j. differential geometry 62 (2002) 289-349

SCALAR CURVATURE AND STABILITY OF TORIC VARIETIES

S.K. DONALDSON

Abstract

We define a stability condition for a polarised algebraic variety and state a conjecture relating this to the existence of a Kahler metric of constant scalar curvature. The main result of the paper goes some way towards verifying this conjecture in the case of toric surfaces. We prove that, under the stability hypothesis, the Mabuchi functional is bounded below on invariant metrics, and that minimising sequences have a certain convergence property. In the reverse direction, we give new examples of polarised surfaces which do not admit metrics of constant scalar curvature. The proofs use a general framework, developed by Guillemin and Abreu, in which invariant Kahler metrics correspond to convex functions on the moment polytope of a toric variety.

1. Introduction

This paper is a step towards the solution of the general problem of finding conditions under which a complex projective variety admits a Kahler metric of constant scalar curvature. The pattern of the answer one expects is that this differential geometric condition should be equivalent to some notion of “stability” in the sense of Geometric Invariant Theory. This expectation is probably now an item of folklore: going back to suggestions put forward by Yau in the case of Kahler-Einstein metrics, and the many results of Tian and others in this case; reinforced by a detailed formal picture which makes clear the analogy with the well-established relation between the stability of vector bundles and Yang-Mills connections [5]. Here, we begin the investigation of the problem in the special case of toric varieties, working within a general differential-geometric framework developed by Guillemin [11], [12] and Abreu [1]. While we are not able to achieve a decisive result, the geometry in this comparatively simple situation is surprisingly rich and displays very clearly the interplay between the analysis and the stability condition.

Received 11/13/2002.

While the general idea that constant scalar curvature should be related to stability has been mentioned by a number of authors a precise formulation of a conjecture, with the appropriate stability condition, has not yet appeared in the literature. Thus we begin by addressing this question, in Section 2 below, and define a condition on a polarised variety (V, L) which, following Tian, we call “K-stability”. (Tian gives a very similar definition in the particular case of Fano varieties.) Thus, we believe, the general expectation alluded to above can be given precise form in the following conjecture.

Conjecture. A smooth polarised projective variety (V, L) admits a Kahler metric of constant scalar curvature in the class c_{1}(L) if and only if it is K-stable.

Turning to our main topic, toric varieties: our principle result bears on the two-dimensional case. To state our result we should recall that for any compact Kahler manifold (V, ω_{0}) the Mabuchi functional is a functional M, defined on the metrics in the class [ω_{0}], whose critical points are precisely the metrics of constant scalar curvature [14]. Expressing a metric via a Kahler potential, i.e., as ω_{0} + 2i\overline{∂}∂ψ, we can regard M as a functional on the set H of potentials. Now suppose that (V, L) is a polarised toric surface, with an effective action of the torus T^{c} = (C^{∗})^{2} on L, covering an action on V. Working in the Kahler class c_{1}(L), we may regard the potentials, more invariantly as Hermitian metrics on the line bundle L. Thus T^{c} acts naturally on H. Let H^{T} be the set of potentials which are fixed by the action of the compact subgroup T = (S^{1})^{2} of T^{c}. A critical point of M on H^{T} is a metric of constant scalar curvature, by the principle of symmetric criticality. There is a natural action of the quotient T^{c}/T on H^{T}. To state our main result we introduce a piece of special terminology: we say that a sequence of potentials ψ_{α} is K-convergent if there is a sequence of scalars c_{α} and elements g_{α} of T^{c}/T such that the potentials ψ′_{α} = g_{α}(ψ_{α}) + c_{α} satisfy:

(1) The ψ′_{α} converge pointwise on V \V^{T}, where V^{T} is the finite subset of fixed points of the T -action; (2) On a Zariski-open neighbourhood of any fixed point we can intro-

duce local co-ordinates (w_{1}, w_{2}) in which the action is linear and diagonal and for any η_{1}, η_{2} > 0 there is a constant c\underline{η} such that

ψ′_{α}(w_{1}, w_{2}) ≥ η_{1} log |w_{1}| + η_{2} log |w_{2}| + c\underline{η},

for all α.

(We do not envisage that this notion of convergence has any very general significance: it merely summarises what we are conveniently able to prove.)

Our main result is:

Theorem 1.1. If a polarised toric surface is K-stable then the Mabuchi functional M is bounded below on H^{T} and any minimising sequence has a K-convergent subsequence.

(We will also obtain partial converses to this Theorem (Propositions 7.1.2, 7.1.3), which give, by the way, new examples of complex surfaces which do not admit metrics of constant scalar curvature.)

Of course one would like to go beyond this result to prove that the limit of the minimising sequence is a smooth potential, minimising the Mabuchi functional and hence defining a metric of constant scalar curvature. A slightly different, but related, issue is to show uniqueness: that one gets the same limit for any minimising sequence. We leave these questions, which clearly involve substantial further analytical and PDE problems, for now. We should point out that the arguments in the present paper are all of a fairly elementary nature and we hope they can be seen as preparing the way for an attack on these PDE questions.

We give a brief outline of the proof of Theorem 1.1 and the organisation of this paper. In Section 3 we review the differential geometry of toric varieties, following Guillemin and Abreu. This involves the wellknown correspondence between an n-dimensional toric variety and a polytope \overline{P} in R^{n}. A T -invariant Kahler metric corresponds to a convex “symplectic potential” function u on \overline{P}. The scalar curvature is given by a beautiful formula of Abreu

S(u) = − ∑_{ij} \frac{∂^{2}u^{,ij}}{∂x_{i}∂x_{j}} ,

where (u^{,ij}) is the inverse of the Hessian matrix u_{,ij} = \frac{∂^{2}u}{∂x_{i}∂x_{j}}. Thus the constant scalar curvature equation which motivates our study, S(u) = constant, is a fully nonlinear, fourth order, PDE which stands in a similar relation to the elliptic Monge-Ampere equation, det(u_{ij}) = 1 as, in linear PDE, the biharmonic equation stands to the Laplace equation. Our main concern in this paper is not directly with this equation but with the corresponding functional. For any function A on the polytope \overline{P} we define

F_{A}(u) = − ∫_{P} log det(u_{,ij}) dμ + L_{A}(u),

where L_{A} is the linear functional

L_{A}(u) = ∫_{∂P} u dσ − ∫_{P} Au dμ.

Here dμ is standard Lebesgue measure on R^{n} and dσ is a certain measure on the boundary ∂P. The Euler-Lagrange equation δF_{A} = 0 is the PDE S(u) = A. We are primarily interested in the case when A is a constant, and then the functional F_{A} corresponds to the Mabuchi functional (restricted to T -invariant metrics), as considered in Theorem 1.1.

In Section 4 we take up the algebro-geometric point of view and we show that any rational, piecewise-linear convex function on P defines a “test configuration” in the sense of our definition of K-stability in Section 2. We show that, if the toric variety is K-stable, then for any nontrivial function u of this kind L_{A}(u) ≥ 0 (here A is a constant): in fact L_{A}(u) is essentially the “Futaki invariant” which enters into the definition of K-stability.

Section 5 is the heart of the paper. Here we relate the nonlinear functional F_{A} to its linear component L_{A}. We show that if L_{A} is positive on a suitable class of convex functions then F_{A} is bounded below. All of the discussion up to this point applies equally well in any dimension n. The restriction to the case n = 2 in our main Theorem enters here in the analysis of the semi-positive case, where we are able to show that the extremal function is piecewise-linear. We expect that a similar result will be true in higher dimensions, but the arguments become substantially harder. Finally, in Section 6, we show that the extremal function is actually a rational piecewise linear function: again we expect this to be true in higher dimensions, but an extension of the direct attack on the algebra that we use in the two dimensional case seems to become very complicated. In Section 7 we put together the various components to complete the proof of Theorem 1.1, and conclude with a discussion of a number of ideas related to the work in the paper.

There is a substantial existing literature, going back to [15], on the existence of Kahler-Einstein metrics on toric Fano varieties. Of course in the Fano case—with the anticanonical polarisation—the constant scalar curvature and Kahler-Einstein conditions are equivalent. It will be interesting to relate the techniques we develop here to the results already known in the Fano case.

2. K-stability of polarised varieties

2.1 The definition

In this section we define a notion of stability for a pair (V, L) where L is an ample line bundle over a compact complex manifold V. The main point of the definition is that it involves another space V_{0}. In simple cases this will given by another complex structure on the differentiable manifold underlying V, but we also need to admit singularities, so we allow V_{0} to be a general scheme. Suppose Λ is an ample line bundle over a projective scheme W, and suppose we have a fixed C^{∗}-action on the pair (W, Λ). For each positive integer k we have a vector space

H_{k} = H^{0}(W; Λ^{k})

with a C^{∗}-action. From this we obtain integers d_{k} = dim H_{k} and w_{k}, the weight of the induced action on the highest exterior power of H_{k}. By general theory the integers d_{k}, w_{k} are, for large k, given by polynomial functions of k, with rational co-efficients: d_{k} = Q(k), w_{k} = P (k) say. We write F (k) = w_{k}/kd_{k}. The ring ⨁ H_{k} is finitely generated, and it follows easily that w_{k}/kd_{k} is bounded: hence the degree of P is at most the degree of Q plus 1. So for large enough k we have an expansion

F (k) = F_{0} + F_{1}k^{−1} + F_{2}k^{−2} + ⋯,

with rational co-efficients F_{i}. We define the Futaki invariant of the C^{∗}-action on (W, Λ) to be the co-efficient F_{1}.

Definition 2.1.1. A test configuration for (V, L), of exponent r consists of:

(1) a scheme V with a C^{∗}-action;

(2) a C^{∗}-equivariant line bundle L → V; (3) a flat C^{∗}-equivariant map π : V → C, where C^{∗} acts on C by

multiplication in the standard way;

such that any fibre V_{t} = π^{−1}(t) for t ≠ 0 is isomorphic to V and the pair (V, L^{r}) is isomorphic to (V_{t}, L|_{V_{t}}).

There are a few obvious points to note about this definition. First, all the fibres V_{t} for nonzero t are necessarily isomorphic, due to the action. Second, the C^{∗}-action on V induces an action on the central fibre: the scheme V_{0} = π^{−1}(0). Third, a special class of test configurations arises when (V, L) has a C^{∗}-action and we take the product configuration V = V × C.

Definition 2.1.2. The pair (V, L) is K-stable if for each test configuration for (V, L) the Futaki invariant of the induced action on (V_{0}, L|_{V_{0}}) is less than or equal to zero, with equality if and only if the configuration is a product configuration.

This definition is close to that given by Tian in [16], but there are a number of differences. Tian considers Fano manifolds and works with vector fields rather than group actions. More importantly, Tian restricts attention to configurations with smooth, or at least normal, central fibres. Despite these differences of detail we use the same terminology “K-stable” introduced by Tian, since it does not seem worth introducing new terminology into the literature. A semantic point is that what we have called K-stable would be called “properly K-semistable” by Tian.

2.2 Differential-geometric formula

Here we explain the relation between the “Futaki invariant” of the previous section and the usual differential geometric formulation, in the case when the central fibre V_{0} is smooth. Thus we suppose that (V_{0}, L) is a smooth polarised variety with a fixed C^{∗}-action. Let ω_{0} be any Kahler metric on V_{0} in the class 2πc_{1}(L), induced by a choice of Hermitian metric on L. Let v̂ be the vector field on the total space of L which generates the C^{∗} action, covering a vector field v on V_{0}. We can write

v̂ = \overline{v} + if\underline{t},

where \overline{v} is the horizontal lift of v, \underline{t} is the canonical vector field on the total space of L, defined by the action of scalar multiplication, and f is a smooth function on V_{0}. The condition that v̂ is holomorphic gives

\overline{∂}f = − (i_{v}(ω_{0}))^{0,1} .

(See [7], p. 490). Let S be the scalar curvature of ω_{0} and a be the average value of S over V_{0}. Set

ν = ∫_{V} (S − a)f \frac{ω_{0}^{n}}{n!} .

Proposition 2.2.1. The number ν is independent of the choice of metric ω_{0}, and hence is an invariant of the C^{∗}-action on (V_{0}, L).

This is essentially the original result of Futaki [8]: see also the exposition of Calabi [3]. In fact if we let g = G(S − a), where G is the Green’s operator, we can rewrite the integral as

∫_{V} (∂g, v)ω_{0}^{n}

which is the form given in [3]. This formulation also has the advantage that it uses only the holomorphic vector field v, and in fact the condition that the action lifts to L is not needed.

Proposition 2.2.2. The invariant F_{1} is given by

F_{1} = −\frac{1}{4Vol(V_{0})}ν.

To prove this we use Riemann-Roch formulae to compute the dimensions d_{k} and weights w_{k} of the action on H^{0}(V_{0}, L^{k}); this is very similar to the discussion in Chapter 5 of [9]. We can suppose that k is large enough for all the higher cohomology groups to vanish so d_{k} is given by the Riemann-Roch formula

d_{k} = ∫_{V_{0}} ch(L^{k})Td(V_{0}) = ∫_{V_{0}} e^{kω_{0}}Td(V_{0}).

The Todd class is represented by the differential form Td(V_{0}) = 1 + \frac{1}{2}ρ + … where ρ is the Ricci form, so we have

d_{k} = Ck^{n} + Dk^{n−1} + O(k^{n−2}),

where C = ∫_{V_{0}} \frac{ω_{0}^{n}}{n!} is the volume of V_{0} and

D = \frac{1}{2(n − 1)!} ∫_{V_{0}} ρω_{0}^{n−1} = ∫_{V_{0}} \frac{S}{4} \frac{ω_{0}^{n}}{n!} .

To compute the weights we use the equivariant Riemann-Roch formula. Here we suppose that ω_{0} is preserved by the circle subgroup of C^{∗} and let v be the vector field generating this action. Recall that the equivariant cohomology H_{S^{1}}^{∗}(V_{0}) is the cohomology of the homotopy quotient V_{0} ×_{S^{1}} ES^{1} and there is an integration-over-the-fibre map

∫_{V_{0}} : H_{S^{1}}^{∗}(V_{0}) → H^{∗}(BS^{1}) = Q[t],

where t is a generator in H^{2}. The equivariant Riemann-Roch formula asserts that w_{k} is given by the co-efficient of t in

∫_{V_{0}} ch(L^{k})Td(V_{0}),

where now the Chern and Todd classes are regarded as equivariant cohomology classes. To make the calculation we use the de Rham model of equivariant cohomology, with the complex (Ω_{V_{0}}^{∗})^{S^{1}} [t] and differential d+ti_{v}, see [2]. The equivariant Chern class of L is represented by ω_{0}+tf in this model. Let ρ + tR be the representative for c_{1}(V_{0}), where R is a function on V_{0}. Then we obtain

w_{k} = Ak^{n+1} + Bk^{n} + O(k^{n−1}),

where

A = ∫_{V_{0}} f \frac{ω^{n}}{n!} , B = ∫_{V_{0}} (\frac{fS}{4} + R) \frac{ω^{n}}{n!} .

Lemma 2.2.3. The function R is the divergence of the vector field Iv on V.

To prove this we use the characterisation

L_{v}(θ) = ∇_{v}(θ) + iRθ,

for any local section θ of the canonical bundle K_{V_{0}}. In particular we may consider a local holomorphic section θ, in which case we obtain

L_{Iv}(θ) = ∇_{Iv}(θ) − Rθ. (2.2.4)

Write θ ∧ \overline{θ} = σ \frac{ω^{n}}{n!} , then

L_{Iv}(θ ∧ \overline{θ}) = (∇_{Iv}σ + σdiv(Iv))\frac{ω^{n}}{n!}
= ∇_{Iv}(θ ∧ \overline{θ}) + σdiv(Iv)

On the other hand (2.2.4) gives

L_{Iv}(θ ∧ \overline{θ}) = (∇_{Iv}θ ∧ \overline{θ}) + θ ∧ ∇_{Iv}\overline{θ}) − 2Rθ ∧ \overline{θ},
= ∇_{Iv}(θ ∧ \overline{θ}) − 2Rθ ∧ \overline{θ},

and the result follows from these.

It follows from Lemma (2.2.3) that the integral of R over V_{0} vanishes so

B = ∫_{V_{0}} \frac{fS}{4} \frac{ω^{n}}{n!} .

Then

\frac{w_{k}}{kd_{k}} = \frac{A + Bk^{−1} + O(k^{−2})}{C + Dk^{−1} + O(k^{−2})} = \frac{A}{C} + \frac{AD − BC}{C^{2}} k^{−1} + O(k^{−2}),

and AD − BC = −\frac{C}{4} ν as required.

2.3 Relation with Hilbert-Mumford stability

We will discuss briefly the relation between the notion of K-stability defined above and “Hilbert-Mumford” stability, as studied in the algebraic geometry literature. This discussion is very similar to the corresponding one in [16]. Suppose (V, L) is a polarised variety as usual and that the linear system |L^{r}| gives a projective embedding of V as a projective variety with Hilbert polynomial P (n) = χ(L^{rn}). The Hilbert scheme H_{P} parametrising projective schemes in CP^{N} with the given Hilbert polynomial P is constructed as a subscheme of a Grassmannian of mplanes in the symmetric power s^{n}(C^{N+1}) by assigning to a scheme X the subspace of polynomials of degree n which vanish on X. In turn this Grassmannian is embedded in a projective space by the Plucker embedding, so we have a point in

P (Λ^{m}(s^{n}(C^{N+1})))

where m is dim s^{n}(C^{N+1}) − P (n). The group SL(N + 1, C) acts on Λ^{m}s^{n}C^{N+1} and the construction naturally assigns to (V, L^{r}) an orbit under the induced action on the Hilbert scheme in the projective space. The pair (V, L) is said to be Hilbert-Mumford stable, with the given values of r and n, if this is a stable orbit in the sense of Geometric Invariant theory i.e if the corresponding orbit O in the vector space is closed. The Hilbert criterion asserts that this is true if and only if for any 1-parameter subgroup C^{∗} ⊂ SL(N +1, C) and any z ∈ O the C^{∗}-orbit of z is closed. This readily translates into the following numerical criterion. For any 1-parameter subgroup ρ : C^{∗} → SL(N+1, C) the points [ρ(t)(z)] in the projective space converge to some limit [z_{0}] as t → 0, where [z_{0}] is fixed by ρ(C^{∗}). Thus there is an integer weight W (ρ) ∈ Z with ρ(t)(z_{0}) = t^{W} z_{0}. The stability condition is that W (ρ) ≤ 0 for all 1parameter subgroups ρ, with equality if and only ρ fixes z.

The connection with our previous discussion arises from the fact that there is essentially a one-to-one correspondence between test configurations in the sense of 2, with exponent r, and 1-parameter subgroups in SL(H^{0}(L^{r})). Starting with a test configuration we take a point [z_{0}] in the Hilbert scheme representing the central fibre V_{0}. The C^{∗}-action on V_{0} induces an action on C^{N+1} = H^{0}(V_{0}, L). Making a base change if necessary we can suppose that this is the product of a scalar action and a 1-parameter subgroup in SL(N + 1, C) and we are in just the position considered above. In the other direction, starting with a 1-parameter subgroup, the limit point [z_{0}] lies in the Hilbert scheme so the orbit extends to a map from C to the Hilbert scheme and we get a test configuration by pulling back the universal family. The essential task then is to see what the numerical criterion boils down to in this case, in terms of the C^{∗}-action on (V_{0}, L). The line spanned by z_{0} is, by definition, the highest exterior power of the kernel of the surjection

s^{n}(H^{0}(V_{0}, L)) → H^{0}(V_{0}, L^{n}).

Thus it can be identified with

Λ^{max}s^{n}(H^{0}(V_{0}, L)) ⊗ Λ^{max}H^{0}(V_{0}, L^{n})^{−1}.

For any vector space E there is a canonical isomorphism

Λ^{max}s^{n}(E) ≅ (Λ^{max}E)^{n dim s^{n}(E)/ dim E},

so the line in question can be identified (writing s^{n} for s^{n}(H^{0}(L))) with

Λ^{max}H^{0}(V_{0}, L)^{n dim s^{n}/ dim H^{0}(L)} ⊗ Λ^{max}H^{0}(L^{n})^{−1}.

However, we must take care of the fact that the 1-parameter subgroup we need is obtained by writing the natural action as a product of a scalar part and a part in SL(N + 1). The scalar part is given by the the weight of the natural action on Λ^{max}H^{0}(L) to the power 1/ dim H^{0}(L). Thus the action of this scalar part on the line spanned by z_{0} has weight equal to u^{mn/ dim H^{0}(L)} where u is the weight of the natural C^{∗}-action on Λ^{max}H^{0}(L). (This is, in general, rational rather than integral, corresponding to the fact that we may need to lift to a covering of the C^{∗}-action. However we can ignore this covering if we are only interested in finding the sign of W.) Putting this together, the weight W we need is the weight of the natural action on the line

(Λ^{max}H^{0}(L)^{(n dim s^{n}−mn)/ dim H^{0}(L)} ⊗ Λ^{max}H^{0}(L^{n})^{−1}.

Since dim s^{n} − m = P (n) = d_{n} we can write this weight as

W = \frac{nd_{n}w_{1}}{d_{1}} − w_{n} = nd_{n}(F (1) − F (n)).

So we conclude that, with the given values of r, n, the pair (V, L) is Hilbert-Mumford stable if and only if for each test configuration with exponent r the central fibre satisfies the numerical condition F (n) − F (1) ≥ 0, with equality if and only if the configuration is a product. To relate this to the K-stability condition, note that any configuration of exponent r_{0} defines configurations of exponent pr_{0} for all positive integers p— replacing L by L^{p}. Suppose we know that, for some r_{0} and all large p, all configurations of exponent r = pr_{0} arise in this way. Then the condition becomes that

F (pn) − F (n) ≥ 0

where F is the function defined by a configuration of exponent r_{0}. Clearly, if the variety is K-stable this condition will hold, for each such configuration, once n and p are large enough. Thus the essential difference between the notions of Hilbert-Mumford stability, and K-stability is that the first depends on the values of the function F at specific values of the parameter, while the second involves the asymptotics as the parameter becomes large. This difference is analogous to the difference between the notions of “Gieseker stability” and “Mumford stability” in the theory of vector bundles.

3. Toric geometry

3.1 The symplectic potential

In this subsection we review standard material on toric differential geometry, following Guillemin [11], [12] and Abreu [1]. Another paper which adopts a similar point of view is [10]. The main theme is the interplay between symplectic and complex structures. We begin on the symplectic side, so we consider a compact symplectic manifold V, ω of real dimension 2n and a line bundle L → V with a connection whose curvature is −2πiω. We suppose that the torus T^{n} = (S^{1})^{n} acts effectively on this data with a moment map m : V → R^{n}. The image of the moment map is an integral polytope \overline{P} in R^{n} i.e., it is the convex hull of a finite set of points in the lattice Z^{n}. The polytope is defined by a finite number of linear inequalities

h_{k}(x) ≥ c_{k}, (3.1.1)

where h_{k} are linear maps from R^{n} to R which induce primitive maps from the integer lattice Z^{n} to Z. We denote the interior of the polytope by P. The fibres of the moment map m are orbits under the T^{n}-action. The action is free on the dense open subset V_{0} = m^{−1}(P) and the universal cover of V_{0} can be identified with P × R^{n}. We introduce standard coordinates (x_{i}, η_{i}), i = 1, … n on this universal cover (which, in the familiar way we also think of as coordinates on V_{0}) in which the symplectic form is given by ω = ∑ dx_{i}dη_{i}; the moment map is given by projection to the x-coordinates and the group acts by translation in the η-coordinates.

We now turn to the complex side, so we suppose instead that V is a compact complex n-manifold with an action of the complex torus T_{C}^{n} = (C^{∗})^{n} which has a dense, free, open orbit V_{0}, and the action lifts to a positive holomorphic line bundle L → V. Thus we can identify V_{0} with (C^{∗})^{n} and we have standard complex co-ordinates w_{1}, …, w_{n} ∈ C\{0}. Again, we work in the covering space where we can take co-ordinates z_{i} = log w_{i} = ξ_{i}+\sqrt{−1}η_{i} in C, so the action is represented by translation in the z_{i} variables. We consider Kahler metrics on V which are invariant under the action of T^{n} ⊂ T_{C}^{n}. Such a metric can be represented by a Kahler potential over V_{0}; ω = 2i\overline{∂}∂ϕ so ϕ can be viewed as a function of the complex variables z_{1}, … z_{n}. The T -invariance means that we may restrict attention to functions which only depend on the real parts ξ_{1}, …, ξ_{n}. In sum, our Kahler metric is given by

ω = \frac{\sqrt{−1}}{2} ∑_{ij} ϕ_{,ij}dz_{i}d\overline{z}_{j}, (3.1.2)

where

ϕ_{,ij} = \frac{∂^{2}ϕ}{∂ξ_{i}∂ξ_{j}} ,

and ϕ = ϕ(ξ_{1}, …, ξ_{n}) is a strictly convex function on R^{n}.

To match up these points of view one observes that the moment map for the T^{n} action on the Kahler metric (3.1.2) is given by

m(z_{1}, … z_{n}) = (\frac{∂ϕ}{∂ξ_{i}} ) .

We define the symplectic potential u to be the Legendre dual of ϕ. Recall that this is defined in the following way. For each point \underline{x} in the image, P, of the moment map there is a unique point \underline{ξ} = \underline{ξ}(\underline{x}) in R^{n} where \frac{∂ϕ}{∂ξ_{i}} = x_{i}. We let

u(\underline{x}) = ∑ x_{i}ξ_{i} − ϕ(\underline{ξ}). (3.1.3)

Then u is a convex function on P and ϕ can be recovered from u by the symmetrical relation between Legendre dual functions. Thus the passage between the complex and symplectic viewpoints amounts to the change from the ξ_{i} to x_{i} coordinates and the metric data is encoded in either the Kahler potential ϕ(\underline{ξ}) on R^{n} or the symplectic potential u(\underline{x}) on P. In the symplectic coordinates the metric is given by

g = ∑ u_{,ij}dx_{i}dx_{j} + ∑ u^{,ij}dξ_{i}dξ_{j},

where the matrix (u^{,ij}) is the inverse of the Hessian matrix u_{,ij} = \frac{∂^{2}u}{∂x_{i}∂x_{j}}. According to Abreu [1], the scalar curvature of this metric is

S(u) = − ∑_{ij} \frac{∂^{2}u^{,ij}}{∂x_{i}∂x_{j}} . (3.1.4)

We digress to explain how this expression for the scalar curvature fits into the picture developed in [5], considering the action of the symplectomorphism group of a symplectic manifold on the space of compatible almost-complex structures. In the presence of the T -action on the symplectic manifold V we restrict attention to the space J^{T} of T -invariant almost-complex structures, with the action of the group G^{T}, the identity component of the group of symplectomorphisms of V which commute with T. The Lie algebra of G^{T} can be identified with certain smooth functions on \overline{P} modulo constants i.e., the symplectomorphisms are generated by Hamiltonian functions of the x_{i} variables, Elements of J^{T} are represented by maps from P to the Siegel generalised upper half-space H_{n} = Sp(2n, R)/U(n) of linear complex structures on R^{2n}, compatible with the standard symplectic form. The crucial point is that H_{n} has an invariant Kahler structure. Explicitly, we can represent any almostcomplex structure at a point by a pair of matrices A, B, describing the action of J on the tangent space to the T -orbits:

J (\frac{∂}{∂η_{i}} ) = ∑ A_{ij} \frac{∂}{∂η_{j}} + ∑ B_{ij} \frac{∂}{∂x_{j}} .

This defines a unique complex structure provided that B is symmetric and positive definite and B^{−1}A is symmetric. We write

X = U + iV = B^{−1}A + iB^{−1}.

This gives the standard representation of H_{n} as the set of complex symmetric matrices X with positive definite imaginary part, making evident the complex structure on H_{n}. The invariant Kahler form Ω is defined by the formula:

Ω(δ_{1}U + iδ_{1}V, δ_{2}U + iδ_{2}V) (3.1.5)

= Tr (δ_{1}UV^{−1}δ_{2}V V^{−1} − δ_{2}UV^{−1}δ_{1}V V^{−1})
= −Tr(δ_{1}Uδ_{2}B − δ_{2}Uδ_{1}B).

We see then that elements of J^{T} can be represented by matrix-valued functions U, V as above on P. The action of a symplectomorphism generated by a Hamiltonian function f on P is just

X ↦ X + f_{,ij} (3.1.6)

where f_{,ij} = \frac{∂^{2}f}{∂x_{i}∂x_{j}}. We define a map μ from J^{T} to functions on P by

μ(U, V) = − ∑ \frac{∂^{2}V^{ij}}{∂x_{i}∂x_{j}} ,

where (V^{ij}) = V^{−1}. It follows from (3.1.5) and (3.1.6) that μ is a moment map for the action of the compactly supported elements of G^{T} on J^{T}, where the latter space is given the symplectic form induced in a natural way by Ω, and integration over P. This just amounts to the integration-by-parts formula

∫_{P} f ∑ \frac{∂^{2}β^{ij}}{∂x_{i}∂x_{j}} = ∫_{P} ∑ f_{ij}β^{ij},

for compactly supported f and arbitary β.

In [5] we considered a formal complexification of the action of the symplectomorphisms on the space of almost-complex structures. In the present situation, this extension can be made in a completely straightforward way—we just use the same formula (3.1.6) with a complex-valued function f on P. The integrability condition for an almost-complex structure represented by a complex matrix-valued function X_{ij} is just

\frac{∂X_{ij}}{∂x_{k}} = \frac{∂X_{ik}}{∂x_{j}} ,

and all the structures for which this holds lie in the same orbit of the complexified action: i.e., we can write X_{ij} = f_{ij} for some complexvalued function f on P. In particular

V_{ij} = u_{ij} = \frac{∂^{2}u}{∂x_{i}∂x_{j}} ,

where u is the imaginary part of f. Taking inverses, we see that the moment map μ on this complex orbit is given by Abreu’s formula ∑ \frac{∂^{2}u^{ij}}{∂x_{i}∂x_{j}}. This ties up Abreu’s calculation with the discussion in [5] where it is shown, in general, that the scalar curvature furnishes a moment map for the action of the symplectomorphisms on the space of almost-complex structures; and concludes our digression.

Throughout this subsection we have focussed attention on the interior of P, corresponding to the free orbits of the T -action. We turn now, following Guillemin and Abreu, to the question of the boundary behaviour of the symplectic potential. Recall that the polytope P is defined by linear inequalities h_{k}(x) > c_{k}. We write δ_{k}(x) = h_{k}(x) − c_{k}, so the δ_{k} are positive functions on P. Let

u_{0}(x) = ∑_{k} δ_{k}(x) log δ_{k}(x),

so u_{0} is continuous function on \overline{P}, smooth on the interior. This gives the model for the boundary behaviour required to generate a smooth metric on the compact manifold V. We define S to be the set of continuous, convex, functions u on \overline{P} such that u − u_{0} is smooth on \overline{P}. Then the discussion of this section can be summed up the following Proposition.

Proposition 3.1.7. Let V be a symplectic toric manifold defined by a polytope \overline{P} in R^{n}. There is a one-to-one correspondence between the T -invariant Kahler potentials ψ on V; and the symplectic potentials u in S. Changing ψ to ψ + c for c ∈ R changes u to u − c. The action of T^{c}/T on the Kahler potentials coresponds to the action of the linear functions, by addition, on the symplectic potentials.

Of course the Kahler potentials ϕ that we are considering here, corresponding to symplectic potential u, are not quite the same as those discussed in Section 1 where we represent the metrics in a cohomology class as ω_{0} + 2i\overline{∂}∂ψ. However it is easy to pass between the two points of view. The potentials ϕ, regarded as distributions on the compact manifold V, satisfy an equation 2i\overline{∂}∂ϕ = ω + H, where H is a fixed current, independent of ϕ, supported on the divisor at infinity. Thus if we have a reference metric ω_{0} we can represent another metric ω as ω_{0} + i\overline{∂}∂ψ where ψ is the smooth function ϕ − ϕ_{0}.

One point to notice about this correspondence is that it behaves well under restriction to the faces of the polytope P. These faces correspond to toric subvarities of V and the restriction of any function in S to the interior of a face is smooth and yields a symplectic potential for the corresponding subvariety.

3.2 The Mabuchi functional

We begin by considering a general compact Kahler manifold (V, ω_{0}) of complex dimension n. The Mabuchi functional M is a real-valued function on the set of Kahler metrics in the same Kahler class [ω_{0}], defined up to the addition of an overall constant [14]. The functional is defined through the formula for its variation at a metric ω = ω_{0} +2i\overline{∂}∂ψ with respect to an infinitesimal change δψ in the Kahler potential:

δM = − ∫_{V} (S − a)δψ \frac{ω^{n}}{n!} , (3.2.1)

where S is the scalar curvature of ω and a is the average value of the scalar curvature, a topological invariant of the data. Thus δM is not changed if one adds a constant to δψ, and so depends only on the variation of the metric. From a more abstract point of view, the formula (3.2.1) defines a 1-form on the space of Kahler metrics in the cohomology class and one checks that this 1-form is closed, hence is the derivative of a function M. The Mabuchi functional is related to the Futaki invariant in the following way. Let v be a holomorphic vector field on V and the (0, 1) component of i_{v}(ω) be −\overline{∂}f as in (2.2). Then the infinitesimal change in the Kahler potential generated by the action of v is just the imaginary part of f. It follows that the infinitesimal change in the Mabuchi functional, under the action of v is given by the imaginary part of the Futaki invariant. That is, if σ_{t} : V → V is the 1-parameter group of holomorphic automorphisms of V generated by v,

\frac{d}{dt}M(σ_{t}^{∗}(ω)) = Im(ν(v)). (3.2.2)

We also want to recall the definition of another functional I_{V} on the set of Kahler potentials. This is defined in the same manner as above by the formula

δI_{V} = 2 ∫_{V} δψ \frac{ω^{n}}{n!} , (3.2.3)

see [6]. In this case the variation does not vanish when δψ is constant, so I is defined on the potentials rather than the metrics.

Now suppose that D is a positive divisor in V representing c_{1}(V). Thus there is a meromorphic n-form χ on V with no zeros and with a simple pole along D. We assume that D is a sum of smooth subvarieties D_{r}, meeting with normal crossings in V. Thus there are functionals I_{D_{r}} defined on the restriction of potentials to D_{r} and we set I_{D} = ∑ I_{D_{r}}. Given a Kahler metric ω we define a function ν on V by ν = |χ|^{−2} so

C_{n}ω^{n} = ν χ ∧ \overline{χ},

for a universal numerical factor C_{n} which will not be important.

Proposition 3.2.4. For any metric ω = ω_{0} + 2i\overline{∂}∂ψ on V

M(ω) = −I_{D}(ψ) + aI_{V} (ψ) + ∫_{V} log ν \frac{ω^{n}}{n!} .

Note here that the combination I_{D}(ψ) − aI_{V} (ψ) is unchanged if ψ is changed by a constant, so it is well-defined on the Kahler metrics, rather than potentials. In the formula of the Proposition both sides are well-defined up to overall constants.

To prove the Proposition, we write

S \frac{ω^{n}}{n!} = \frac{2}{(n − 1)!}ρ ∧ ω^{n−1},

where ρ is the Ricci form. We have an equation of currents

ρ = D − i\overline{∂}∂ log ν,

so

δM = −2 ∫_{D} δψ \frac{ω^{n−1}}{(n − 1)!} + 2 ∫_{V} i\overline{∂}∂ log ν δψ \frac{ω^{n−1}}{(n − 1)!} + 2a ∫_{V} δψ \frac{ω^{n}}{n!} .

Hence

δM = δ(−I_{D}(ψ) + aI_{V} (ψ)) + 2 ∫_{V} i\overline{∂}∂ log ν δψ \frac{ω^{n−1}}{(n − 1)!}.

Integrating by parts, this gives

δM = δ(−I_{D}(ψ) + aI_{V} (ψ)) + n ∫_{V} log ν(δω)\frac{ω^{n−1}}{n!} .

On the other hand, differentiating the equation C_{n}ω^{n} = νχ∧\overline{χ}, we have

n(δω)ω^{n−1} = C_{n}^{−1}δν χ ∧ \overline{χ},

and

δ (∫_{V} ν log νχ ∧ \overline{χ}) = ∫_{V} (log ν + 1)δν χ ∧ \overline{χ}.

The contribution

∫_{V} δν χ ∧ \overline{χ}

is

C_{n}^{−1}δ ∫_{V} ω^{n},

which vanishes; so we have

δ ∫_{V} ν log ν χ ∧ \overline{χ} = C_{n} ∫_{V} log ν(δω)ω^{n−1},

and hence

δM = δ (−I_{D}(ψ) + aI_{V} (ψ) + ∫_{V} log ν \frac{ω^{n}}{n!} ) ,

which completes the proof.

Proposition (3.2.4) is related to a formula of Chen [4]. Indeed one gets a similar formula given any current representing c_{1}(V): in Chen’s case this is the Ricci form of some fixed metric while in our case it is the divisor D.

We now specialise to the case when V is a toric variety, as before. Up to scale, there is a unique T -invariant meromorphic n-form χ on V given in our co-ordinates z_{i} by χ = dz_{1}dz_{2} … dz_{n}. This has a simple pole along the divisor D = m^{−1}(∂P) and the constant a is

a = \frac{Vol(D)}{Vol(V)}. (3.2.5)

We define a measure dσ on ∂P as follows. On the face defined by the equation h_{r}(x) = c_{r} we let dσ be the constant (n − 1)-form such that dh_{r} ∧ dσ is, up to sign, the standard Euclidean volume form dμ. For example, if P is a polyhedron in R^{2} and an edge of P has slope p/q, where p, q are coprime integers, then the measure on this edge is given by

dσ = \frac{|dx_{2}|}{p} = \frac{|dx_{1}|}{q}.

Now let ω_{0} be some invariant metric on V and let ϕ_{0} be a Kahler potential defining ω_{0} in our standard co-ordinates on the open T^{c}-orbit, as in 3.1. Any other invariant metric ω_{0} +i\overline{∂}∂ψ is given by a Kahler potential ϕ = ϕ_{0} + ψ on the open orbit, which corresponds to a symplectic potential u on P.

Lemma 3.2.6.

(1)  I_{V} (ψ) = −(2π)^{n} ∫_{P} udμ;
(2)  I_{D}(ψ) = −(2π)^{n} ∫_{∂P} u dσ.

Notice here that I_{V} and I_{D} are a priori defined only up to an overall constant, and this is the way in which the formulae should be read. However the expressions on the right-hand side of the formulae have no such ambiguity; thus in this toric situation there is a natural way to normalise the functionals.

To prove the Lemma we recall first that the push forward of the volume form under the moment map m is (2π)^{n} times the standard volume form dμ on R^{n}. Now recall that the symplectic potential u is defined by

u(\underline{x}) = ∑ x_{i}ξ_{i} − ϕ(\underline{ξ}), (3.2.7) where \underline{ξ} = \underline{ξ}(\underline{x}) is implicity defined by the equation \frac{∂ϕ}{∂ξ_{i}} = x_{i}. Consider an infinitesimal variation ϕ̃ in the Kahler potential, leading to variations ˜\underline{ξ} and ũ in \underline{ξ} and u. Differentiating (3.2.7), we have

ũ(\underline{x}) = ∑ x_{i}ξ̃_{i} − ϕ̃(ξ_{i}) − ∑ \frac{∂ϕ}{∂ξ_{i}} ξ̃_{k},

and the defining equation for \underline{ξ} gives simply ũ(\underline{x}) = −ϕ̃(\underline{ξ}). Thus the variation in the functional I_{V}, which is by definition,

δI_{V} = ∫_{V} ϕ̃\frac{ω^{n}}{n!} ,

can be written as

δI_{V} = −(2π)^{n} ∫_{P} ũdμ,

by the property of the push-forward measure. This is the same as the variation in

−(2π)^{n} ∫_{P} udμ,

thus proving item (1) of the Lemma. Item (2) follows in just the same way, when one replaces P by its codimension 1-faces.

Proposition 3.2.8. The Mabuchi functional is given by M(ω) = (2π)^{n}F_{a}(u) where

F_{a}(u) = − ∫_{P} log det(u_{ij}) + ∫_{∂P} udσ − a ∫_{P} udμ.

Given the preceding Lemma, this is just a matter of transforming the term

∫_{V} log ν \frac{ω^{n}}{n!}

in (3.2.4) into the symplectic setting. Differentiating the equations defining \underline{ξ}(\underline{x}) and u we find that the Hessian u_{ij} of u is the inverse of the Hessian \frac{∂^{2}ϕ}{∂ξ_{i}∂ξ_{j}} of ϕ (at the points \underline{x} and \underline{ξ}(\underline{x}) respectively). Thus the function log ν is log ν = log det(ϕ_{ij}) = − log det(u_{ij}), and

∫_{V} log ν \frac{ω^{n}}{n!} = −(2π)^{n} ∫_{P} log det(u_{ij})dμ

by the property of the push-forward measure.

Notice that the constant a, given by (3.2.5), can also be written as

a = \frac{Vol(∂P, dσ)}{Vol(P, dμ)}.

Thus a is fixed so that F_{a}(u) is unchanged if one adds a constant to u. In a similar vein, recall from Section 2 that the Futaki invariant gives, in our situation, a linear map from the Lie algebra of the holomorphic automorphism group T^{c} to C. This Lie algebra can be represented by the R-linear complex valued functions on P ⊂ R^{n}.

Lemma 3.2.9. The Futaki invariant maps a linear function f to

∫_{∂P} fdσ − a ∫_{P} fdμ.

This follows immediately from (3.2.8) and the relation (3.2.2) between the Mabuchi functional and the Futaki invariant. Thus the toric variety has Futaki invariant zero precisely when the centre of mass of P in R^{n}, with the measure adμ coincides with that of ∂P, with the measure dσ, and in this case the functional F_{a}(u) is unchanged if one adds any affine-linear function to u.

3.3 General properties of the functional

In this section we will consider a more general functional F_{A} defined on suitable convex functions u on P, with

F_{A}(u) = − ∫_{P} log det(u_{ij}) + L_{A}(u), (3.3.1)

where A is a bounded function on P and

L_{A}(u) = ∫_{∂P} udσ − ∫_{P} Audμ. (3.3.2)

From the point of view of toric varieties it is natural to consider the functions u in the space S, which correspond to smooth Kahler metrics. But it also useful and interesting to extend the domain to include more general convex functions. Let C_{∞} denote the set of continuous convex functions on \overline{P} which are smooth in the interior. Clearly L_{A} is welldefined on C_{∞}, but the situation for the nonlinear term is not so clear. For any function u in C_{∞} let log^{+} det(u_{ij}) be max(0, log det(u_{ij})), so in particular log^{+} det(u_{ij}) is defined to be zero if det(u_{ij}) vanishes.

Lemma 3.3.3. For any u in C_{∞}, log^{+} det u_{ij} is integrable on P.

Given this, we can define F_{A}(u) for any u ∈ C_{∞}, taking values in (−∞, ∞].

Proposition 3.3.4. Suppose A ∈ L^{∞}(P) and v ∈ S satisfies the equation

∑ \frac{∂^{2}v^{ij}}{∂x_{i}∂x_{j}} = −A

in P. Then v is an absolute minimiser for F_{A} on C_{∞} i.e., for any u ∈ C_{∞}, F_{A}(u) ≥ F_{A}(v) .

The proofs of (3.3.3) and (3.3.4) hinge on the following Lemma. Note first that for any u ∈ S the function (u^{,ij})_{,ij} is bounded on P: this is clear from its identification with the scalar curvature on the toric variety, or by direct calculation.

Lemma 3.3.5. Suppose that u ∈ S and f ∈ C_{∞}. Then u^{,ij}f_{,ij} is integrable on P and

∫_{P} u^{,ij}f_{,ij}dμ = ∫_{P} (u^{,ij})_{,ij}fdμ − ∫_{∂P} fdσ.

For δ > 0 let P_{δ} be the interior polytope with faces parallel to those of P and separated by a distance δ. The function f is smooth over the closure of P_{δ} so we can integrate by parts

∫_{P_{δ}} (u^{,ij}f_{,ij} − (u^{,ij})_{,ij}f) dμ = ∫_{∂P_{δ}} u_{,i}^{,ij}f + u^{,ij}f_{,i}. (3.3.6)

Here the integrand on the right-hand side is written as a vector field, which defines an (n − 1)-form by contraction with the volume form dμ. We want to show that

∫_{∂P_{δ}} u_{,i}^{,ij}f → ∫_{∂P} fdσ, (3.3.7)

and

∫_{∂P_{δ}} u^{ij}f_{i} → 0, (3.3.8)

as δ → 0. Consider the second term. Let x be a point of ∂P_{δ}, in the interior of an (n − 1)-dimensional face, and v = v(x) be the vector v^{i} = u^{,ij}ν_{j} where ν is the unit normal to the face containing x. Thus the integrand in (3.3.8) can be written as ∇_{v}f; the derivative of f at x in the direction v. We let y = y(x) be the closest point to x on the intersection of the ray {x + tv : t > 0} and the boundary ∂P. We claim that there are constants c_{1}, c_{2} such that for all small δ and all x,

|v| ≤ c_{1}|y − x| ; |y − x| ≤ c_{2}δ. (3.3.9)

We assume this for the moment. The convexity of f implies that

|∇_{v}f| ≤ c_{1}|f(y) − f(x)|,

using the first inequality of (3.3.9). So

|∫_{∂P_{δ}} u^{,ij}f_{,i}| ≤ c_{1} ∫_{∂P_{δ}} |f(y(x)) − f(x)|
≤ c_{1}Vol(∂P_{δ}) max_{x∈∂P_{δ}}|f(y(x)) − f(x)|.

Then the second inequality of (3.3.9), and the fact that f is uniformly continous in \overline{P} implies the desired result (3.3.8). To prove (3.3.9) we consider first the model case when u = ∑ x_{i} log x_{i} on the region x_{1}, … x_{n} > 0 and x is a point (δ, x_{2}, …, x_{n}). Then u^{ij} is the diagonal matrix diag(δ, x_{2}, …, x_{n}) and the normal is ν = (1, 0, …, 0). Thus the vector v is (δ, 0, …, 0) so y = (0, x_{2}, …, x_{n}) and

|v| = |y − x| = δ.

The proof in the general case is straightforward, checking that changing u by adding a smooth function makes a small change to this model case.

We now turn to (3.3.7). Here we want to show that the (n − 1)- form u_{i}^{ij} converges to the measure dσ (defined in the obvious way) on ∂P_{δ}. Exact equality holds for the model case and it is straightforward to deduce the assertion from this.

Corollary 3.3.10. The same assertion as in Lemma (3.3.5) holds when f is the difference of two functions in C_{∞}.

This is obvious by linearity.

The other ingredient in the proofs of (3.3.3) and (3.3.4) is the convexity of the function − log det J on the space of positive definite matrices. Thus if j(t) = − log det((1 − t)J_{0} + tJ_{1}) we have

j′(0) = −Tr(J_{0}^{−1}(J_{1} − J_{0})), j′′(0) = Tr(J_{0}^{−1}(J_{1} − J_{0}))^{2}.

We now give the proof of (3.3.3). First, by considering the function u+|x|^{2} we may reduce to the case where det(u_{ij}) ≥ 1 so log^{+} det(u_{ij}) = log det(u_{ij}). We choose any function v ∈ S and write f = u − v. Then we apply (3.3.10) to f, so that v^{ij}f_{ij} is integrable on P. However the convexity of − log det implies that

log det(u_{ij}) = log det(v_{ij} + f_{ij}) ≤ log det(v_{ij}) + v^{ij}f_{ij}.

So log det(u_{ij}) is integrable.

To prove (3.3.4), the convexity of − log det immediately implies that the functional F_{A} is convex on C∞. Let u = v + f as above and let g(t) = F_{A}(v + tf). Thus g is a convex function on [0, 1]. We claim that g is differentiable at t = 0. For

\frac{∂}{∂t} log det(v_{ij} + tf_{ij}) = v^{ij}f_{ij}

which is integrable over P, thus the result follows from the standard test for differentiation under the integral and

g′(0) = − ∫_{P} ((v^{ij})_{ij} + A)fdμ,

which vanishes by the hypothesis of (3.3.4). Now the convexity of g implies that g(1) ≥ g(0), as required.

Proposition 3.3.11. For any A ∈ L^{∞} the infimum of F_{A} on S is equal to the infimum of F_{A} on C_{∞}.

Since S ⊂ C_{∞} we need to show that for any v ∈ C_{∞} and ϵ > 0 there is a u ∈ S with F_{A}(u) ≤ F_{A}(v) + ϵ. First we define v_{η}, for η < 1, by v_{η}(x) = v(ηx). Thus v_{η} is smooth on \overline{P} and it is easy to see that F_{A}(v_{η}) → F_{A}(v) as η → 1. Thus it suffices to consider the case when v itself is smooth on \overline{P}. We can choose a family of smooth functions l_{δ} on [0, ∞), for δ > 0, with l_{δ}(x) = x log x if x ≤ δ, l′′_{δ} ≥ 0 and 0 ≥ |l_{δ}(x)| ≥ 2δ log δ for all x. Then we can define a function U_{δ} on P by

U_{δ} = ∑_{k=1}^{r} l_{δ}(h_{k}(x) − c_{k}).

Thus U_{δ} ∈ S but ∥U_{δ}∥_{L^{∞}} ≤ 2rδ log(δ^{−1}). So

F_{A}(v + U_{δ}) − F_{A}(v) = O(rδ log(δ^{−1})) → 0

as δ → 0.

4. K-stability for toric varieties

4.1 Basic theory

In this section we take up the algebro-geometric point of view. Our goal is to give a criterion for a toric variety to be K-stable. We begin by recalling the basic facts about the algebraic geometry of toric varieties.

Let P be any integral polytope in R^{n}. For positive integers k let

B_{k,P} = k\overline{P} ∩ Z^{n}

and let U_{k,P} be a complex vector space with a basis labelled by the elements of B_{k,P}. This space carries an obvious representation of the n-torus T^{c}, since the integer points label the one-dimensional representations. The basic fact we need is:

Proposition 4.1.1. There is a projective toric variety V_{P} associated to P with a positive, T^{c}-equivariant line bundle L → V_{P} such that

H^{0}(V_{P}; L^{k}) = U_{k,P}

as representations of T^{c}.

To define V_{P} one can just take the closure of a generic T^{c}-orbit in the projectivization of U_{1,P}^{∗}. Note that V_{P} will only be a smooth variety if P satisfies the “Delzant” condition, but this will not be important for us.

By Proposition (4.1.1), the dimension of the space H^{0}(V_{P} , L^{k}) is given by counting the number of lattice points, N(kP) say, in k\overline{P}. We need a simple result describing the asymptotics of N(kP) as k tends to infinity. The most basic fact is that

N(kP) = k^{n}Vol(P) + O(k^{n−1}), (4.1.2)

the refinement we need is:

Proposition 4.1.3. The number N(kP) of lattice points in k\overline{P} satisfies

N(kP) = k^{n}Vol(P) + \frac{k^{n−1}}{2} Vol(∂P) + O(k^{n−2})

as k → ∞. Here the volume of ∂P is computed using the measure dσ.

Proposition (4.1.3) is almost a standard fact—compare the much more general results of [13], for example. We give a proof, for completeness, in the Appendix.

4.2 Constructing degenerations

Continuing with notation as above, suppose that f is a convex, rational, piecewise-linear, function on R^{n}. That is

f = max(λ_{1}, … λ_{p}),

where λ_{r} are affine-linear functions with rational co-efficients. Fix an integer R such that f ≤ R on \overline{P}. Given this data, we define a polytope Q ⊂ R^{n+1} = R^{n} × R to be

Q = {(x, t) : x ∈ P, 0 < t < R − f(x)}.

The polytope kQ is defined by integral equations provided k is a multiple of a suitable denominator d. Thus, taking k = d, we get an (n + 1)- dimensional toric variety W with a line bundle L → W. The face \overline{Q} ∩ (R^{n} × \{0\}) = P × \{0\} is a copy of P so we get a natural embedding of i : V → W such that the restriction of L to V is isomorphic to L^{d}. We write the torus T^{n+1} acting on W as T^{n} × C^{∗}, where the T^{n} action restricts in the obvious way to i(V). The result we need can be summed up as follows.

Proposition 4.2.1. There is a C^{∗}-equivariant map p : W → CP^{1} with p^{−1}(∞) = i(V) such that the restriction of p to W \ i(V) is a test configuration for (V, L) with Futaki invariant

F_{1} = −\frac{1}{2Vol(P)} (∫_{∂P} fdσ − a ∫_{P} fdμ) ,

where a = \frac{Vol(∂P)}{Vol(P)}.

First, there is no loss in supposing that the functions λ_{r} defining f have integral co-efficients, so d = 1. We consider the sections H^{0}(L) over W. This space has a basis s_{I,i} where I is a lattice point in P ∩ Z^{n} and 0 ≤ i ≤ R − f(i), and the C^{∗} action acts with weight i on s_{I,i}. It follows that the ratios s_{I,i}/s_{I,i+1} give C^{∗}-equivariant maps from the region where they are defined (outside the common zeros of s_{I,i}, s_{I,i+1}) to CP^{1}. By rescaling the basis elements we can suppose that these maps all agree on the intersections of their domains of definition. For each point in W there is some section s_{I,i} which does not vanish, so we get a well-defined equivariant map p : W → CP^{1}. Now the set i(V) is fixed by the C^{∗} action, and the action is trivial on the restriction of L to i(V). Thus all the sections s_{I,i} for i > 0 must vanish on i(V) and so p maps i(V) to ∞ ∈ CP^{1}. We claim that there are no other points in p^{−1}(∞). First, if w is a point with a nontrivial stabiliser then by the same argument as above there is a section s_{I,i} which does not vanish at w with (I, i) in the boundary of Q but i ≠ 0. This implies that p(w) ≠ ∞. Second if w is a point in the free T^{n+1}-orbit in W then the closure of the C^{∗}-orbit of p(w) must be the whole of the image of p, and so p(w) cannot be the fixed point ∞. Now C^{∗} acts with same weight −1 on the normal bundle to i(V) and the tangent bundle to CP^{1} at ∞. It follows that the derivative of p cannot vanish on i(V), so the nearby fibres p^{−1}(t) for large t are diffeomorphic to V. However these fibres are toric varieties and so by the rigidity of toric varieties we see that p^{−1}(t) is in fact isomorphic to V for large t, and hence for all nonzero t because of the group action.

The only remaining task is to compute the Futaki invariant of this test configuration. The divisors V_{0} = p^{−1}(0) and i(V) are each defined by the vanishing of sections σ_{0}, σ_{1} of the line bundle p^{∗}(O(1)) over W. Hence, at least when k is large enough, we have exact sequences:

0 → H^{0}(W, L^{k}(−1)) → H^{0}(W, L^{k}) → H^{0}(V_{0}, L^{k}) → 0
0 → H^{0}(W, L^{k}(−1)) → H^{0}(W, L^{k}) → H^{0}(i(V), L^{k}) → 0

where the inclusion maps are multiplication by σ_{0}, σ_{1}. From this we see first that the dimension d_{k} of H^{0}(V_{0}, L^{k}) is the same as that of H^{0}(V, L^{k}). The C^{∗}-action acts with weight 0 on σ_{0} and weight 1 on σ_{1} so it follows that the weight w_{k} of the action on Λ^{d_{k}}H^{0}(V_{0}, L^{k}) is given by the weight of the action on Λ^{d_{k}}H^{0}(i(V), L^{k}) plus the dimension of H^{0}(W, L^{k}(−1)). But the action on H^{0}(i(V), L^{k}) is trivial, so we have

d_{k} = dim H^{0}(W, L^{k}(−1)) = dim H^{0}(W, L^{k}) − dim H^{0}(i(V), L^{k}). (4.2.2)

Thus

d_{k} = N(kP), w_{k} = N(kQ) − N(kP).

For the first term we immediately have from (4.1.3):

d_{k} = k^{n}Vol(P) + \frac{k^{n−1}}{2} Vol(∂P) + O(k^{n−2}).

For the second term, we divide the boundary of Q into three parts:

(1) points (x, 0) for x ∈ \overline{P}; (2) points (x, R − f(x)) for x ∈ \overline{P};

(3) points (x, t) with x ∈ ∂P and 0 < t < R − f(x).

Clearly the number of lattice points in each of the first two parts is just N(kP). The number ν_{k} of lattice points in the third class is estimated, applying (4.1.2) to each face, by the volume so

ν_{k} = k^{n} ∫_{∂P} R − fdσ + O(k^{n−1}).

Then, applying (4.1.3) to Q, we have

w_{k} = k^{n+1} ∫_{P} (R − f)dμ + \frac{k^{n}}{2} ∫_{∂P} (R − f)dσ. (4.2.3)

The formulae (4.2.2) and (4.2.3) imply that

\frac{w_{k}}{kd_{k}} = \frac{A}{C} + \frac{1}{2kC^{2}} (BC − AD) + O(k^{−2}),

where

A = ∫_{P} R − fdμ;
B = ∫_{∂P} R − fdσ;
C = Vol(P);
D = Vol(∂P).

By definition, the Futaki invariant is the co-efficient (BC − AD)/2C^{2} of the k^{−1} term, which is

−\frac{1}{2Vol(P)} (∫_{∂P} fdσ − \frac{Vol(∂P)}{Vol(P)} ∫_{P} fdμ) ,

as asserted in the Proposition.

5. The main argument

5.1 Reduction to the linear functional

We will now relate the nonlinear functional F_{A} to its linear part L_{A}. Here we can take any bounded function A on P which satisfies the moment condition, for each affine-linear function f,

∫_{P} fAdμ = ∫_{∂P} fdσ. (5.1.1)

When this condition holds the functionals L_{A}(u), F_{A}(u) are not changed if one adds an affine-linear function to u, and it is convenient to work with functions which are normalised in the following way. We fix a point p ∈ P and say u is normalised if u ≥ 0 and u(p) = 0. The main result is:

Proposition 5.1.2. Suppose there is a constant λ > 0 such that for all normalised functions u ∈ C_{∞} we have

L_{A}(u) ≥ λ ∫_{∂P} u dσ.

Then F_{A} is bounded below on C_{∞}.

The proof of this is surprisingly easy. First, observe the following:

Lemma 5.1.3. There is a constant C such that

∫_{P} u dμ ≤ C ∫_{∂P} u dσ

for all normalised functions u.

This is clear when one works in polar co-ordinates centred at the point p. Now fix some arbitrary smooth invariant metric on V, thus some function v in S. We define a function B on P by

B = − ∑ \frac{∂^{2}v^{ij}}{∂x_{i}∂x_{j}} .

We know that B represents the scalar curvature on V, hence is certainly bounded (a fact we could readily verify directly). According to Proposition (3.3.4), the function v gives an absolute minimum for F_{B} on C_{∞}, so for all functions u ∈ C_{∞}

− ∫_{P} log det(u_{ij})dμ + L_{B}(u) ≥ C,

where C = F_{B}(v). Now replace u in this inequality by ru, for a constant r > 0. The inequality becomes

− ∫_{P} log det(u_{ij})dμ + rL_{B}(u) ≥ C_{r}, (5.1.4)

where C_{r} = C + log rVol(P). The hypothesis of the Proposition gives

L_{A}(u) ≥ λ ∫_{∂P} u dσ, (5.1.5)

for all normalised u, and Lemma (5.1.3) implies that

|L_{A}(u) − L_{B}(u)| ≤ C∥A − B∥_{L^{∞}} ∫_{∂P} udσ. (5.1.6)

Combining (5.1.5) and (5.1.6) we have

|L_{A}(u) − L_{B}(u)| ≤ C′L_{A}(u), (5.1.7)

where C′ = λ^{−1}C∥A − B∥_{L^{∞}}. Thus

L_{B}(u) ≤ (C′ + 1)L_{A}(u)

and (5.1.4) gives

− ∫_{P} log det(u_{ij})dμ + r(C′ + 1)L_{A}(u) ≥ C_{r}.

Finally, choose r = r_{0} = (C′ + 1)^{−1}, to get F_{A}(u) ≥ C_{r_{0}}.

The point here is that C_{∞} is preserved by multiplication by positive scalars. The same is not true for S and this is the only reason for introducing the larger domain for F_{A} in Section (3.3). We use the same idea to prove

Proposition 5.1.8. Under the same hypothesis as (5.1.2), there is constant K such that if u^{(α)} is any sequence of normalised functions in C_{∞} which is a minimising sequence for F_{A} on C_{∞} then

∫_{∂P} u^{(α)} ≤ K,

for large enough α.

To prove this we consider the scaling of any function v in C_{∞};

F_{A}(rv) = − ∫_{P} log det(v_{ij})dμ − log rVol(P) + rL_{A}(v).

Fixing v, this is a function of r which is minimised when r = r_{1} = L_{A}(u)/Vol(P), and L_{A}(r_{1}u) = Vol(P). Given any minimising sequence of normalised functions u^{(α)} we can write u^{(α)} = r_{α}v^{(α)} where

L_{A}(v^{(α)}) = Vol(P).

Then

F_{A}(u^{(α)}) = F_{A}(v^{(α)}) + Vol(P)(r_{α} − log r_{α} − 1).

It follows that r_{α} − log r_{α} − 1 → 0 as α → ∞, and thence that r_{α} → 1. This means that L_{A}(u^{(α)}) → Vol(P) and then our result follows from (5.1.3).

5.2 Compactness

Here we study the linear functional L_{A}, for an arbitrary bounded function A on P. Our goal is to obtain a criterion under which the bound in (5.1.2) of the previous subsection will hold. Let P^{∗} be the union of P and its codimension-one faces.

Definition 5.2.1. The set C_{1} is the set of positive convex functions f on P^{∗} such that

∫_{∂P} f dσ < ∞.

Note that the integral of any function in C_{1} over the boundary makes sense, even though f is not defined on the whole boundary, because the measure dσ is supported on the codimension-one faces. The same proof as for Lemma (5.1.3) shows that any f ∈ C_{1} is integrable on P, so L_{A}(f) is defined. Our main result is:

Proposition 5.2.2. Either there is a λ > 0 such that

L_{A}(f) ≥ λ ∫_{∂P} fdσ

for all normalised functions f in C_{∞} or there is a function f in C_{1} which is not affine linear and such that L_{A}(f) ≤ 0.

The general scheme of proof is a compactness argument using gradient bounds derived from the following Lemma. For any convex function f on the polytope P and any point x in P we let D_{x}(f) be the supremum of the slope of supporting hyperplanes to f at x; i.e.,

D_{x}(f) = sup{|λ| : f(y) ≥ λ(y − x) + f(x) for all y ∈ P }.

Lemma 5.2.3. There is a universal constant κ such that for any positive convex function f on P with

∫_{P} fdμ < ∞

and for any point x ∈ P we have

D_{x}f ≤ κd_{x}^{−(n+1)} ∫_{P} fdμ,

where d_{x} is the distance from x to the boundary of P.

To prove this, we may suppose that x is the origin and that f(y) ≥ Dy_{1} + f(0), where y_{1} denotes the first component of y ∈ P ⊂ R^{n}. Let B^{+}(d_{0}) denote the intersection of the ball of radius d_{0} with the half-space {y_{1} ≥ 0} so B^{+}(d_{0}) ⊂ P and f ≥ Dy_{1} on B^{+}(d_{0}). Thus

∫_{P} fdμ ≥ D ∫_{B^{+}(d_{0})} y_{1}dμ = Dκ^{−1}d_{0}^{n+1},

for a constant κ = (n + 1)/2Vol(B^{n−1}).

Lemma 5.2.4. For any convex function f on P and any two points x, y in P:

|f(x) − f(y)| ≤ max(D_{x}(f), D_{y}(f)) |x − y|.

This is clear in the one-dimensional case and the general case reduces to this by considering restriction to the line segment between x and y.

Now for small d > 0 let P_{d} denote the set of points in P such that d_{x} ≥ d. Clearly

min_{P_{d}} f ≤ \frac{1}{Vol(P_{d})} ∫_{P} fdμ,

and Lemmas (5.2.3) and (5.2.4) imply that

max_{P_{d}} f ≤ min_{P_{d}} f + Rκd^{−(n+1)} ∫_{P} fdμ,

where R is the diameter of P. Thus a bound on the integral of f over P gives a Lipschitz bound on f in any interior region, and Ascoli-Arzela implies:

Corollary 5.2.5. Any sequence of positive convex functions f_{n} on P with

∫_{P} f_{n}dμ ≤ C

has a subsequence which converges uniformly over compact subsets of P.

Proposition 5.2.6. Suppose that f_{n} is a sequence of normalised functions in C_{∞} with

∫_{∂P} f_{n} ≤ C.

Then there is a subsequence which converges, uniformly over compact subsets of P, to a convex function which has a continuous extension to a function f^{∗} on P^{∗}, defining an element of C_{1} with

∫_{∂P} f^{∗}dσ ≤ lim inf ∫_{∂P} f_{n}dσ.

In the proof we will assume our base point p ∈ P is the origin.

First, Lemma (5.1.3) yields a bound on the integrals of the f_{n} over P, so we are in the situation of Corollary (5.2.5) and we can suppose that the f_{n} converge uniformly on compact subsets of P to a limit f. Next, we can apply Corollary (5.2.5), replacing P by one of its codimension 1 faces, so we may suppose that the f_{n} converge pointwise over all of P^{∗}. We must beware however because we may not be able to take f^{∗} = lim f_{n}. Instead, we proceed as follows. It is a simple fact that any convex function on an open interval has a well-defined limit, possibly infinite, at each end point. We apply this to the restriction of f to the rays through the origin, so for any point z in a codimension-1 face of the boundary of P we define f^{∗}(z) ∈ [0, ∞] to be the limit of f(tz) as t tends to 1 from below. For each η < 1 consider the set η∂P ⊂ P, with the obvious measure induced by dσ. It is clear that the integral of f_{n} over η∂P is at most C so Lemmas (5.2.3), (5.2.4) give a Lipschitz bound on the restriction of f to ηK for any compact subset K of a codimension face of P. It follows then that f^{∗}(z) is the limit of f(x_{n}) for any sequence x_{n} in P converging to z.

We claim now that, for z in a codimension-one face,

f^{∗}(z) ≤ lim f_{n}(z). (5.2.7)

(This statement includes the assertion that f^{∗}(z) is finite.) Indeed, since the f_{n} are convex and normalised,

f_{n}(tz) ≤ tf_{n}(z),

so taking limits as n → ∞

f(tz) ≤ t lim f_{n}(z),

and now (5.2.7) follows by taking limits as t → 1. Thus f^{∗} is a continuous function on P^{∗}, convex on P, and hence convex on P^{∗} by continuity. We also have

∫_{∂P} f^{∗}dσ ≤ ∫_{∂P} lim f_{n}dσ ≤ lim inf ∫_{∂P} f_{n}dσ,

using (5.2.7) for the first inequality and Fatou’s Lemma for the second. This completes the proof.

We can now prove our main result (5.2.2). Suppose the first alternative does not hold, so there is a sequence of normalised functions f_{n} ∈ C_{∞} with

∫_{∂P} f_{n}dσ = 1

and L_{A}(f_{n}) → 0 as n → ∞; that is

∫_{P} Af_{n}dμ → 1.

We apply Proposition (5.2.6), so we can suppose that there is a function f^{∗} in C_{1} such that f_{n} converges to f^{∗}, uniformly on compact subsets of P. For η < 1, we consider the “collar” region P \ ηP. It is clear that there is a constant c such that for any normalised convex function g

∫_{P \ηP} gdμ ≤ c(1 − η) ∫_{∂P} gdσ.

Thus

∫_{P \ηP} Af_{n}dμ ≤ c∥A∥_{L^{∞}}(1 − η).

It follows from this, and the uniform convergence over compact subsets, that

∫_{P} Af^{∗}dμ = lim ∫_{P} Af_{n}dμ = 1.

On the other hand we know that

∫_{∂P} fdσ ≤ lim ∫_{∂P} f_{n}dσ = 1,

so we have L_{A}(f^{∗}) ≤ 0, as desired.

Recall that a piecewise linear convex function on R^{n} is a function of the form

max (g_{1}, …, g_{N})

where the g_{j} are affine-linear functions.

Proposition 5.2.8. For any f ∈ C_{1} there is a sequence of piecewise linear convex functions f_{n} such that

∫_{∂P} |f_{n} − f|dσ → 0; ∫_{P} |f_{n} − f|dμ → 0,

as n → ∞.

We will construct f_{n} such that 0 ≤ f_{n} ≤ f everywhere, so the conditions we need are

∫_{∂P} f − f_{n} → 0, ∫_{P} f − f_{n} → 0.

First note that it suffices to satisfy these conditions separately, since if the sequence f_{n}^{∂P} satisfies the first condition, with f_{n}^{∂P} ≤ f, and the sequence f_{n}^{P} satisfies the second condition, with f_{n}^{P} ≤ f; the sequence of functions f_{n} = max(f_{n}^{P} , f_{n}^{∂P}), are still PL convex functions and satisfies both conditions. Now consider a decomposition

P = ∪_{i}Σ_{i} where the Σ_{i} are the intersections of P with a standard mesh of closed cubes in R^{n} with side length δ. Let σ_{i} ∈ Σ_{i} be a point where f is minimal. Let \underline{f}_{δ} be the function on P with \underline{f}_{δ}(x) = f(σ_{i}) for x in the interior Σ_{i}, defined arbitrarily on the boundaries. The integral of this function corresponds to the usual “lower sum” in the theory of the Riemann integral, and it is clear using (5.2.3) and the continuity of f in P that

∫_{P} f − \underline{f}_{δ}dμ → 0

as δ → 0.

The convex sets in R^{n+1} = R^{n} × R

Σ_{i} × (−∞, f(σ)), {(x, t) ∈ P × R : t ≥ f(x)},

have disjoint interiors. It follows from a standard separation theorem that there is a hyperplane in R^{n+1} such that the two sets lie in different closed half-spaces. In other words, this means that there is an affinelinear function g_{i} such that

f(x) ≥ g_{i}(x)

for all x ∈ P, and

g_{i}(x) ≥ f(σ_{i})

for all x ∈ Σ_{i}. Thus if we define g to be max g_{i} we have f ≥ g ≥ \underline{f}_{δ} and so

∫ f − g ≤ ∫ f − \underline{f}_{δ}

and ∫ f − g → 0 with δ.

To handle the boundary integral we first consider the restriction of f to η∂P for η < 1. We may consider this as a function f_{η} on ∂P in an obvious way and the dominated convergence theorem implies that f_{η} tends to f|_{∂P} in L^{1}(∂P) as η → 1. Let g be any PL function with 0 ≤ g ≤ f on P. The fact that f(0) = 0 implies that g(x) ≥ g(ηx) for all x ∈ ∂P. Thus

∫_{∂P} f − gdσ ≤ ∫_{∂P} f − f_{η}dσ + ∫_{η∂P} f − gdσ.

Since the first term can be made arbitrarily small by taking η close to 1, it suffices to show that we can find a PL function g of this kind with

∫_{η∂P} f − g dσ

as small as we please. To see this we follow the same argument as before. We subdivide η∂P into small convex pieces and use the separation theorem to find a PL function g ≤ f everywhere on P but with g bounded below by the Riemann lower sum function on η∂P. Note there is no loss in supposing g ≥ 0, as above, because we can always replace g by max(g, 0).

5.3 Extremal functions

In this subsection we analyse the extremals of the linear functional in the semi-positive case. We suppose the dimension n is 2 throughout the subsection (although some of the arguments go over to higher dimensions). We define a simple piecewise linear function on R^{2} to be a function of the form

f(x) = max(λ(x) + c, 0),

where λ : R^{2} → R is a linear function with |λ| = 1. We call the line λ(x) = −c on which f is not smooth the crease of f. Let A be a continuous, strictly positive function on \overline{P} ⊂ R^{2} satisfying the moment conditions (5.1.1). The goal of this subsection is to prove:

Proposition 5.3.1. Suppose L_{A}(u) ≥ 0 for all u ∈ C_{1} but there is a function u ∈ C_{1}, which is not affine linear, with L_{A}(u) = 0. Then there is a simple piecewise linear function f whose crease meets P and with L_{A}(f) = 0.

The proof will require a number of elementary lemmas. First note that, while u is only defined a priori on P^{∗}—the complement of the vertices in P —it is easy to see that it extends to a continous map from \overline{P} to (−∞, ∞], i.e., u(x) has a well-defined limit, finite or infinite, as x approaches a vertex. Now suppose that u ≥ 0 on \overline{P} and that u = 0 somewhere in P. Let K = u^{−1}(0) ⊂ \overline{P}. The convexity of u implies that K is a closed convex set.

Lemma 5.3.2. K is the convex hull of K ∩ ∂P.

To prove the Lemma we need to show that any point x_{0} of K lies inside the convex hull of K ∩ ∂P. Suppose not, then there is a line separating x_{0} from K ∩ ∂P. It follows that there is an affine-linear function g with g(x_{0}) > 0 and g(y) ≤ −c < 0 for all y ∈ K ∩ ∂P. For small ϵ > 0 consider the convex function

u_{ϵ}(x) = max(u(x), ϵg(x)).

A straightforward compactness argument shows that when ϵ is sufficiently small, u(y) ≥ ϵg(y) for all y ∈ ∂P. Thus u_{ϵ} = u on ∂P while u_{ϵ}(x) > u(x). It follows that L_{A}(u_{ϵ}) < L_{A}(u) (because the boundary contributions is not changed but the interior contribution is strictly smaller, since A > 0). This gives a contradiction.

Of course, adding an affine-linear function to u, we deduce an analogue of (5.3.2) for any supporting hyperplane of u. One approach to proving (5.3.1) would be to show that this implies that u can be written as an integral

u = ∫_{T} f_{t}dτ,

where T is some parameter space, dτ is a positive measure on T, and for each t the function f_{t} is a simple PL function. The hypothesis of (5.3.1) would then imply that L_{A}(f_{t}) = 0 for each t which would imply the desired result. However this line of argument seems to involve some relatively sophisticated analysis, so instead we will follow a more elementary line in the proof below.

Lemma 5.3.3. Let f be a continous function on an interval [a, b] in R and suppose a ≤ c ≤ c′ ≤ b.

(1) Suppose that f ≥ 0 on [a, b], that f is convex on the sub-intervals [a, c] and [c′, b] and that f = 0 on the interval [c, c′]. Then f is convex on [a, b].

(2) Suppose that f = μ_{1} on the segment [a, c] and f = μ_{2} on the segment [c′, b], where μ_{1}, μ_{2} are affine-linear functions. Suppose that f is convex on [c, c′] and that f ≥ μ_{1}, μ_{2} throughout [a, b]. Then f is convex on [a, b].

The proof is left to the reader.

Lemma 5.3.4. Either the conclusion of Proposition (5.3.1) is true or for each point x_{0} in P the supporting hyperplane to u at x_{0} is unique; that is, there is exactly one affine-linear function g with g(x_{0}) = u(x_{0}) and g ≤ u on P.

Suppose that there are two distinct supporting hyperplanes at x_{0}. By adding an affine-linear function to u, we may suppose that one of them is the zero hyperplane, i.e., that u(x_{0}) = 0 and u ≥ 0 on \overline{P}. Thus there is a simple piecewise linear convex function f with crease a line L through x_{0} such that u ≥ ϵf for some fixed ϵ > 0. The crease L meets ∂P in two points p, q say. The set K = u^{−1}(0) is contained in the half-space where f ≥ 0 bounded by L, and so Lemma (5.3.2) implies that p and q are both in K (because x_{0} is in K). Hence K contains the line segment L ∩ \overline{P}. Define a function v on P by v = u − ϵf. We claim that v is convex. For v is certainly convex on the intersection of \overline{P} with each of the two half-spaces defined by the crease (since it differs from u by an affine linear function on each piece); v ≥ 0 and v vanishes on the intersection L ∩ P of the two pieces. Then the convexity of v follows from (1) of Lemma (5.3.3) (with c = c′), applied to any segment in \overline{P}.

Now we write u = ϵf + v. The hypothesis of Proposition (5.3.1) implies that L_{A}(v), L_{A}(f) ≥ 0 so we must have L_{A}(v) = L_{A}(f) = 0 and the Lemma is proved.

Let us now suppose–arguing for a contradiction–that the conclusion of Proposition (5.3.1) does not hold. The conclusion of Lemmas (5.3.2) and (5.3.3) can be expressed as follows. We define an equivalence relation on points of P by saying x is related to x′ if u has the same supporting hyperplane at x and x′. Then each equivalence class is of the form K ∩ P where K ⊂ \overline{P} is the convex hull of K ∩ ∂P. It follows then that each equivalence class in P is either a line segment or intersection of a closed polygon with P. Let K_{1} ∩ P, K_{2} ∩ P be two distinct equivalence classes. Thus there are distinct affine-linear functions g_{1}, g_{2} such that u = g_{i} on K_{i}. There are open half-spaces H_{1}, H_{2} in R^{2} with the following properties:

(1) H_{1} ∩ \overline{P} , H_{2} ∩ \overline{P} are disjoint convex subsets of \overline{P}.

(2) K_{i} is contained in the closure of H_{i}.

(3) The topological boundary of H_{i} ∩ \overline{P} in \overline{P} is contained in K_{i}.

We define a function \underline{u} on \overline{P} by

\underline{u}(x) = g_{1}(x) if x ∈ H_{1} ∩ \overline{P};

= g_{2}(x) if x ∈ H_{2} ∩ \overline{P}; = u(x) otherwise.

The conditions on the half-planes, stated above, show that \underline{u} is continuous on P. We let v = u − \underline{u}.

Lemma 5.3.5. The functions \underline{u} and v are convex.

As for Lemma (5.3.4), this follows from Lemma (5.3.3), by restricting to segments in P. (Item (1) of (5.3.3) applies to v and item (2) to \underline{u}.) Clearly the functions \underline{u} and v both lie in C_{1} so, as before, the extremal condition implies:

Corollary 5.3.6. L_{A}(\underline{u}) = L_{A}(v) = 0.

Return now to to consider the decomposition of \overline{P} into closed, convex equivalence classes defined by the extremal function u. Choose any equivalence class K ∩ P and a point p in K ∩ P which does not lie in the interior of K. Thus p lies in a line segment in K with end points a, b in ∂P. We consider a point q in P, close to p but not in K. Let K_{q} be the equivalence class containing q. Clearly, if q is close enough to p it must lie on a line segment in K_{q} with end points a′, b′ where a′ and a lie on some common edge E of ∂P and b′ and b lie on another common edge F of ∂P. It may happen that a′ is equal to a or b′ is equal to b (both cannot occur since p and q are in different equivalence classes). Let I be the line segment in P with end points p, q.

Lemma 5.3.7. If the conclusion of Proposition (5.3.1) does not hold, there is a constant c_{1} > 0 such that L_{A}(f) ≥ c_{1} for all simple convex piecewise linear functions whose crease meets I.

The proof is easy: L_{A} is a continuous function on the compact set of simple piecewise-linear functions whose crease meets I.

Suppose that K_{1}, K_{2} are distinct equivalence classes as above, and each meets the line segmentI. Let d be the minimal distance between points of the sets K_{1}∩I, K_{2}∩I. Continuing with the notation as above, if we define h = max(g_{1}, g_{2}) then u ≥ h and we can write h = g_{1}+ϵf where ϵ > 0 and f is a simple piecewise-linear function. Thus L_{A}(h) = ϵL_{A}(f), since L_{A} vanishes on the affine-linear functions, and

L_{A}(h) ≥ c_{1}ϵ (5.3.8)

by Lemma (5.3.7).

Lemma 5.3.9. There is a constant c_{2} > 0 such that for any K_{1}, K_{2} as above, (\underline{u} − h)(x) ≤ c_{2}ϵd.

Assuming this Lemma we can complete the proof of (5.3.1). The Lemma implies that, for any K_{1}, K_{2}

|L_{A}(\underline{u} − h)| ≤ c_{3}ϵd,

where

c_{3} = c_{2} (∫_{P} |A|dμ + Vol(∂P, dσ)) .

Thus, assuming the conclusion of Proposition (5.3.1) is false,

L_{A}(\underline{u}) ≥ c_{1}ϵ − c_{3}dϵ

by (5.3.8). On the other hand, by (5.3.6), L_{A}(\underline{u}) = 0, so we must have d ≥ d_{I} = c_{1}/c_{3}. Now to finish the proof: we subdivide the segment I into N equal pieces by a chain of points p_{0} = p, p_{1}, … p_{N} = q and choose N so large that the distance between consecutive points p_{i}, p_{i+1} is less than d_{I}. It follows then that consecutive points must lie in the same equivalence class, hence the supporting hyperplanes at p and q are in fact equal, a contradiction.

It only remains to give the proof of Lemma (5.3.9). We suppose that neither K_{1} nor K_{2} is equal to K or K_{q}—the argument works equally well in the case of equality, except for a change of notation. Likewise we asssume that a, a′, b, b′ are all distinct—again the argument works just as well if a = a′ with a change of notation. The complement \overline{P} \ (K ∪ K_{q}) has a connected component which is a quadrilateral with two opposite edges given by segments \overline{aa′} and \overline{bb′} in the edges E, F of P, but otherwise disjoint from the the boundary of P. It follows easily that K_{1} ∩ ∂P has exactly two connected components—one contained in E and the other in F —and likewise for K_{2}. The function \underline{u} − h is supported in Q = \overline{P} \ (H_{1} ∪ H_{2}) and it follows in turn that Q is another quadrilateral with two opposite edges given by segments \overline{αα′} ⊂ \overline{aa′}, \overline{ββ′} ⊂ \overline{bb′} say, and that (after possibly interchanging K_{1}, K_{2}) the other edges \overline{αβ} and \overline{α′β′} lie in K_{1}, K_{2} respectively. This means that any point of Q lies on a line segment J in Q with one endpoint in K_{1} and one endpoint in K_{2} and with the length of J at most equal to D, where

D = max(d(α, α′), d(β, β′).

Elementary geometry shows that there is a constant C, depending only on the line segments I, E, F such that

D ≤ Cd.

To finish the proof we again use a simple lemma about convex functions of one real variable.

Lemma 5.3.10. Let f be a positive convex function on an interval [0, L] with f(0) = 0. Let

h̃(x) = max(0, λ(x − L) + f(a)),

where λ > 0 is such that f(x) ≥ h̃(x) for all x ∈ [0, L]. Then for all x in [0, L] we have

f(x) − h̃(x) ≤ \frac{λL}{4}.

We leave the proof of this to the reader. Lemma (5.3.9), with c_{2} = C/4, follows from (5.3.10) and the preceding discussion, by considering restriction to the line segment J.

6. Rationality of extremal functions

In this section we consider an integer polytope P ⊂ R^{2} with the canonical boundary measure, which satisfies the moment conditions (5.1.1). We say a simple PL function f is rational if it has the form

f(x) = ρ max(0, λ(x) + c)

where ρ is real, c is rational and λ is a linear function on R^{2} with rational co-efficients. This is the same as saying that the crease of f is a rational line. Here we prove:

Proposition 6.1. If there is a simple PL function g_{0}, whose crease meets P, such that L_{a}(g_{0}) ≤ 0 then there is a rational simple PL function f, whose crease meets P, with L_{a}(f) ≤ 0.

To prove this we consider three cases—clearly the crease of g_{0} meets ∂P in precisely two points p, q and the cases are:

(1) one or both of p, q is a vertex of P;

(2) neither p nor q is a vertex of P and the faces of P containing p and q are parallel;

(3) neither p nor q is a vertex of P and the faces of P containing p, q are not parallel.

We begin with the third case. We consider the function L_{a} on the manifold of simple PL functions. Under the hypothesis (3) this is a smooth function in a neighbourhood of g_{0}. The rational simple PL functions form a dense subset, so the desired result will obviously hold if either L_{a}(g_{0}) < 0 or L_{a}(g_{0}) = 0 but the derivative of L_{a} does not vanish at g_{0}. So we need to show that if L_{a} and its derivative both vanish at g_{0} then g_{0} is rational. Let P^{+} be the support of g_{0} in P and P^{−} be the complement in P: likewise let (∂P)^{+} be the support of g_{0} in ∂P and (∂P)^{−} be its complement in ∂P.

Lemma 6.2. The function L_{a} and its derivative both vanish at g_{0} if and only if P^{+} and (∂P)^{+} have the same mass and centre of mass: that is

∫_{(∂P)^{+}} αdσ = a ∫_{P^{+}} αdμ

for all affine-linear functions α on R^{2}.

We leave the proof to the reader. Notice that the hypothesis (5.1.1) implies that we get the same condition if we replace P^{+}, (∂P)^{+} by P^{−}, (∂P)^{−}.

We will now write down this condition more explicitly. By making a rational affine transformation of R^{2} we may suppose that the faces of P containing p and q are segments of the co-ordinate axes in R^{2} and that P lies in the positive quadrant {(x_{1}, x_{2}) : x_{1}, x_{2} > 0}. Let p = (t_{1}, 0) and the face containing p be

I_{1} = {(τ_{1}, 0) : a_{1} < τ_{1} < b_{1}},

and let q = (0, t_{2}) and the face containing q be

I_{2} = {(0, τ_{2}) : a_{2} < τ_{2} < b_{2}}.

The crease of g_{0} is the line with equation \frac{x_{1}}{t_{1}} + \frac{x_{2}}{t_{2}} = 1. Let Δ be the triangular region defined by the inequalities

x_{1} > 0, x_{2} > 0, \frac{x_{1}}{t_{1}} + \frac{x_{2}}{t_{2}} < 1.

We may suppose, interchanging P^{+} and P^{−} if necessary, that P^{+} is contained in Δ. Thus P^{+} = Q ∪ R say, where R is the region defined by the inequalities

x_{1} > 0, x_{2} > 0, \frac{x_{1}}{t_{1}} + \frac{x_{2}}{t_{2}} < 1, \frac{x_{1}}{a_{1}} + \frac{x_{2}}{a_{2}} > 1,

and Q = P^{+} \ R. The set Q is a union of rational triangles: likewise Δ = S ∪ R where S = Δ \ R is a rational triangle. The mass M(Δ) of Δ, with the measure adμ, is \frac{1}{2}at_{1}t_{2}. The moments

M_{i}(Δ) = ∫_{Δ} x_{i}adμ

are

M_{i}(Δ) = \frac{1}{6}at_{1}t_{2}t_{i}.

The mass M(P^{+}) differs from M(Δ) by M(Q) − M(S) and each of M(Q), M(S) is rational, since a is rational and Q and S are made up of rational triangles. Thus

M(P^{+}) ≅ M(Δ), M_{i}(P^{+}) ≅ M_{i}(Δ) mod Q.

We can argue in the same way for the boundary. Let the measure dσ be λ_{i}dτ_{i} on the face I_{i}, where λ_{1}, λ_{2} are positive rational numbers. Let J_{1}, J_{2} be the segments

J_{1} = {(τ_{1}, 0) : 0 < τ_{1} < t_{1}}, J_{2} = {(0, τ_{2}) : 0 < τ_{2} < t_{2}}.

The mass of J_{1} ∪ J_{2}, with the measure λ_{i}dτ_{i} on J_{i}, is

M(J_{1} ∪ J_{2}) = λ_{1}t_{1} + λ_{2}t_{2}

and the moments are

M_{i}(J_{1} ∪ J_{2}) = \frac{1}{2}λ_{i}t_{i}^{2}.

The same argument as before shows that the mass and moments of (∂P)^{+} differ from those of J_{1} ∪ J_{2} by rational numbers. Thus the condition that P^{+} and (∂P)^{+} have the same mass and moments gives

(6.3)  \frac{1}{2}at_{1}t_{2} − (λ_{1}t_{1} + λ_{2}t_{2}) = r_{0}
\frac{1}{6}at_{1}^{2}t_{2} − \frac{1}{2}λ_{1}t_{1}^{2} = r_{1}
\frac{1}{6}at_{1}t_{2}^{2} − \frac{1}{2}λ_{2}t_{2}^{2} = r_{2},

where r_{0}, r_{1}, r_{2} are rational.

Lemma 6.4. Let a, λ_{1}, λ_{2}, r_{0}, r_{1}, r_{2} be rational numbers with a, λ_{1}, λ_{2} ≠ 0. If (t_{1}, t_{2}) is a solution of the equations (6.3) then either t_{1}, t_{2} are both rational or λ_{1}r_{1} = λ_{2}r_{2} but λ_{1}t_{1} ≠ λ_{2}t_{2}.

To prove the Lemma, we may first suppose that a = 2 (multiplying the equations by an overall factor). Now put s_{i} = λ_{i}t_{i} and L = λ_{1}λ_{2}. The equations become:

(6.5)  s_{1}s_{2} = Ls_{1} + Ls_{2} + r′_{0}
\frac{1}{3}s_{1}^{2}s_{2} = \frac{1}{2}Ls_{1}^{2} + r′_{1}
\frac{1}{3}s_{2}^{2}s_{1} = \frac{1}{2}Ls_{2}^{2} + r′_{2}

for rational r′_{i}. Now multiply the first equation by s_{1}/3 and use the second equation to get:

\frac{1}{6}Ls_{1}^{2} = \frac{1}{3}Ls_{1}s_{2} + \frac{r′_{0}}{3} s_{1} − r′_{1}.

Substitute for s_{1}s_{2} using the first equation and divide by L/3 to obtain

\frac{1}{2}s_{1}^{2} = (L + \frac{r′_{0}}{L} ) s_{1} + Ls_{2} + k_{1},

where k_{1} = r′_{0} − \frac{3r′_{1}}{L} is rational. Symmetrically

\frac{1}{2}s_{2}^{2} = Ls_{1} + (L + \frac{r′_{0}}{L} ) s_{2} + k_{2},

where k_{2} = r′_{0} − \frac{3r′_{1}}{L}. Thus if we let α = L + r′_{0}/L we have

\frac{1}{2}s_{1}^{2} = αs_{1} + Ls_{2} + k_{1}
\frac{1}{2}s_{2}^{2} = Ls_{1} + αs_{2} + k_{2}
s_{1}s_{2} = Ls_{1} + Ls_{2} + r′_{0}.

Now put u = s_{1} + s_{2} and v = s_{1} − s_{2}. The equations above become

\frac{1}{2}u^{2} = (2L + α)u + (K + L(α − L))
\frac{1}{2}v^{2} = αu + (K − L(α − L))
\frac{1}{2}uv = (α − L)v + J

where K = k_{1} + k_{2}, J = k_{1} − k_{2}. The first equation is quadratic in u and we have

u = (2L + α) ± \sqrt{D},

where

D = (2L + α)^{2} + 2(K + L(α − L)).

We want to prove that \sqrt{D} is rational. Suppose the contrary; the second and third equations above yield:

(6.6)  v^{2} = 2αu + (D − 8αL − α^{2})
uv = 2(α − L)v + 2J.

So

(6.7)  v = \frac{2J}{u − 2(α − L)}.

Note that the denominator here cannot vanish, since u is not rational, by hypothesis. Now

u − 2(α − L) = (4L − α) + \sqrt{D}

so

(u − 2(α − L))^{−1} = μ ((4L − α) − \sqrt{D}) ,

where μ = (4L − a)^{2} − D is rational. Thus (6.7) implies

v = ν ((4L − α) − \sqrt{D}) ,

where ν = 2Jμ ∈ Q. This gives

v^{2} = ν^{2} ((4L − α)^{2} + D − 2(4L − α)\sqrt{D}) ,

whereas (6.6) gives

v^{2} = 2α ((2L + α) + \sqrt{D}) + (D − 8αL − α^{2}).

If \sqrt{D} is irrational we may equate the co-efficients of \sqrt{D} in these two expressions for v^{2} so we get two equations:

α = −ν^{2}(4L − α)
2α(2L + α) + D − 8αL − α^{2} = ν^{2} ((4L − α)^{2} + D) .

The second of these simplifies to

α^{2} − 4αL + D = ν^{2}((4L − α)^{2} + D).

Thus if we put β = α − 4L our equations become

α = ν^{2}β
αβ + D = ν^{2}(β^{2} + D)

from which we deduce D = ν^{2}D. Thus ν^{2} = 1 and α = β, which implies L = 0, a contradiction to our hypothesis (since L was defined to be λ_{1}λ_{2}).

We have now shown that \sqrt{D} and hence u is rational. Equation (6.7) shows that v is rational so long as u ≠ 2(α − L). If u and v are both rational then so are t_{1}, t_{2}. On the other hand if u = 2(α − L) and v is irrational then J = 0—which gives k_{1} = k_{2} and so λ_{1}r_{1} = λ_{2}r_{2}—while λ_{1}t_{1} ≠ λ_{2}t_{2}. This completes the proof of the Lemma.

To dispose of the second alternative allowed by Lemma (6.4), we consider the second derivatives of the function L_{a} at a critical point.

Lemma 6.8. Suppose, in the notation above, that L_{a} and its first derivative both vanish at g_{0}, that λ_{1}r_{1} = λ_{2}r_{2} but λ_{1}t_{1} ≠ λ_{2}t_{2}. Then g_{0} is not a local minimum of L_{a}.

To see this, we introduce local co-ordinates u_{1}, u_{2} on a neighbourhood of g_{0} in the space of simple PL functions. For u_{1}, u_{2} > 0 we have a simple PL function

G_{u_{1},u_{2}}(x_{1}, x_{2}) = ρ max (0, 1 − λ_{1}u_{1}x_{1} − λ_{2}u_{2}x_{2}) ,

where ρ = 1/\sqrt{λ_{1}^{2}u_{1}^{2} + λ_{2}^{2}u_{2}^{2}}. Thus g_{0} is the function G_{(λ_{1}t_{1})^{−1},(λ_{2}t_{2})^{−1}}. Straightforward calculations show that

(6.9)  F_{a}(G_{u_{1},u_{2}}) = ρ (\frac{1}{2}F (u_{1}, u_{2}) − λ_{1}r_{1}u_{1} − λ_{2}r_{2}u_{2} − r_{0})

where

F (u_{1}, u_{2}) = \frac{1}{u_{1}} + \frac{1}{u_{2}} − \frac{C}{u_{1}u_{2}}

with C = a/3λ_{1}λ_{2}. Notice that the factor ρ in (6.9) is not relevant to the analysis of the critical point and that the terms involving r_{i} are linear, so their second derivatives vanish. The hypotheses of the Lemma imply that the partial derivatives \frac{∂F}{∂u_{1}} , \frac{∂F}{∂u_{2}} , evaluated at the point u_{1} = (λ_{1}t_{1})^{−1}, u_{2} = (λ_{2}t_{2})^{−1}, are equal but that u_{1} ≠ u_{2}. Now

\frac{∂F}{∂u_{1}} = −\frac{1}{u_{1}^{2}} (1 − \frac{C}{u_{2}} ) , \frac{∂F}{∂u_{2}} = −\frac{1}{u_{2}^{2}} (1 − \frac{C}{u_{1}} ) ,

so we have

u_{1}^{2} − Cu_{1} = u_{2}^{2} − Cu_{2}.

Since the u_{i} are positive it is clear from the graph of the function u^{2}−Cu that we must have u_{1}, u_{2} < C. Now consider the second derivative

\frac{∂^{2}F}{∂u_{1}^{2}} = \frac{2}{u_{1}^{3}} (1 − \frac{C}{u_{2}} ) .

If u_{2} < C then \frac{∂^{2}F}{∂u_{1}^{2}} is negative so F does not have a local minimum at (u_{1}, u_{2}) and this implies that L_{a} does not have a local minimum at g_{0}.

Lemmas (6.4) and (6.8) together complete the proof of Proposition (6.1) in case (3). We next move on to case (1). If both p and q are vertices there is nothing to prove, since g_{0} is then itself rational. Suppose q is a vertex and p is not. We restrict L_{a} to the set of simple PL functions whose crease passes through the fixed point q. We may suppose that we are in the same situation as that considered for case (3) above, except that now t_{1} is a boundary point of the face J_{1}; so t_{1} is rational. The conditions that F_{a} and its derivative with respect to u_{2} vanish tell us that

\frac{1}{u_{2}} − \frac{C}{u_{1}u_{2}}

is rational. Thus u_{2} is rational unless u_{1} = C. In this latter case the derivative \frac{∂F_{a}}{∂u_{2}} vanishes identically for all u_{2}. Thus any simple PL function with a crease through q and any point of the face J_{2} is also an extremal, and in particular we can find a rational extremal.

Finally we consider case (2) when the crease of g_{0} passes through two parallel faces of P. Making a rational affine transformation, we may suppose that these are segments in the lines x_{2} = 1, x_{2} = −1, each lying in the half-plane x_{1} > 0. The meaure dσ is given by λdx_{1} on these segments for some λ > 0. Let the crease of g_{0} be defined by the equation

x_{1} − s = θx_{2}.

Let S be the quadrilateral defined by the inequalities

x_{1} > 0, −1 < x_{2} < 1, x_{1} − s < θx_{2}.

We consider the measure dσ on the boundary of S, given by λdx_{1} on the “horizontal” faces and zero elsewhere. The same argument as before shows that the mass and moments of P^{+}, (∂P)^{+} are equal to those of S, ∂S modulo rationals. Let r_{0} be the difference between the mass of S and the mass of ∂S and let r_{1}, r_{2} be the differences between the moments. A short calculation shows that the centre of mass of S is the point

(\frac{1}{6s}(3s^{2} + θ^{2}), \frac{θ}{3s}) .

From this one readily shows that

r_{0} = 2(λ − a)s, r_{2} = 2 (λ − \frac{a}{3}) θ, r_{1} = 2(λ − a)s^{2} + 2 (λ − \frac{a}{3}) θ^{2}.

As before, we know that r_{0}, r_{1}, r_{2} are rational. If λ is not equal to a or a/3 the first two of these immediately give that s, θ are rational and we are done. The two degenerate possibilities are similar to the final possibility considered above in case (1). If λ = a then θ is rational but we get no constraint on s. However in this case the r_{i} do not depend on s and this means that the functional F_{a}(g) does not change as we vary g through simple PL functions with crease parallel to that of g_{0}, provided we do not cross any vertices of P. Thus while the original g_{0} may not be rational we can find a rational zero of F_{a}. Similarly if λ = a/3 the functional does not change as we vary g through simple PL functions with crease through the fixed point (s, 0) and we can find a rational zero.

7.

7.1 Conclusion of proof

We will now put together the various components in the proof of Theorem 1.1. As a preliminary, we examine the notion of “K-convergence” of Kahler potentials over a toric surface, which enters into the statement of the Theorem. Thus we suppose we have a sequence u_{α} of symplectic potentials which correspond to Kahler potentials ψ_{α} via the Legendre transform.

Lemma 7.1.1. The sequence ψ_{α} has a K-convergent subsequence if and only if there is a subsequence α′ and affine-linear functions λ_{α′} on R^{n} such that the symplectic potentials u_{α′} = u_{α′} + λ_{α′} satisfy the following two conditions.

(1) There is a constant C such that when Q is either P or a face of

P,

C ≥ min_{Q} u_{α′} ≥ −C,

for all α′.

(2) For each x ∈ P there is a C_{x} such that

u_{α′}(x) ≤ C_{x}.

To prove this Lemma, first recall that the actions of addition of constants and of the complex torus T^{c} considered in the definition of Kconvergence go over to the action of addition of affine-linear functions on the symplectic potentials. Thus there is no loss in ignoring these actions on either side. Now consider the Kahler potentials ϕ_{α} on the open orbit in V. The derivatives of these take values in the bounded set P, so are bounded. Thus a necessary and sufficient condition for there to be a subsequence converging pointwise on the open orbit is that there is a subsequence with ϕ_{α′}(0) bounded. But, by definition,

ϕ_{α}(0) = min_{P} u_{α},

so this is equivalent to min_{P} u_{α′} bounded. Likewise, a necessary and sufficient condition that there be a subsequence of the ϕ_{α} converging pointwise on the codimension-1 orbit corresponding to a face Q of P is that there be a subsequence α′ with min_{Q} u_{α′} bounded. Thus the pointwise convergence criterion in the definition of K-convergence corresponds exactly to the condition in item (1) of the Lemma. To complete the proof we need to see that the second condition in the defintion of K-convergence corresponds to the second item in the statement of the Lemma. Thus consider a fixed point of the action on V, corresponding to a vertex of P. By applying an integral, affine-linear, transformation, there is no loss in generality in supposing that this vertex is the origin in R^{2}, and that, near to the origin, P is the positive quadrant. This means that the local co-ordinates (w_{1}, w_{2}) around the fixed point are the same as our standard co-ordinates on the orbit and

ξ_{i} = log |w_{i}|.

Now consider a point (η_{1}, η_{2}) near to the origin in P. By definition

u_{α}(x) = max_{ξ} (∑ x_{i}ξ_{i} − ϕ(ξ)) .

Thus u_{α}(x) ≤ C if and only if

ϕ(ξ) ≥ ∑ x_{i}ξ_{i} − C,

which is in turn equivalent to

ϕ(w_{1}, w_{2}) ≥ x_{1} log |w_{1}| + x_{2} log |w_{2}| − C.

This effectively finishes the proof: we leave for the reader the task of working out how the calculation above transforms for the other fixed points.

We are ready for the proof of Theorem 1.1. Given the toric variety (V, L) we consider the linear functional L_{a} on the normalised functions in C_{∞} and two cases:

(1) L_{a}(f) > 0 for all nonzero f. Then according to (5.2.2) there is an

ϵ > 0 such that

L_{a}f ≥ ϵ ∫_{∂P} fdσ,

and by (5.1.2) the functional F_{a} is bounded below, i.e the Mabuchi functional is bounded below on the space of T -invariant potentials H^{T}. Let ϕ^{(α)} be a minimising sequence, as considered in Theorem 1.1. Modulo the actions of T^{c} and of addition of constants, we can suppose that the ϕ^{(α)} are the Legendre duals of normalised symplectic potential functions u^{(α)}. These form a minimising sequence for F_{a} on S and hence also a minimising sequence for F_{a} on C_{∞} by (3.3.11). Thus they satisfy a bound

∫_{∂P} u^{(α)}dσ ≤ K

by (5.1.8). By Corollary (5.2.5) we can suppose, taking a subsequence, that the u^{α} converge uniformly on compact subsets of P and on the interior of each face of ∂P. This means that the two criteria of Lemma (7.1.1) are satisfied so, by that Lemma, the ϕ_{α} have a K-convergent subsequence.

(2) L_{a}f ≤ 0 for some nonzero f. If strict equality holds then, by the density result (5.2.8), we can suppose that f is a PL function and then, clearly, that it is also a rational PL function. If there is no f for which strict inequality holds, i.e., if L_{a}g ≥ 0 for all g but L_{a}f = 0, then by (5.3.1) and (6.1) we can suppose that f is a rational simple PL function. In either case there is a nontrivial rational PL function f with L_{a}(f) ≤ 0. This function defines a toric test configuration by with positive Futaki invariant by (4.2.1), and hence (V, L) is not K-stable. This completes the proof of our main result (1.1).

Although we are not yet able to prove a full converse to Theorem 1.1, there is a simple partial converse. First we have:

Proposition 7.1.2. Suppose there is a function f ∈ C_{1} with L_{a}(f) < 0. Then the Mabuchi functional is not bounded below on the invariant metrics and the manifold V does not admit any Kahler metric of constant scalar curvature in the given cohomology class.

To see this, we can suppose (by an approximation argument) that f lies in C_{∞}. If u is some fixed element of S we consider the sequence u_{k} = u + kf . Then

F_{a}(u_{k}) ≤ F_{a}(u) + kL_{a}(f) → −∞

hence F_{a} is not bounded below on C_{∞}. By (3.3.11) the functional is not bounded below on S either, hence the Mabuchi functional is not bounded below on the invariant metrics. The last assertion follows from (3.3.4) together with Lichnerowicz’s Theorem (which implies that any constant scalar curvature metric must be T -invariant).

To formulate a sharper result, we say that a polarised toric variety is K-stable with respect to toric degenerations if there is no destabilising configuration of the kind constructed in Section 4. Then we have:

Proposition 7.1.3. Suppose (V, L) is a toric variety such that the Mabuchi functional is bounded below on the invariant metrics H^{T} and any minimising sequence has a K-convergent subsequence. Then (V, L) is K-stable with respect to toric degenerations.

Thus the gap which needs to be filled to give a complete analytic criterion for K-stability is to show that this condition (K-stability) is equivalent to “K-stability with respect to toric degenerations”.

To prove (7.1.3), suppose the variety is not K-stable with respect to toric degenerations. This means that there is a non-affine rational, piecewise linear function with L_{a}(f) ≤ 0. Strict inequality cannot hold, since then the Mabuchi functional would not be bounded below by the Proposition above. Thus L_{a}(f) = 0 and we can suppose f is a simple PL function by (5.3.1). We approximate f by a sequence of convex functions f_{α} which are smooth up to the boundary, such that

∥f_{α} − f∥_{L^{∞}(\overline{P})} ≤ 1/α^{2}.

Let u_{α} be functions in S forming a minimising sequence for F_{a} and define

ũ_{α} = u_{α} + αf_{α}.

Then

F_{a}(ũ_{α}) ≤ F_{a}(u_{α}) + C/α,

for some C, so ũ_{α} is another minimising sequence. By the hypothesis, and Lemma (7.1.1), we can without loss of generality suppose that the min_{P} u_{α} are bounded and for each x ∈ P the u_{α}(x) are bounded above. This implies that, for each x, the u_{α}(x) are bounded above and below. Likewise, we can find affine-linear functions λ_{α} such that the ũ_{α} + λ_{α} are bounded above and below for each x ∈ P. Thus, for each x ∈ P, (λ_{α} + αf_{α})(x) is a bounded sequence. It follows that the λ_{α}(x) are bounded, for each x in the set where f = 0. Since this set has nonempty interior, we deduce that the λ_{α} are bounded on \overline{P}, since they are affinelinear. But this contradicts the fact that αf_{α} tends to infinity at some points of P.

Note that our notion of “K-convergence” does not capture the full strength of the convergence we have proved for the symplectic potentials, but in this respect our main Theorem is anyway of a provisional nature, since we expect that much stronger results are true.

7.2 Discussion

(1) Our main result deals with the two dimensional case but the general scheme of the argument works in all dimensions. To have a Theorem like (1.1) in higher dimensions one needs to extend the results of Sections (5.3) and 6. These are in principle elementary questions about convex functions, which one can address without knowing anything about toric and Kahler geometry.

(2) We give an explicit example to illustrate the theory we have developed. We start with the complex projective plane, represented as a toric variety by the triangle

P_{0} = {(x_{1}, x_{2}) : x_{1}, x_{2} > 0, x_{1} + x_{2} < 1}.

We obtain new toric varieties by repeatedly blowing up the points corresponding to the vertices of P_{0}. We recall that, in general, if P is a polygon corresponding to a toric surface, and v is a vertex of P, there are vectors e_{1}, e_{2} forming a Z-basis for Z^{2} such that the two edges of P meeting at v lie in the rays {v + te_{i} : t ≥ 0}. Blowing up the surface at the point corresponding to v gives a new toric variety with an associated polygon P̃ obtained by removing the triangle with vertices v, v+ϵe_{1}, v+ϵe_{2} from P. Here ϵ is any sufficiently small rational number (so that some multiple of P̃ is an integral polygon). For all this, see [12]. If we apply this procedure n times to the origin, and to the vertices that are generated from it, we get a polygon with vertices (1, 0), (0, 1) and a chain of 2^{n} vertices

((α_{1}, 0), (α_{2}, β_{2}), …, (0, β_{2^{n}}),

say, where the rational numbers α_{i} are decreasing and the β_{i} are increasing. The values of the α_{i} and β_{i} depend on the blow-up parameters chosen at each stage but the sequence of slopes

τ(i) = \frac{β_{i} − β_{i−1}}{α_{i−1} − α_{i}} ,

does not depend on these choices. Conversely, if we have a polygon with a sequence of vertices of this kind, and with this sequence of slopes, the polygon arises from a blow-up, for suitable choices of the parameters. It is easy to check that there is always an index i_{n} such that τ(i_{n}) = \frac{n−2}{2n−3}, and then τ(2^{n} + 2 − i_{n}) = \frac{2n−3}{n−2}.

We define a 9-sided polygon Q_{n} as follows. We replace the vertex (0, 0) of the original triangle P_{0} by three vertices,

(1/4, 0), (r_{n}, r_{n}), (0, 1/4)

where

r_{n} = \frac{n − 2}{4(3n − 5)}.

This value of r_{n} is chosen so that the two new edges have slopes n − 2/2n − 3 and 2n − 3/n − 2. There is obviously a cyclic symmetry of the original triangle P_{0}, permuting the vertices—this just corresponds to the symmetry between the three homogeneous co-ordinates on the projective plane. We replace each of the other vertices of P_{0} by a triple of new vertices as above, maintaining this symmetry. Now Q_{n} has a canonical measure dσ on its boundary, and we define a_{n} to be the ratio of the mass of the boundary and the area of the interior, as usual. The moment conditions are satisfied because of the symmetry condition. Thus we have a functional L = L_{a_{n}}, which vanishes on affine-linear functions.

Lemma 7.2.1. For sufficiently large n the linear functional L is nonpositive. In fact if f is a simple convex function with crease along the line through (1/4, 0), (0, 1/4) then L(f) < 0 for large n.

To see this we choose f to be zero on the region {x_{1} + x_{2} ≥ 1/4}. Thus L(f) has two contributions:

(1) the integral of f over the edges of slope (n−2/2n−3), (2n−3/n−2) with respect to the measure dσ;

(2) minus a_{n} times the integral of f over the triangle T with two edges as above and the third edge the line through (1/4, 0) and (0, 1/4).

Now, for the first contribution, the integers n − 2, 2n − 3 are coprime so the measure dσ on these edges is O(n^{−1}) times the Euclidean measure, thus the first contribution is O(n^{−1}). For the second contribution, first note that the a_{n} are bounded away from zero, since the total mass of the boundary of Q_{n} is at least 6 times that of the segment (1/4, 1/2) along the x_{1}-axis, which is independent of n, and the area of Q_{n} is less than the area of P_{0}. The triangle T contains the triangle with vertices (1/4, 0), (1/12, 1/12), (0, 1/4) so the integral of f over T is also bounded away from zero. So when n is large enough the magnitude of the second contribution must exceed that of the first and the Lemma is proved.

(In fact a little calculation shows that the conclusion of Lemma (7.2.1) holds for n ≥ 5.)

Now fix n large enough for the conclusion of Lemma (7.2.1) to hold. The polygon Q_{n} does not itself arise from a blow-up of the plane but since the slopes of the new sides were chosen to arise in the sequence (τ(i)) we can approximate Q_{n} arbitrarily closely by such a polygon, in an obvious way. That is, we succesively blow up the three fixed points of the projective plane and choose the blow-up parameters defining all the other sides to be very small. We make the whole construction symmetrical under the cyclic symmetry of P_{0} to ensure that the moment condition is satisfied. By continuity, the linear functional defined by this polygon must be negative on f, if the approximation to Q_{n} is close enough. This means, by (7.1.2), that the corresponding toric variety cannot have a T -invariant metric of constant scalar curvature, even though its Futaki invariant vanishes (or, in other words, it has no extremal Kahler metric).

(3) Most of the ideas in the body of this paper apply in a more general setting. We consider a bounded open convex set Ω ⊂ R^{n} with Lipschitz boundary, a positive measure dσ on ∂Ω in the measure-class of (n − 1)-dimensional Lebesgure measure and a bounded function A on Ω. Then we can define a space C_{∞} and functionals L_{A} and F_{A} in the obvious way. We expect that the following is true:

Conjecture 7.2.2. If L_{A}(f) > 0 for all non-affine functions in C_{∞} then F_{A} is bounded below and the minimum is attained in C_{∞} by a function which satisfies the equation S(u) = A.

(4) A generalisation which is pertinent to Kahler geometry arises in the study of extremal metrics. Here the function A is an affine-linear function on R^{n}. That is, given a toric variety V corresponding to a polytope P ⊂ R^{n} with measure dσ on ∂P, there is obviously a unique affine-linear function A = A_{P} satisfying the moment conditions (5.1.1). The existence of an extremal Kahler metric on V is equivalent to the existence of a function in S minimising F_{A}. Most of the discussion of the paper goes over to this case, with an appropriate modification of the definition of K-stability. However there is one point of difficulty: the results of (5.3) apply to positive functions A, but for some polytopes A_{P} is not positive throughout P. Thus we leave the extension of our results to extremal Kahler metrics for the future.

Suppose now that Q is a polyhedron in R^{2} and dσ is a measure supported on some of the faces of ∂Q, equal to the canonical measure considered before on the faces on which it does not vanish. Let V_{Q} be the toric variety defined by Q. The faces of ∂Q correspond to curves in V_{Q}: we let D be the divisor defined by the union of the curves for which dσ vanishes on the corresponding face of the boundary. There is a unique affine-linear function A = A_{Q} satisfying the moment conditions (5.1.1). Let L be the linear functional defined by A_{Q} and dσ.

Conjecture 7.2.3. If L is strictly positive on convex, non-affine functions then F is bounded below and there is a smooth minimiser, which corresponds to a complete extremal metric on V_{Q} \ D.

The general picture one might expect is that if a toric variety is not K-stable then there should be a canonical decomposition of the associated polygon into subpolygons. On the boundary of each of these subpolygons we have a measure—the usual one on the faces which come from the boundary of P and zero on the faces created by the decomposition. These subpolygons should be of two kinds; either they satisfy the hypothesis of the conjecture or, exceptionally, they should be parallelograms in which two opposite faces are faces of P. We expect that the “Calabi flow” (or other minimising procedure) generates a sequence of metrics with diameter tending to infinity, breaking the manifold into pieces. On the pieces corresponding to subpolygons of the first kind the metrics should converge to the complete extremal metrics of Conjecture (7.2.3). On the pieces corresponding to the parallelograms the metrics should collapse, with behaviour modelled on a product S^{1} × B, where the length of the circle tends to zero.

(4a) There is an alternative variational formulation of the PDE S(u) = A, which is related to almost-complex structures. Suppose we are in the setting of (2) above, with a function A on a convex set Ω and measure dσ on ∂Ω. Suppose we can find a symmetric tensor field V^{ij} on Ω, with V^{ij} > 0 everywhere, which represents the linear functional L_{A} in the sense that

L_{A}(u) = ∫_{Ω} V^{ij}f_{,ij}dμ,

for all f ∈ C_{∞}. (Here we use the summation convention over repeated indices, and u_{,ij} denotes the second partial derivative.) Essentially this means that V^{ij} satisfies the distributional equation

(V^{ij})_{,ij} = A − [dσ]

where [dσ] is regarded as a distribution supported on ∂Ω. Clearly this implies that L_{A} > 0 on C_{∞}. It is reasonable to expect that a converse is true, so that if L_{A} is positive on C_{1} we can find such a tensor V^{ij}. This is clearly a weaker conjecture than (7.2.2), which requires a solution of the special form V_{ij} = u_{,ij}. In any case, let us suppose that we do have a solution V^{ij}, and let V be the space of all such solutions: a convex subset in an affine space. We consider the functional

N (V) = ∫_{Ω} log det V^{ij} dμ

on V. This is a convex function and so has at most one critical point, a global minimum, on V. To find the appropriate Euler-Lagrange equations we use:

Lemma 7.2.4. The general solution of the equation V_{,ij}^{ij} = 0, for symmetric tensor fields V^{ij}, on Ω has the form

V^{ij} = T_{,k}^{ijk} + T_{,k}^{jik}

for an arbitrary tensor field T^{ijk} which is skew in the indices j, k.

This is a simple exercise, using two applications of the Poincaré Lemma.

Now the derivative of the functional N at a point V^{ij} in the direction ϵ^{ij} is given by

∫_{Ω} V_{ij}ϵ_{ij} dμ.

Thus if V^{ij} is a critical point for N on V we must have

∫_{Ω} V_{ij}(T_{,k}^{ijk} + T_{,k}^{jik}) dμ = 0,

for all compactly supported tensors T^{ijk}, skew in j, k. Integrating by parts, this tells us that V_{ij,k} is symmetric in j, k. Using the Poincaré Lemma again, we can find a tensor field θ_{i} such that V_{ij} = θ_{i,j}. Now the condition that V is symmetric means, by the Poincaré Lemma, that θ_{i} = u_{,i} for some function u on Ω. In sum, solving the equation of Conjecture (7.2.2) is equivalent to finding a minimiser of the functional N on V.

In the case when the domain is a polytope corresponding to a toric variety V, the discussion above corresponds to considering T -invariant almost-complex structures on V, as explained in (3.1). We call the function V_{,ij}^{ij} the Hermitian scalar curvature of the almost-complex structure defined by V_{ij}. Thus finding a Kahler metric of constant scalar curvature is equivalent to minimising the function N on the set of almost-complex structures of constant Hermitian scalar curvature. Notice also that, in the converse direction, our examples of manifolds which are not K-stable give examples where there is not even an almost-complex structure of constant Hermitian scalar curvature.

(5) Our main result, Theorem (1.1), yields a new real-valued invariant of K-stable toric surfaces: the infimum of the functional L_{a}. This seems to be special to the toric case, since in general the Mabuchi functional is only defined up to an overall constant, so even if it is bounded below, its infimum is not well-defined as a real number.

Appendix: Proof of Proposition (4.1.3)

If X ⊂ R^{n} is a finite union of compact polytopes we define ν(X) to be the number of lattice points in the interior of X plus one half the number of lattice points in the boundary. By applying the basic result (4.1.2) to a polytope P and to its boundary faces one sees that Proposition (4.1.3) is equivalent to:

Proposition A1. For an integer polytope P ⊂ R^{n},

ν(kP) = k^{n}Vol(P) + O(k^{n−2}).

This is the statement that we prove here. If X_{1} and X_{2} have disjoint interiors then one easily sees that

ν(k(X_{1} ∪ X_{2})) = ν(kX_{1}) + ν(kX_{2}) + O(k^{n−2}),

since the points that are counted differently in the sums lie in the (n−2) skeleton of k(X_{1} ∪ X_{2}). Since any integer polytope can be decomposed into a union of integer simplices it suffices to prove (A1) in the case when P is a simplex. We now use induction on the dimension n. The statement is readily verified for n = 1 so we suppose n ≥ 2 and that the result has been proved for lower-dimensional simplices.

Lemma A2. Suppose a function f on the positive integres satisfies

f(k + 1) − f(k − 1) = W k^{n−1} + O(k^{n−3}).

Then f(k) = \frac{W}{2n}k^{n} + O(k^{n−2}).

We leave the proof of this to the reader. Using this Lemma, and (4.1.2), it suffices to show that

ν((k + 1)P) − ν((k − 1)P) = W k^{n−1} + O(k^{n−3}),

for some constant W. We can suppose that one vertex of our n-simplex P is the origin in R^{n} and that the other vertices lie on the hyperplane {x_{n} = h}, where (x_{1}, … x_{n}) are standard co-ordinates on R^{n} and h is a positive integer. Thus P is a cone on a copy of an integer (n−1)-simplex Q ⊂ R^{n−1}. Comparing the counts involved in the definition of the two terms we have

ν((k + 1)P)) − ν((k − 1)P)

= ν(kQ) + \frac{1}{2} (ν((k + 1)Q) + ν((k − 1)Q))

+ ∑_{i=1}^{h−1} (ν((k + i/h)Q) + ν(k − i/h)Q))

+ \frac{1}{4}(N((k + 1)∂Q − N((k − 1)∂Q)),

where N((k +1)∂Q) denotes the number of lattice points on the boundary of (k + 1)Q. The first term ν(kQ) has the desired form, by the induction hypothesis, and the difference N((k + 1)∂Q) − N((k − 1)∂Q) is O(k^{n−3}), so it suffices to prove that for 1 ≤ i ≤ h:

ν((k + i/h)Q) + ν((k − i/h)Q) = W_{i}k^{n−1} + O(k^{n−3}). (A3)

We claim that, in fact, for any real number ρ

ν((k + ρ)Q) + ν((k − ρ)Q) = 2ν(kQ) + O(k^{n−3}), (A4) this implies (A3), by the induction hypothesis. To see (A4), consider a face F_{α} of Q, defined by an equation λ_{α} = c_{α}, where λ_{α} is a primitive integral linear function on R^{n−1}. We can find an integer vector e_{α} such that λ_{α}(e_{α}) = 1. This choice defines a reflection map R_{k,α} on R^{n−1}, preserving the integer lattice, with fixed set the hyperplane containing the corresponding face of kQ. For simplicity of exposition, we suppose that the origin in R^{n−1} lies in the interior of Q—the reader may check that the argument works without this assumption. Write A_{k} for the complement of the interior of (k−1)Q in (k+1)Q. For δ > 0 let A_{k}^{δ} be the complement of the δ-neighbourhood of the (n − 3)-skeleton of (k + 1)Q in A_{k}. We can fix a δ, independent of k, such that A_{k}^{δ} decomposes into a disjoint union of connected components A_{k}^{δ}(α) associated to the faces F_{α}; thus A_{k}^{δ}(α) is contained in a region {kc_{α} − ρ_{α} ≤ λ_{α}(x) ≤ kc_{α} + ρ_{α}}. Finally, let

B_{k}^{δ}(α) = A_{k}^{δ}(α) ∩ R_{α}(A_{k}^{δ}(α)),

and let B_{k} = ⋃_{α} B_{k}^{δ}(α). Define an involution R on B_{k} to be given by R_{α} on B_{k}^{δ}(α). The complement A_{k} \ B_{k} is contained in an ϵ-neighbourhood of the (n − 3)-skeleton of kQ, for some fixed ϵ independent of k. Thus ν((k + ρ)Q − ν(kQ) and ν(kQ) − ν((k − ρ)Q) can be computed, up to O(k^{n−3}), by counting only those points in B_{k}. The map R matches up the points in B_{k} counted in computing ν((k + ρ)Q) − ν(kQ) with those counted in computing ν(kQ) − ν((k − ρ)Q), so the two terms agree up to O(k^{n−3}), as asserted in (A4).

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