∞. Then

F = ⨁_{k} Î_{k}

is called the Fourier ring. F is naturally a graded ring.

5.2

Suppose we are given elements f, g of S =C[z, z^{−1}]. We define â_{n}(f, g) ∈ C to be the coefficient of z^{n} in f and b̂_{n}(f, g) to be the coefficient of z^{n} in g. If P ∈ F, we can extend these definitions to define P (f, g) ∈ C[ζ, ζ^{−1}]. So if f = ∑ α_{n}z^{n} and g = ∑ β_{n}z^{n} and P ∈ I_{k}, then P (f, g) is the result of substituting α_{n} for â_{n} and β_{n} for b̂_{n} in P. Note that P (f, g) is well-defined. To check that two elements of F are equal, all we have to do is to check that they induce identical functions on S^{2}. Also, if P ∈ F[Z], then we can define P (f, g) ∈ C[Z, ζ, ζ^{−1}]. We denote by F_{0} the analogous construction for R̂_{0}. We can naturally map Ψ : F → F_{0}[[ϵ]] by sending ζ to exp(2πiϵ) considered as a formal power series in ϵ.

Suppose we have f ∈ S and let N be a positive integer, which we will think of as being large and let

ζ_{N} = exp (\frac{2πi}{N}) .

Then we can define

T_{N}(f)(n) = f(ζ_{N}^{n})

so T_{N}(f) : Z/N → C. Note that if N is sufficiently large depending on f and k, then we can recover f from T_{N}(f), namely

\frac{1}{N} ∑ T_{N}(f)(n)ζ_{N}^{−nk}

is the coefficient of z^{k}.

Now suppose we have a polynomial P ∈ S_{1} (see Definition 2.2.1 for definition of S_{1}) and let C_{N} be the set of C valued functions on Z periodic of order N. We can then define

(32) P (F, G)(n) = P ( …, F (n − 1), F (n), F (n + 1), …;

…, G(n − 1), G(n), G(n + 1))

Given f ∈ C_{N}, we define

f̂(n) = \frac{1}{N} ∑_{k∈Z/N} f(k)ζ_{N}^{−nk}.

Now suppose we are given two elements P_{1}, P_{2} of S_{1}. We can find a continuous derivations E_{P_{1},P_{2}} of F with the property that for all f, g ∈ S if P_{i}(T_{N}(f), T_{N}(g)) = h_{i,N},

(E_{P_{1},P_{2}}(â_{n})(f, g))_{ζ=ζ_{N}} = ĥ_{1,N}(n)

and

(E_{P_{1},P_{2}}(b̂_{n})(f, g))_{ζ=ζ_{N}} = ĥ_{2,N}(n)

for all N sufficiently large depending on f and g and n.

We can construct a series of maps

f_{n} : R[[ϵ]] → F_{1}

with the properties

f_{n}(v^{(k)}) = (2πin)^{k}â_{n},
f_{n}(w^{(k)}) = (2πin)^{k}b̂_{n},
f_{n}(1) = δ_{n,0}

and

f_{n}(F G) = ∑_{l∈Z} f_{l}(F)f_{n−l}(G).

Then we have

f_{n}(∂P) = 2πinf_{n}(P).

Suppose we are given a tame derivation D of R[[ϵ]] and a derivation D̂ of R̂_{0}[[ϵ]].

Definition 5.2.1. We say D and D̂ are compatible if

f_{n}(D(P)) = D̂(f_{n}(P)).

Proposition 5.2.2. Given D = D_{P,Q} for P, Q ∈ S_{1}, then there is a unique compatible D̂. D̂ maps the image of Ψ to itself and restricts to E_{P,Q} on the image of Ψ.

5.3

Suppose we have a finite collection P of elements of S_{1}

P_{−k}, …, P_{0}, …, P_{k}, Q_{−k}, …, Q_{0}, …, Q_{k}, R_{−k}, …, R_{0}, …, R_{k}.

Any polynomial with higher index is considered to be 0. Under some conditions on P, we can attempt define a Poisson bracket { , }_{P} on the functions G_{N} on C_{N}^{2} by asking that the bracket be a derivation in each slot, be anti-symmetric and satisfy Jacobi’s identity. Further, let A_{k}, B_{k} ∈ G_{N} be defined by

A_{k}(f, g) = f(k)

and

B_{k}(f, g) = g(k).

Then we can define

{A_{k}, A_{l}}_{P} = P_{k−l}(…, Â_{k}, A_{k+1}, …; …, Â_{l}, …)
{A_{k}, B_{l}}_{P} = Q_{k−l}(…, Â_{k}, A_{k+1}, …; …, B̂_{l}, …)

and

{B_{k}, B_{l}}_{P} = R_{k−l}(…, B̂_{k}, B_{k+1}, …; …, B̂_{l}, …).

Note the ˆ in the above equations is the place holder. We will suppose that { , }_{P} defines a Poisson bracket on G_{N} for N sufficiently large. We can define a modified bracket { , }_{P,N} by

{A_{k}, B_{l}}_{P,N}

= Q_{k−l} (…, −2 + \frac{1}{N^{2}} A_{k}, −2 + \frac{1}{N^{2}} A_{k+1}, …; …, 1 + \frac{1}{N^{2}} B_{l}, …) .

Next define

Â_{k,N} = \frac{1}{N} ∑_{l∈Z/N} ζ_{N}^{−kl}A_{l}

and

B̂_{k,N} = \frac{1}{N} ∑_{l∈Z/N} ζ_{N}^{−kl}B_{l}.

Proposition 5.3.1. Suppose that { , }_{P} defines a Poisson bracket on G_{N} for all N sufficiently large. Then there is a Poisson bracket { , }_{P} : F × F → F[Z] so that for f, g ∈ S, then

({â_{k}, â_{l}}_{P}(f, g))_{ζ=ζ_{N} ,Z=1/N} = {Â_{k,N}, Â_{l,N}}_{P,N}(T_{N}(f), T_{N}(g))

for all N sufficiently large with analogous formulas for {â_{k}, b̂_{l}}_{P}(f, g) and {b̂_{k}, b̂_{l}}_{P}(f, g).

Given P ∈ S_{1}[Z], define P_{N} : C_{N} × C_{N} → C by

P_{N}(F, G) = \frac{1}{N} ∑_{n∈Z/NZ} P (−2 + \frac{F}{N^{2}} , 1 + \frac{G}{N^{2}} ) (n)_{Z=1/N}.

Lemma 5.3.2. Suppose P ∈ S_{1}[Z]. Then there is a unique H_{P} in F[Z] so that

H_{P} (f, g)_{ζ=ζ_{N} , Z=1/N} = P_{N}(T_{N}(f), T_{N}(g)).

for N sufficiently large.

Proposition 5.3.3.

({â_{k}, H_{P} }(f, g))_{ζ=ζ_{N} , Z=1/N} = {Â_{k}, P_{N}}(T_{N}(f), T_{N}(g))

and

({b̂_{k}, H_{P} }(f, g))_{ζ=ζ_{N} , Z=1/N} = {B̂_{k}, P_{N}}(T_{N}(f), T_{N}(g)).

Suppose that P ∈ S_{1}[Z] and f and g are in S. The function of ϵ defined by

H(ϵ) = P (f, g)_{ζ=exp(2πiϵ), Z=ϵ}

is an analytic function of ϵ and for N sufficiently large,

H (\frac{1}{N} ) = P_{N}(T_{N}(f), T_{N}(g)).

If

|H (\frac{1}{N} )| < K(f, g)N^{−l}, (33)

for all N sufficiently large, then the first l − 1 derivatives of H vanish. We can attach a formal power series Ψ(P) to P in F_{0}[[ϵ]] by setting Z = ϵ and setting ζ = 1 + 2πiϵ + ⋯ = exp(2πiϵ). If (33) holds for all f, g ∈ S, then the first l − 1 derivatives of Ψ(L) vanish.

5.4

We next calculate two Poisson brackets. Our first bracket is given by:

{B_{n}, A_{n}}_{P_{1}} = −2B_{n}
{B_{n−1}, A_{n}}_{P_{1}} = 2B_{n−1},

with all other brackets between the A_{i} and B_{j} being zero, except for the obvious antisymmetric versions of these two formulas. We can now compute:

(34) \{Â_{n,N}, B̂_{m,N}\}_{P_{1}} = \frac{1}{N^{2}} \{∑_{l} ζ_{N}^{−ln}A_{l}, ∑_{k} ζ_{N}^{−mk}B(k)\}

= \frac{1}{N^{2}} ∑_{k,l} ζ_{N}^{−(ln+mk)}\{A_{l}, B_{k}\}

= \frac{1}{N^{2}} ∑_{l} ζ_{N}^{−l(n+m)}2B_{l} − ∑_{k} ζ_{N}^{−k(n+m)+n}2B_{k}

= \frac{2}{N} B̂_{n+m}(1 − ζ_{N}^{n}),

238 d. gieseker

and all the other brackets zero except as obviously required by antisymmetry.

For our second Poisson bracket { , }_{P_{2}} we take

{A_{k}, A_{k+1}}_{P_{2}} = B_{k}

{B_{k}, A_{k+1}}_{P_{2}} = B_{k}A_{k+1}
{B_{k}, A_{k}}_{P_{2}} = −B_{k}A_{k}
{B_{k}, B_{k+1}}_{P_{2}} = B_{k}B_{k+1}.

Now we can calculate the appropriate Fourier brackets:

{B̂_{n,N}, B̂_{m,N}} (35)

= \frac{1}{N^{2}} \{∑_{l} B_{l}ζ^{−lk}, ∑_{k} B_{k}ζ^{−km}\}

= \frac{1}{N^{2}} ∑_{k,l}\{B_{l}, B_{k}\}ζ_{N}^{−ln−km}

= \frac{1}{N^{2}} ∑_{k}\{B_{k+1}, B(k)\}ζ_{N}^{−(k+1)n−km} + ∑_{k}\{B_{k−1}, B_{k}\}ζ_{N}^{−(k−1)n−km}

= \frac{1}{N^{2}} ∑_{k} B_{k}B_{k−1}ζ^{−(k−1)n−km} − B_{k}B_{k+1}ζ^{−(k+1)n−km}

= \frac{1}{N^{2}} ∑_{k} B_{k}B_{k+1}(ζ^{−kn−(k+1)m} − ζ^{−(k+1)n−km})

= \frac{1}{N^{2}} ∑_{k,r,s} B̂_{r}B̂_{s}(ζ^{(−kn−(k+1)m)+rk+s(k+1)} − ζ^{−(k+1)n−km+rk+s(k+1)})

= \frac{1}{N} ∑_{n+m=r+s} B̂_{r}B̂_{s}(ζ^{−m+s} − ζ^{−n+s}).

Next, {B̂_{n,N}, Â_{m,N}} (36)

= \frac{1}{N^{2}} \{∑_{l} B_{l}ζ^{−lk}, ∑_{k} A_{k}ζ^{−km}\}

= \frac{1}{N^{2}} ∑_{k,l}\{B_{l}, A_{k}\}ζ_{N}^{−ln−km}

= \frac{1}{N^{2}} (∑_{l}\{B_{l}, A_{l+1}\}ζ_{N}^{−(ln+(l+1)m)} + ∑_{l}\{B_{l}, A_{l}\}ζ_{N}^{−ln−lm})

= \frac{1}{N^{2}} (∑_{l} B_{l}A_{l+1}ζ_{N}^{−(ln+(l+1)m)} − ∑_{l} B_{l}A_{l}ζ_{N}^{−ln−lm})

= \frac{1}{N^{2}} (∑_{l} B_{l}(A_{l+1}ζ_{N}^{−m} − A_{l})ζ_{N}^{−ln−lm})

= \frac{1}{N^{2}} ∑_{l,r,s} B̂_{r}Â_{s} (ζ_{N}^{−m+rl+s(l+1)−ln−lm} − ζ_{N}^{+rl+sl−ln−lm})

= \frac{1}{N} ∑_{r+s=n+m} B̂_{r}Â_{s} (ζ_{N}^{−m+s} − 1) .

Finally, we compute

{Â_{n,N}, Â_{m,N}} (37)

= \frac{1}{N^{2}} \{∑_{l} A_{l}ζ_{N}^{−lk}, ∑_{k} A_{k}ζ_{N}^{−km}\}

= \frac{1}{N^{2}} ∑_{k,l}\{A_{l}, A_{k}\}ζ_{N}^{−ln−km}

= \frac{1}{N^{2}} (∑_{l}\{A_{l}, A_{l+1}\}ζ_{N}^{−ln−(l+1)m} + \{A_{l}, A_{l−1}\}ζ_{N}^{−ln−(l−1)m})

= \frac{1}{N^{2}} (∑_{l} B_{l}ζ_{N}^{−ln−(l+1)m} − B_{l−1}ζ_{N}^{−ln−(l−1)m})

= \frac{1}{N^{2}} (∑_{l,r} B̂_{r}ζ_{N}^{−ln−(l+1)m+rl} − B̂_{r}ζ_{N}^{−ln−(l−1)m+r(l−1)})

= \frac{1}{N} B̂_{n+m}(ζ_{N}^{−m} − ζ_{N}^{−n}).

We will now investigate the bracket { , }_{2} defined by { , }_{2} = { , }_{P_{2}}. Note that if we have a continuous bracket

{ , } : F × F → F[Z]

then we have an induced bracket on F_{0}[[ϵ]] obtained by replacing Z by ϵ and ζ by the formal power series exp(2πiϵ). By abuse of notation, we continue to call the induced bracket by the some name as the original bracket.

Let W_{l} = Y_{0}^{l} and consider

X_{R} = ∑_{l=1}^{R} \frac{(−1)^{l}H_{W_{l}}Z^{2l−2}}{l + 1} .

Proposition 5.4.1. Let

X = lim_{R→∞} Ψ(X_{R}).

Then X is a Casimir for { , }_{P_{2}}. The leading term of X in ϵ is b̂_{0}.

Proof. We will look at {â_{p}, X_{R}} = V_{R}. Now

V_{R}(f, g)_{ζ=ζ_{N} , Z=1/N}

= ∑_{r∈Z/NZ} (∑_{l=0}^{R−1} (\frac{(−1)^{l}g(ζ_{N}^{r})^{l}}{N^{2l}} ) \{Â_{p}, B_{r}\}_{P_{2}, N}(T_{N}(f), T_{N}(g))) .

On the other hand,

∏_{k=0}^{N} (1 + \frac{B_{k}}{N^{2}} )

is a Casimir for { , }_{P_{2},N}, as was pointed out to me by Ali Kisisel. In particular for N sufficiently large,

(38) 0 = ∑_{r=0}^{N} \frac{\{Â_{p}, B_{r}\}_{P_{2},N}}{1 + \frac{B_{r}}{N^{2}}} (T_{N}(f), T_{N}(g))

= ∑_{r=0}^{N} \frac{\{Â_{p}, B_{r}\}_{P_{2},N}}{1 + \frac{g(ζ_{N}^{r})}{N^{2}}} (T_{N}(f), T_{N}(g))

= ∑_{r=0}^{N} (∑_{l=0}^{∞} (\frac{(−1)^{l}g(ζ_{N}^{r})^{l}}{N^{2l}} ) \{Â_{p}, B_{r}\}_{P_{2},N}(T_{N}(f), T_{N}(g))) .

Thus if we fix f and g and l, we can find a constant K(f, g) so that

|V_{R}(f, g)_{ζ=ζ_{N} , Z=1/N}| < K(f, g)N^{−l}

for N sufficiently large. Thus the first l−1 derivatives of V_{R}(f, g) vanish. Since this is true of any f and g, the first l − 1 derivatives of V_{R} vanish.

Thus {â_{p}, X} = 0. A similar argument shows that {b̂_{p}, X} = 0, so X is a Casimir. q.e.d.

Suppose that P_{1}, P_{2} and P are in S_{1}[Z]. Suppose further that

{A_{k}, P_{N}}_{P,N} = P_{1}(…, A_{k−1}, Â_{k}, A_{k+1}, …; …, B_{k−1}, B̂_{k}, B_{k+1}, …)

and

{B_{k}, P_{N}}_{P,N} = P_{2}(…, A_{k−1}, Â_{k}, A_{k+1}, …; …, B_{k−1}, B̂_{k}, B_{k+1}, …),

with the ˆ indicating place holder, not Fourier.

Proposition 5.4.2. E_{P_{1},P_{2}}(â_{p}) = {â_{p}, H_{P} }_{P} with a similar formula for b̂_{k}.

5.5

We get a series of derivations D_{k} of F[[ϵ]] compatible with D_{k} (see Proposition 2.4.1). The D_{k} all preserve the ideal I_{Q} generated by

f_{n}(v^{(0)} − Q) = L_{n}.

On the other hand, we have two compatible Poisson brackets on F_{0}[[ϵ]]. Let

Z(n) = exp(2πinϵ).

The first is defined by

{â_{n}, b̂_{m}}_{1} = (δ_{n,−m} + ϵ^{2}b̂_{n+m})(1 − exp(2πϵin))

with all other terms zero except as dictated by the Poisson bracket axioms. Thus we obtain

{â_{n}, b̂_{m}}_{1} = (−2πiϵn)δ_{n,−m} + higher order terms in ϵ.

In particular,

{L_{n}, L_{−n}}_{1} = δ_{n,−m}(−4πϵin) + higher order terms in ϵ.

The second is defined by

{b̂_{n}, b̂_{m}}_{2} (39)

= \frac{1}{ϵ} ∑_{n+m=r+s} Φ′(b̂_{r})Φ′(b̂_{s})(exp(2πϵi(−m + s) − exp(2πϵi(−n + s))

= \frac{1}{ϵ^{2}} \{b̂_{n}, b̂_{m}\}_{P_{2}}

with analogous expression for {b̂_{n}, â_{m}}_{2} and {â_{n}, â_{m}}_{2} from the Fourier expressions for the second bracket { , }_{P_{2}}. First suppose that n+m = 0. We can then compute

ϵ{b̂_{n}, b̂_{−n}}_{2} = (exp(2πϵi(n) − exp(2πϵi(−n))(1 + ϵ^{2}b̂_{0})^{2} (40)

+ ∑_{r+s=0,r≠0} ϵ^{4}b̂_{r}b̂_{s}(exp(2πϵi(n + s) − exp(2πϵi(−n + s)).

If n + m ≠ 0, then

(41) ϵ{b̂_{n}, b̂_{m}}_{2} = ϵ^{2}b̂_{n+m}(Z(n) − Z(−n) − Z(m) + Z(−m)) + higher order terms in ϵ.

So we get

{b̂_{n}, b̂_{m}}_{2} = δ_{n+m,0}(4πin) + higher order terms in ϵ.

We have similar results of {â_{n}, b̂_{m}}_{2} and {â_{n}, â_{m}}_{2}.

Now

L_{n} = â_{n} − b̂_{n} + higher order terms in ϵ.

So

{L_{n}, L_{−n}}_{2} = 16πinδ_{n,−m} + higher order terms in ϵ.

5.6

Ideally, our object would be to define induced brackets on F[[ϵ]]/I_{Q} = F_{0}[[ϵ]]. We will define brackets on a somewhat different ring S. First, let J_{Q} be the ideal generated by the L_{q} for q ≠ 0 and I′_{Q} the ideal generated by J_{Q} and the Casimir

X_{1} = \frac{X}{ϵ^{2}} .

Our aim is to define a well-defined bracket { , }_{2} on F_{0}[[ϵ]]/I′_{Q}. Now for each t ∈ C, the ring F[[ϵ]] has an automorphism ϕ_{t} defined by

ϕ(â_{n}) = â_{n} + tδ_{n,0}

together with

ϕ(b̂_{n}) = b̂_{n}.

Our first bracket behaves in the following way under ϕ:

{ϕ(a), ϕ(b)}_{1} = ϕ{a, b}_{1}.

Further, there are H_{k} ∈ F[[ϵ]] so that the vector fields obtained by bracketing with the H_{k} with either bracket have the same span as the derivations of F[[ϵ]] compatible with D_{T_{k}} (see Equation (3)). Further, the H_{k} all commute with respect to either bracket and because of Theorem 1.0.1, bracketing with H_{k} with respect to either bracket preserves the ideal I_{Q}. Further,

ϕ_{t}(H_{k}) = H_{k} + ∑_{l<k} C_{k,l}(t)H_{l},

where the C_{k,l} ∈ C[[ϵ]][t].

Proposition 5.6.1. Bracketing with the H_{k} with respect to either bracket preserves the ideal J_{Q}.

Proof. Note that ϕ_{t}(L_{n}) = L_{n} + tδ_{n,0}. We want to show that {H_{k}, L_{p}}_{1} ∈ J_{Q} for p ≠ 0. We work by induction on k. We have that the first H, namely H_{1} = â_{0} is a Casimir for the first bracket. So we have the first step of the induction. But modulo J_{Q}, G = {H_{k}, L_{p}}_{1} is then invariant under ϕ_{t}. But G = XL_{0} + Y where Y ∈ J_{Q}. Let J_{N} be the closure of the ideal generated by J_{Q} and all the b̂_{m} for |m| > N. ϕ_{t}(G) reduces to a polynomial of positive degree in t in the ring F[[ϵ]]/J_{N}[t] if X ∉ J_{N}. Note F[[ϵ]]/J_{N} is an integral domain. In fact, F[[ϵ]]/J_{N} is just C[â_{0}, b̂_{−N}, …, b̂_{N}][[ϵ]]. But ϕ_{t}(G) is invariant, so X ∈ J_{N} for all N, which implies that X ∈ J_{Q} and so G ∈ J_{Q}. q.e.d.

If P ∈ F_{0}[[ϵ]] and P ∈ F_{0}[[ϵ]]/J_{Q} is the image of P, then we define a good extension of P modulo ϵ^{n} to be an element P′ ∈ F_{0}[[ϵ]] so that {L_{q}, P′}_{k} ∈ (J_{Q} + ϵ^{n}) for q ≠ 0. An extension is good if it is good modulo ϵ^{n} for all positive n. It is easy to see that good extensions exist. Suppose we have constructed a good extension P_{n} modulo ϵ^{n}. Then we can try

P_{n+1} = P_{n} + ϵ^{n} ∑_{q≠0} W_{q}L_{q}

for W_{q} ∈ F_{0}. Bracketing through by L_{r} for r ≠ 0 allows us to choose the W_{q} uniquely so that P_{n+1} is good modulo ϵ^{n+1}. Thus we can find a good extension of P . Given P , P′ ∈ F_{0}[[ϵ]]/J_{Q}, we can define their bracket by taking good extensions P and P′ of P and P′ and taking their bracket in F[[ϵ]] and then reducing modulo J_{Q}. This construction gives a well-defined bracket {P, Q}_{2} on F[[ϵ]]/J_{Q}. Now I′_{Q} is obtained from J_{Q} by adding the Casimir X_{1} so we get an induced bracket on F[[ϵ]]/I′_{Q}. For a given n ≠ 0, define

β̂_{n} = b̂_{n} + (\frac{1}{2} + \frac{nϵiπ}{2} − \frac{ϵ^{2}π^{2}n^{2}}{4} ) L_{n} + ϵ^{2} ∑_{k≠0} (−\frac{n}{8} + \frac{3k}{8} ) b̂_{n−k} \frac{L_{k}}{k}

is the good extension of b̂^{n} modulo ϵ^{3} for the second bracket. For n = 0, let

β̂_{0} = \frac{1}{2} (â_{0} + b̂_{0}) + \frac{1}{2} ∑_{k≠0} L_{k}b̂_{−k}.

Theorem 5.6.2.

{β̂_{n}, β̂_{m}}_{2} ≡ i (π(n − m)β̂_{n+m} − δ_{n,−m}π^{3}n^{3} mod ϵ^{3}) .

The ring F[[ϵ]]/I′_{Q} is generated topologically by the images of β̂_{k} and ϵ. Further, all the H_{k} still Poisson commute in this extension of the Virasoro algebra.

6. Convergence?

6.1

Let Q be the element of R_{0}[[ϵ]] we have constructed in Theorem 4.6.2. As in [5], we can compute the coefficients Q_{n} ∈ R_{0} of Q. If g is a periodic function analytic on R, we can ask when the power series

∑_{k=0}^{∞} Q_{n}(g)(z)ϵ^{n} (42)

converges for z ∈ R. Suppose that

|g^{(n)}(z)| < n!, (43)

where g^{(n)} indicates the n^{th} derivative of g. For n ≤ 23, I calculated Q_{n}(g) using Maple and got a bound |Q_{n}(g)(z)| < K_{n} for z ∈ R by replacing each of the terms in Q_{n} by the obvious estimate using (43). Here are the decimal values of K_{n}.

\begin{matrix}n & K_{n} \\ 2 & .500 \\ 3 & .375 \\ 4 & .359 \\ 5 & .312 \\ 6 & .300 \\ 7 & .289 \\ 8 & .283 \\ 9 & .288 \\ 10 & .285 \\ 11 & .305 \\ 12 & .312 \\ 13 & .348 \\ 14 & .387 \\ 15 & .452 \\ 16 & .634 \\ 17 & .756 \\ 18 & 1.70 \\ 19 & 1.95 \\ 20 & 7.81 \\ 21 & 8.46 \\ 22 & 53.2 \\ 23 & 55.2 \end{matrix}

In order for 42 to converge for ϵ < 1/T all we would need is that K_{n} < T^{n}. On the basis of the fact that we have constructed many functions g coming from algebraic geometry for which 6.1.1 converges and the fact that the K_{n} appear to be growing not too fast, I believe there should be some general convergence property of Q. (Calculating the case n = 23 used over a gigabyte of memory and took over 500 hours on a Sun Enterprise.)

References

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[5] D. Gieseker, The Toda hierarchy and the KdV hierarchy, Comm. Math. Phys. 181 (1996) 587–603, MR 97g:58076, Zbl 0867.35096.

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Department of Mathematics UCLA