Constant scalar curvature metrics on toric manifolds (draft)

Constant scalar curvature metrics on toric manifolds (draft) Julius Ross November 2012 In one word he told me secret of success in mathematics: Plagiarize! Tom Lehrer 1 Introduction 1 2 Toric Kähler manifolds 2 3 Symplectic Potentials 4 4 Abreu s formula for the scalar curvature 6 5 K-stability 8 6 An Integration by Parts Formula 10 7 A simplification of the K-stability condition on Toric Surfaces 13 8 Further topics 17 1 Introduction These notes contain a presentation of the following selection of topics concerning constant scalar curvature Kähler (csck) metrics on toric manifolds: 1. The correspondence between (invariant) Kähler metrics on toric manifolds and symplectic potentials on the associated moment polytope 2. Abreu s formula for the scalar curvature of a metric in terms of its symplectic potential 3. The concept of K-stability for toric manifolds, and the toric Yau-Tian- Donaldson conjecture 4. A discussion of the K-stability condition on toric surfaces in terms of simple piecewise linear functions. The aim is to present these core ideas to those familiar with the basic tools of toric geometry, Kähler geometry and symplectic geometry but who are not assumed to be experts in the combination of the three. Where possible we have taken as a direct approach as possible, leaving aside how these ideas fit into the picture for non-toric manifolds. Moreover we have chosen to miss out entirely the algebro-geometry picture connecting with geometric invariant theory and projective embeddings. 1

2 Even with this point of view there remain many topics that could easily have been included if the time allowed. A major piece of work due to Donaldson proves the equivalence between the existence of a csck metric and K-stability on toric surfaces. The author had originally planned to include some details of this, and still hopes to do so in a future version of these notes. Moreover the Mabuchi functional, extremal Kähler metrics, the Calabi and Ricci flows, and orbifold metrics are all interesting in the toric setting. In the final section we discuss very briefly some of these providing a mini-guide to further reading. We stress that the author does not claim any originality for the material in these notes other than that of presentation, and has relied heavily on papers of many others in particular Abreu, Guillemin, Donaldson, Wang-Zhou. Many references are included, but there has been no attempt to give a complete survey of the field. Acknowledgments: These notes were written to support a minicourse given by the author at the MACK5 workshop in Rome in November 2012. I wish to thank the organizers S. Diverio, S. Trapani and the scientific committee S. Boucksom, V. Guedj, P. Eyssidieux for this opportunity. I also thank various participants of this workshop for discussions concerning this material, in particular Robert Berman and Gabor Szkelyhidi. 2 Toric Kähler manifolds We are interested in Kähler metrics on toric manifolds, and following Abreu [3] we make this our central definition: Definition 2.1. A toric Kähler manifold consists of 1. A connected 2n-dimensional Kähler manifold (M, ω, J) where J the complex structure on M and ω H 1,1 (M) denotes Kähler form, and 2. An effective action τ of the real n-dimensional torus T n = (S 1 ) n on M preserving ω and J. Two toric manifolds (M i, ω i, J i, τ i ) to be considered isomorphic if they are equivariantly isomorphic as Kähler manifolds. By abuse of notation we shall refer to a toric Kähler manifold simply as M if the other structure is understood. For simplicity we shall assume always that M is compact. There are two important ways to consider a toric Kähler manifold. First, by forgetting the complex structure, any toric Kähler manifold can be thought of as a toric symplectic manifold, and second by forgetting the Kähler form it can be thought of as a smooth toric variety. We discuss these each in turn. 2.1 Symplectic Structure Let (M, ω, J, τ) be a toric Kähler manifold. By forgetting J we can think of ω as a symplectic form in H 2 (M). By hypothesis the T n -action τ preserves

3 ω, and so gives a Hamiltonian action. Thus (M, ω, τ) is a toric symplectic manifold (i.e. a connected 2n-dimensional symplectic manifold with an effective Hamiltonian action of T n ). Now associated to any Hamiltonian action there is a moment map µ : M Lie(T n ) = R n where we have identified R n with the dual of the Lie algebra of T n in the standard way. By a famous theorem of Atiyah [4] and Guillemin-Sternberg [14], the image of µ is a closed polytope P R n called the moment polytope. The points in P parametrize the orbits of the T n action. Let P denote the interior of P. Then if y P the fibre µ 1 (y) is a copy of T n. In fact M := µ 1 (P ) = P T n is the open dense subset of points where the T n action is free. The boundary of M in M is then described by the boundary of P. For example, if y lies in the interior of an (n 1)-dimensional face of P, the µ 1 (y) consists of points for which the T n action has a one dimensional stabilizer T = S 1, and thus is a copy of T n 1, and there is a similar description for the fibres over the lower dimensional faces. 2.2 Delzant s Theorem We next state a theorem of Delzant which characterizes those polytopes that arise as the moment polytope of a toric symplectic manifold. As will be seen below, Delzant s theorem is actually much stronger in that it provides a canonical representative of a toric symplectic manifold among all those with the same moment polytope. Definition 2.2. We say that P is Delzant if at each vertex of P the normal vectors of the faces through that vertex can be taken to be a basis of Z n. Note that in particular this says that each vertex meets precisely n edges, and in this case the condition is that the normal vectors can be taken to an integral basis. The connection with toric manifolds is through the following theorem, which in particular gives an easy method for producing examples: Theorem 2.3. (Delzant) The moment polytope of a symplectic toric manifold is Delzant, and conversely associated to any Delzant polytope P there is a canonical toric symplectic manifold (M P, ω P, τ P ) with moment polytope P. Moreover any other toric symplectic manifold with moment polytope P is equivariantly isomorphic (as a symplectic manifold) to (M P, ω P, τ P ). 2.3 Complex Structure It follows from the construction used to prove Delzant s theorem that there is in fact a canonical T n -invariant complex structure J P on M P that makes (M p, ω P, J p, τ P ) into a toric Kähler manifold. Moreover the T n action extends

4 to a holomorphic action τp C of Tn C = (C ) n on (M P, J P ) and there is an equivariant holomorphic embedding of M into projective space. Thus, by forgetting the Kähler form ω P we see that (M P, J P, τp C ) is a smooth projective complex toric manifold (i.e. a smooth complex manifold M of complex dimension n that contains T n C as a dense open orbit with an action of Tn C that extends the usual action of T n C on itself). So starting with a toric Kähler manifold (M, ω, J, τ) with moment polytope P we have two complex structures, namely J and the one furnished by (the pullback of) J P under the symplectic isomorphism between (M, ω, τ) and (M P, ω P, τ P ). It turns out that there is an equivariant biholomorphism between (M, J, τ) and (M, J P, τ) [3, Proposition A.1], which arises from the fact that both complex structures can be determined by the combinatorics of the fan dual to the polytope. Thus (M, J, τ) can also be considered a projective smooth complex toric manifold. 3 Symplectic Potentials The next goal is to parametrize compatible Kähler metrics on a toric symplectic manifold (M, τ, ω) through functions on the moment polytope. The point is that there are two natural coordinates that one can take on the open dense torus in M, coming from the symplectic and complex points of view. Starting with the symplectic point of view, M is the set on which the T n action is free, and M = P T n. Thus a point in M is of the form (y, t) where y = (y 1,..., y n ) P and t = (e iη1,..., e iηn ) for some η i R that are defined modulo 2π. It is convenient then to consider y i, η i as coordinates on the universal cover P R n, which are referred to either as symplectic coordinates or moment coordinates. Observe that these coordinate depend on the moment map, and thus on the symplectic form. In fact one can check that in these coordinate the symplectic form is ω = dy i dη i. i Now thinking of M is a complex manifold, we can think of M as the complex torus T n C = (C ) n. Then we have coordinates w i C and so one the universal cover we have complex coordinates z i = log w i = ξ i + 1η i where ξ i, η i R. Now suppose we have a Kähler metric ω on M that is invariant under T n. Then on M there is an invariant potential φ for ω satisfying ω = 2 1 1 φ = φ,ij dz i dz j 2 ij

5 where φ,ij = 2 φ z i z j. Observing that T n acts by translation on η i we see that invariance implies φ depends only ξ = (ξ 1,..., ξ n ). Thus in fact φ,ij = 2 φ ξ i ξ j. and the positivity of ω says precisely that the matrix φ,ij is positive definite (i.e. φ is a strictly convex function on R n ). Using the moment map one can translate the potential to the moment polytope. Using the potential we can write ( φ µ(z 1,..., z n ) = ξ φ =,..., φ ), ξ 1 ξ n and by convexity, for each y P there is a unique ξ = ξ(y) R n such that so y = µ(ξ(y)) y i = φ ξ i (ξ(y)). Definition 3.1. The symplectic potential of φ is defined to be the Legendre transform of φ, which is the function given by u(y) = y ξ(y) φ(ξ(y)). Since the Legendre transform is an involution, the symplectic potential uniquely determines φ. Definition 3.2. The set S = S P of symplectic potentials is those functions u that arise as the Legendre transform of some invariant Kähler metric on M. It turns out that any u S extends continuously to P, and so we can consider u either as a function on P or P without confusion. We will discuss in Section 6 the precise behaviour of u S at the boundary P, but for we just observe that it is a smooth strictly convex on P. In fact if we let u ij = u y i y j and u ij be the components of the inverse matrix to u ij then by direct calculation u ij (y) = φ,ij (ξ(y)). (3.1)

6 4 Abreu s formula for the scalar curvature We next give Abreu s calculation showing that the scalar curvature of a toric Kähler metric can be written in terms of derivatives of the symplectic potential with respect to the moment coordinates. Theorem 4.1 (Abreu). The scalar curvature of the toric Kähler metric ω associated to u S is Scal(ω) = R µ where R = 1 2 ij 2 u ij y i y j = 1 2 ij u ij,ij. In the following we will refer to R as the scalar curvature of u. Proceeding to the proof, let y = (,..., ), y 1 y n so if f is a function and G is a matrix ( f y f = y G = y 1,..., f y n ), ( G,..., G ). y 1 y n We next isolate the part of the calculation that is merely differentiation: Lemma 4.2. Suppose G ij = u y i y j for some smooth strictly convex function u on an open set P and let G ij denote its inverse. Then the j-th component of ( y log det(g)) G 1 is (( y log det(g)) G 1 ) j = i G ij y i. Proof. If α(g) := log det(g) where G is a symmetric positive definite matrix then Dα G (U) = tr(ug 1 ), (4.1) which follows as for small t, α(g + tu) = log det(g + tu) = log det(g) + log det(i + tug 1 ) = log detg + log(1 + t tr(ug 1 ) + O(t 2 )) = log detg + t tr(ug 1 ) + O(t 2 ).

7 Now by the chain rule, y α = ( Dα G ( G ),..., Dα G ( G ) ) y 1 y n = (Dα G ( y G)) where the last equality can be considered as notation, so we are thinking of the operator Dα G acting componentwise. Thus putting U = G y i for i = 1,...,n into (4.1) gives y α = tr(( y G) G 1 ) where now both the matrix multiplication and the trace are to be taken componentwise. At this point it seems easiest to work with indices. Differentiating GG 1 = Id gives and since partial derivatives commute, G y u G 1 = G G 1 y u, (4.2) G uv y i = for any u, v, i. Hence ( ) G ( y α) i = tr G 1 = y i uv Thus the term we are looking for is as required. ( y α G) j = i = uv 3 u y i y j y u = G iv y u = uv = uv G uv y i G uv (4.3) G iv y u G uv ( y α) i G ij = iuv δ vj G uv y u G iv G uv y u by (4.2). = u Proof of 4.1. Let φ be the potential of ω, and set F ij = φ,ij. G iv G uv y u G ij G uj y u As mentioned above, φ depends only on ξ i = Re(z i ), so the formula for the scalar curvature as the trace of the Ricci curvature of ω is R = 1 2 F ij 2 log(detf) = 1 ξ i ξ j 2 tr(f 1 ξ T ξ log det(f))

8 where T ξ denotes the transpose of ξ. Now the moment coordinates are y i = φ ξ i and so Letting F 1 T ξ = T y. G ij = u ij = u y i y j we recall from (3.1) that G ij = F ij. Thus R = 1 2 tr(f 1 T ξ ξ log det(f)) = 1 2 tr(f 1 T ξ ξ log det(g)) = 1 2 tr( T y ξ log det(g)) = 1 2 tr( T y ( y log det(g)) G 1 ) = 1 ( ) ( ) 2 tr ( y α G 1 ) j = 1 y k jk 2 tr G ji y i k y i jk = 1 G ij 2 y i y j ij giving the desired formula. by Lemma 4.2 5 K-stability We now turn to the definition of K-stability for a Delzant polytope P. Let F be the (n 1)-dimensional face of P defined by the equation h(y) = 0 where h: R n R is the affine linear function h(y) = y u + c where c R and u is a primitive normal to the face F. Definition 5.1. Define a positive measure dσ on F by requiring where dy is the Lebesgue measure. dh dσ = ±dy Clearly this is independent of l, and the sign is determined by the requirement that dσ be positive. For example suppose that l(y) = py 1 qy 2 on R 2, where p, q are coprime integers, so the face is given by the line through the origin of slope p/q. Then Next set dσ = dy 2 p = dy 1 q. a := vol( P) vol(p) where the volumes of P and P are taken with respect to the measures dσ and dy respectively.

9 Definition 5.2. The Donaldson-Futaki invariant of a convex function f on P is L(f) = fdσ a fdy. P Observe that L is linear, and by choice of a the quantity L(f) vanishes if f is constant. Some care must be taken to restrict the class of functions f to ensure that L(f) is finite, and this will be addressed below where needed. Of particular interest is the case that f is piecewise linear, in which case it can written as f = max{f 1,..., f r } for a finite number of affine linear functions f i on P. We say that f is a convex piecewise linear rational function if we can take each of the f i to be rational (i.e. of the form f i (y) = u y + λ where u Q n and λ Q). Definition 5.3. We say that P is K-semistable if L(f) 0 for all convex piecewise linear rational functions f on P. We say that it is K-polystable if L(f) 0 for all such functions, with equality if and only if f is affine linear. Definition 5.4. We say that a toric Kähler manifold is K-(semi/poly)stable if its associated moment polytope is. Fundamental to the field of csck metrics is the so-called Yau-Tian-Donaldson conjecture that relates the existences of a csck metric to an algebro-geometric stability notion. In the toric case this is the following: Conjecture 5.1 (Donaldson). Let (M, ω, τ) be a toric symplectic manifold with moment polytope P. Then M admits a csck metric compatible with ω if and only if it is K-polystable. Observe that the definition of K-stability is purely combinatorial; thus the conjecture in this form gives a condition for the existence of a csck metric on a toric Kähler manifold purely in terms of the combinatorics of its moment polytope. We remark right away that the particular definition of K-stability given above might not quite be the correct one for the conjecture, and we shall return to this issue later in these notes. Below we will prove that K-polystability is a necessary condition for the existence of an invariant csck metric. In a series of papers [10; 11; 12; 12], Donaldson uses a continuity method to prove the converse for toric surfaces, namely that the combinatorial condition implies the existence of a csck metric. Before moving on, observe that if f is affine linear then both f and f are convex. Thus if P is K-semistable then L(f) = 0 for all affine linear functions f, which holds if and only if (P, dy) and ( P, dσ) have the the same volume and same center of mass. Remark 5.2. In fact, the (rational) affine linear functions f come from compatible C actions on M, and L(f) is known to be the (classical) Futaki invariant of this action [13]. P

10 We refer the reader to [9, Sec 4.2.1] for the relation between this definition and the definition of K-stability for general (i.e. non-toric) polarised manifolds. In essence, for large rational constant C the polytope Q = {(y, t) P R : 0 t C f(y)} is an n + 1-dimensional polytope P which gives an n + 1-dimensional complex toric variety X. This can be considered as a one parameter degeneration of X (more precisely, it has the structure of a test configuration in Donaldson s terminology) into a not-necessarily-normal toric variety defined by the roof of the function f. Moreover the Futaki invariant defined here agrees with a purely algebro-geometric definition that can be defined in terms of this degeneration. Thus what we have defined here should perhaps more precisely be defined as K-stability through toric degenerations but we shall not use this terminology since we will not discuss the general picture any further. The above form of the Yau-Tian-Donaldson conjecture includes the assertion that to check for a csck metric on a toric manifold it is sufficient to consider only degenerations of this form. 6 An Integration by Parts Formula In this section we describe a key integration by parts formula due to Donaldson. At this point we need to know the boundary behaviour of symplectic potentials. Suppose P is written as a finite number of half-spaces: P = d {y R n : h r (y) 0} r=1 where h r are affine linear functions given by h r (y) = u r y + c r such that c r R and u r is a primitive element of Z n. Building on the ideas of Guillimen, Abreu [3, Thm 2.8] proves the following. Theorem 6.1 (Abreu). The set S of symplectic potentials consists of functions of the form u = u + h where and u = d l r (y)log l r (y) (6.1) r=1 1. h extends to a smooth function on P, 2. u is strictly convex (i.e. u ij is strictly positive on P ), and

11 3. The function δ(y) = ( det(u ij ) d l r (y) r=1 ) 1 extends to a smooth strictly positive function on P. Now let P δ denote the polygon with faces parallel to P moved inward by a small distance δ > 0 and u S be a symplectic potential. At any point in P δ let ν = (ν 1,..., ν n ) be the unit outward pointing normal. We let dσ 0 denote the Lebesgue measure on the faces of P and extend the definition of dσ in the obvious way on the faces by translating the measure from P. Definition 6.2. Denote by C the space of functions that are convex and continuous on P which are smooth on P. The following is a key lemma: Lemma 6.3 (Donaldson). Let f C be such that (f) = O(d 1 ) where d is the distance to P, and let u S. Then fu ij,i ν jdσ 0 fdσ and, as δ tends to zero. P δ ij P δ ij f,i u ij ν j dσ 0 0 Proof. We sketch a proof of the first statement. Consider first the model case in which P = {x i 0, i = 1,...,n} and the symplectic potential is u = n i=1 y i log y i. By direct computation P u ij = diag(y 1,...,y n ) =:. Therefore at any point in the interior of a face of P, u ij,i ν j = 1 i (the sign comes from the fact that ν is assumed to be outward facing) and so in fact f u ij,i ν jdσ 0 = fdσ. P δ i P Now the general case follows from this, for if u S then for this model polytope u = u + h where h is smooth on P with Hessian H then u ij =

12 1 + H = 1 (1 + H), and so u ij = (1 + H) 1 = (I + J) for some J. From this one computes that ij uij,i ν j = 1 + O(y 1 ). Thus f u ij,i ν jdσ 0 = fdσ + O(δ). P δ i P Since any Delzant polytope can be brought to this form after a change of basis, this proves the first statement. Corollary 6.4. With f, u as above, u ij f,ij dy = P P u ij,ij fdy + P fdσ. Proof. Applying integration by parts twice on P δ (on which everything is smooth) gives (fu ij,ij + f,ju ij i )dy = fu ij,i ν jdσ (6.2) P δ P δ P P δ (u ij f,ij + u ij,j f,i)dy = P δ u ij f,i ν j dσ. (6.3) Subtracting the first from the second gives (u ij f,ij u ij,ij f)dy = (u ij f,i fu ij,i )ν jdσ P δ so taking δ to zero and applying the previous Lemma gives the result. Corollary 6.5. Suppose u S is such that u ij,ij = a is constant. Then a = vol( P) vol(p) Proof. Put f = 1 into the above formula. So if the scalar curvature of a symplectic potential is constant, then this constant is a combinatorial invariant of P (this just reflects the well known fact that the integral of the scalar curvature on any Kähler manifold can be calculated through topological intersection numbers). Thus in this case if f C then L(f) = fdσ a fdy = u ij f,ij P P P which is non-negative by the convexity hypothesis on f, and strictly positive unless f is affine linear.

13 Corollary 6.6. Let u, f be above. Then L(f) = u ij f ij dy (6.4) with equality if and only if f is affine linear. P Proof. We compute This follows as u ij is positive definite as u is strictly convex on P, and f is convex so f ij is positive semidefinite, and positive definite unless f is affine linear. One can view this as a kind of stability but considered for smooth convex function f. Moreover the formula (6.4) extends to cotinuous, convex but not necessarily smooth function, in which case f ij is to be taken in the sense of distributions. Corollary 6.7. If there is a u S that is csck then P is K-polystable. Proof. By a simple approximation argument it follows from (6.4) that L(f) 0 for any piecewise linear function f on P. Moreover if f is not affine linear then there is a non-trivial hyperplane on which f ij is a strictly positive measure (namely along a hyperplane on which f is not smooth), and thus L(f) > 0. See also [26] for a direct proof of this result that does not use distributions. 7 A simplification of the K-stability condition on Toric Surfaces We now show that for K-stability on toric surfaces it is in fact only necessary to consider a particular kind of convex function f. We work throughout in the case that P is a Delzant polytope in R 2 that contains the origin. The material in this section is adapted from Donaldson [9, Sec 5] and Wang-Zhou [24, Sec 4]. Definition 7.1. We say that a function f on P is simple piecewise linear on P if it is of the form f(y) = max{0, h(y)} where h is an affine linear function. Thus a simple piecewise linear function f has a single crease along the line h(y) = 0, and as long as this line intersects P then f is not affine linear on P. Definition 7.2. Let P be the union of P and the interiors of its codimension 1 faces. We define C 1 to be the set of positive convex functions f in P such that fdσ < P

14 The point of this definition is that it is only the value of f on P that contribute to the two integrals in L(f). Now set { } B = f C 1 : f 1. P We wish to consider properties of a minimiser of L over B; that is an f B such that L(f) L(g) for all g B. A key idea in [24, Sec 4] is that such a minimizer can be written as the following envelope: Lemma 7.3. A minimiser f of L over B is continuous on P and f = sup{h(x) : h is affine linear and h f on P }. Proof. By convexity one can easily show that if y P then the limit f(y ) := lim y y,y P f(y) exits. So f = f on P and f f. The values of f on P P do not contribute to the value of L( f), and f B. But if f(y ) < f(y ) for some y then L( f) < L(f) which is contradicts the minimizing property of f. Hence f = f so f is continuous over P as claimed. Now let f + = sup{h(x) : h is affine linear and h f on P }. As f + is a supremum of convex functions, it is convex, and clearly f + f on P. Thus f + B. Moreover taking h to be the supporting hyperplane of f at some point we see f(y) f + (y) for all y P. Finally if f(y) < f + (y) for some y P then L(f + ) = f + dσ a f + dy < fdσ a fdy = L(f) P P P P which contradicts that f is a minimizer of L over B. Now let h denote a supporting hyperplane of f at some point y P and set K = K y,h = {y P : h(y) = f(y)} which is convex by convexity of f. Recall we say that z is an extreme point of a convex set K if there is a hyperplane L such that L Z = {z}. Equivalently z is an extreme point of K if there is an affine linear function h 0 such that K {h 0 0} and K {h 0 = 0} = {z} (7.1) Definition 7.4. Let P be a compact convex set. We say that a closed convex subset Q P is extremal if every extremal point of Q lies in P. Said another way, Q is extremal if it is the convex hull of Q P.

15 Lemma 7.5. If f is a minimizer for L over B then the subset K is extremal. Proof. By adding an affine linear function to f we may as well assume that h = 0 and K = {y P : f(y ) = 0}. Suppose for contradiction z is an extreme point of K that lies in the interior P. Pick an affine linear h 0 so 7.1 holds. Then, as z is an extreme point of K, we see f is strictly positive on the compact set P {h 0 0} and thus by Lemma 7.3 is bounded below on this set by some σ > 0. Thus for sufficiently small ǫ the inequality f > ǫ(h 0 + 1) holds on P and so by Lemma 7.3, f ǫ(h 0 + 1) on all of P. But this implies f(y) > 0 = h(y) which contradicts h being a supporting hyperplane of f at y. Lemma 7.6. Let h be an affine linear function. Suppose that u is a function on P such that (i) u 0 on P (ii) u is continuous (iii) u is convex on {h 0} and on {h 0} and (iv) u = 0 on P {h = 0}. Then u is convex. Proof. The proof is simple by restricting to line segments, and noting that any such segment with endpoints in {h < 0} and {h > 0} must pass through {h = 0} on which u = 0. We leave the details to the reader. Theorem 7.7. Let P be a two dimensional Delzant polytope. If P is not K-polystable then there exists a simple piecewise linear function f that is not affine linear such that L(f) 0. Proof. Suppose first P is not K-semistable. Then there is a piecewise linear f with L(f ) < 0, which by rescaling we can assume lies in B. By [9, 5.1.3] the function L is bounded from below on B, and using a compactness argument [9, 5.2.6] one sees that the minimum of L over B is realised by some f B which is necessarily not-affine as L(f) < 0. Suppose instead that P is K-semistable but not K-polystable. Then L(g) 0 for all convex piecewise rational functions g, but there exists an f that is not affine linear with L(f) = 0. By an approximation argument [9, 5.2.8] we conclude L(g) 0 for all g B. Thus in both cases there is a minimiser f of L over B, which is not affine linear, and L(f) 0. By adding an affine linear function to f we may as well suppose that h = 0 is a supporting hyperplane to f at the origin in P. Thus K = {y P : f(y) = 0} is a convex set, and by Lemma 7.5 the extreme points of K lie in P. Thus K is either a line segment (with endpoints in P) or the intersection of P with a finite number of closed halfspaces. Pick coordinates y 1, y 2 so that either K = P {y 2 = 0} if K is a line segment, or K {y 2 0} and P {y 2 = 0} = K {y 2 = 0}. Clearly K P since u is not affine linear. Thus the simple piecewise linear function u(y 1, y 2 ) = max{0, y 2 } has a crease which meets the interior of P, and so u is not affine linear.

16 Suppose for contradiction L(u) > 0. There is a constant C such that for any codimension 1 face of P the area of F is bounded by C F dσ. Fix any small positive ǫ < C 1 L(u). Then there is a δ > 0 such that {y P : f(y) < δu(y)} {0 y 2 < ǫ}. (7.2) To see this observe that as f 0 on all of P we certainly have G {y 2 0} since f = 0 if y 2 < 0. Now suppose for contradiction that for arbitrarily small δ there is a point y δ = (y δ 1, yδ 2 ) G with yδ 2 ǫ. Then f(yδ ) < δu(y δ ) and letting δ tend to zero and taking a convergent subsequence gives a (y 1, y 2 ) P with f(y) = 0 and y 2 ǫ, which is impossible as K {y 2 0}. Now consider g = { max{f δu, 0} if y2 0 f if y 2 < 0. (7.3) Observe that g is continuous, g 0 and g = 0 on K {y 2 = 0} = P {y 2 = 0} (it is here that we are using the crucial property of K from Lemma 7.5). Moreover g is convex on both {y 2 > 0} and {y 2 < 0} and thus by Lemma 7.6, g is convex on P. Now using (7.2) we in fact see that g = f δu if y 2 ǫ. Thus and so f g + δu f + δǫ, (7.4) L(g) + δl(u) = L(g + δu) L(f) + δǫc L(g) + δǫc where the first inequality uses (7.4) and the second uses the minimizing hypothesis of u and the fact g B since g f. But this contradicts the choice of ǫ, and therefore L(u) 0 as required. Remark 7.5. Thus in the case of toric surfaces K-stability is computable, by which we mean there is an algorithm that can determine if a given toric surface is K-stable (although it might be hard to implement even for given examples). As far as I am aware, it is an open question as to whether K- stability is computable in higher dimensions. This would of course follow if one could bound the complexity of the piecewise linear functions that one must consider which is a combinatorial problem independent of any considerations of Kähler geometry. Perhaps it is the case that for a polytope of dimension n it is sufficient to consider the Futaki invariant L(f) only over those f that can be written as a maximum of n affine linear functions? 8 Further topics Here we give a mini-guide to the literature for a selection of further related topics:

17 8.1 Extremal Metrics The original interest of csck metrics, due to Calabi, is from a variational point of view. The Calabi-functional on the space of Kähler metrics in a given cohomology class is given by the L 2 -norm of the scalar curvature. From this point of view it is natural to consider extremal metrics which are critical points of the Calabi function (whereas csck metrics are the minimizer if it exits). In the toric situation, a symplectic potential u corresponds to an extremal metric if and only if it satisfies u ij y i y j = A where A is the unique affine linear determined by the requirement that for all affine linear functions h, hdy = Ahdσ. P The above discussion must then be adapted with the Futaki invariant replaced by L(f) = L A (f) = fdσ Afdy. P P The analog of the Yau-Tian-Donaldson conjecture was adapted to extremal metrics by Szkelyhidi [22] and the algebro-geometric condition that is (conjecturally) equivalent is called relative K-stability. The work of Zhou-Zhu [27] is in fact proved for extremal metrics on toric manifolds and relative K-stability, the proof being essentially the same. The proof in Section 7 also works in the extremal setting, but here one must take extra case in the case that A is negative on part of P, which is dealt with in [24] using properties of the real Monge Ampére operator. Many parts of Donaldson continuity method involve extremal metrics, and the existence of extremal metrics on a relatively K-stable toric surfaces can be found in [7]. 8.2 Examples Because of the concrete nature of the topic, it is interesting to ask for specific examples either of stable or unstable polytopes, or of explicit symplectic potentials that give rise to csck metrics. The first example of an unstable polytope appears Donaldson s original work [9, Sec 7.2] in which he provides a polytope which is not K-stable but which is symmetric so that its (classical) Futaki invariant vanishes. Related examples can be found in [24] who show, among other things, that for any m 9 there is an K-unstable 2-dimensional Delzant polytope with m vertices. Examples of unstable polytopes in higher dimensions can be found in [15] and [20]. In terms of potentials, the examples of Calabi of the family of extremal metrics on the blowup of P at two points can be found in the original paper of P

18 Abreu [1]. For orbifold csck metrics there are further explicit examples due to Legendre (see below). 8.3 Kähler-Einstein metrics Kähler-Einstein metrics are a particularly interesting case of csck metrics, and can also be studied in the toric setting. Here the polarisation L is the anticanonical bundle K X which must be assumed to be positive, so we work with Fano toric manifold. A theorem of Wang-Zhu [25] states that there exists a Kähler-Einstein metric on a smooth toric Fano manifold if and only if the classical Futaki invariant vanishes on vector fields (in fact they say more, namely that there always exists a Ricci-Soliton). In the terminology used above this condition is just saying that the Futaki invariant L(f) vanishes for all linear functions f. In turn one can check this holds if any only if the center of mass of P (with respect to the volume form dy) equals the center of mass of P (with respect to the volume form dσ) [9, 3.2.9]. One can extend the scope of this investigation by studying Kähler-Einstein metrics for with the manifold (and/or the metric) is allowed certain mild singularities. This is a topic taken up in the recent work of Berman [6] and Berman-Berndtsson [5] who study the Kähler-Einstein equation through the (real) Monge Amére equation on convex bodies in R n. 8.4 Orbifolds Much of what has been said in these notes can be (and has been) extended to orbifolds (for the algebraic picture see [21]). For orbifold toric manifolds, the moment polytope becomes a polytope in R n along with a labeling of the n 1- dimensional faces. The orbifold version of the Delzant theorem is proved in [19]) and the Guillimen formula, the analog of Theorem 6.1 and Abreu s calculation of the scalar curvature in [2] (see also [8] for an alternative approach that is also for orbifolds). Some explicit extremal and csck metrics on orbifold toric surfaces can be found in [18]. 8.5 Flows There are various ways in which one can witness a csck metric (or a Kähler- Einstein) metric as the limit of some kind of flow on a Kähler manifold. From this point of view, the problem of existence comes down to understanding how these flows degenerate if there is no solution to be found. Very much related to this is the idea of an optimal test configuration, that is the worst in the sense that it makes a given manifold most unstable. For the Calabi-Flow on toric manifolds this has been looked at by Székelyhidi [23] who shows that if the optimal destabilising convex function is piecewise linear, then it decomposes the toric manifold into a number of semistable pieces, analogous

19 to the Harder-Narasimhan filtration for vector bundles. Related work of Huang can be found in [16; 17] Another important flow in this field is the Ricci flow, which has been studied in the toric setting by Zhu [28]. Bibliography [1] Miguel Abreu. Kähler geometry of toric varieties and extremal metrics. Internat. J. Math., 9(6):641 651, 1998. [2] Miguel Abreu. Kähler metrics on toric orbifolds. J. Differential Geom., 58(1):151 187, 2001. [3] Miguel Abreu. Kähler geometry of toric manifolds in symplectic coordinates. In Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001), volume 35 of Fields Inst. Commun., pages 1 24. Amer. Math. Soc., Providence, RI, 2003. [4] M. F. Atiyah. Convexity and commuting Hamiltonians. Bull. London Math. Soc., 14(1):1 15, 1982. [5] Robert Berman. K-polystability of q-fano varieties admitting kahlereinstein metrics. arxiv:1205.6214. [6] Robert Berman and Bo Berndtsson. Real monge-ampere equations and kahler-ricci solitons on toric log fano varieties. Preprint arxiv:1207.6128. [7] Li Sheng Bohui Chen, An-Min Li. Extremal metrics on toric surfaces. Preprint: arxiv:1008.2607. [8] David M. J. Calderbank, Liana David, and Paul Gauduchon. The Guillemin formula and Kähler metrics on toric symplectic manifolds. J. Symplectic Geom., 1(4):767 784, 2003. [9] S. Donaldson. Scalar curvature and stability of toric varieties. Journal of Differential Geometry, 62:289 349, 2002. [10] S. K. Donaldson. Interior estimates for solutions of Abreu s equation. Collect. Math., 56(2):103 142, 2005. [11] S. K. Donaldson. Extremal metrics on toric surfaces: a continuity method. J. Differential Geom., 79(3):389 432, 2008. [12] Simon K. Donaldson. Constant scalar curvature metrics on toric surfaces. Geom. Funct. Anal., 19(1):83 136, 2009. [13] A. Futaki. K ahler-einstein metrics and integral invariants. Springer- Verlag, Berlin, 1988. [14] V. Guillemin and S. Sternberg. Convexity properties of the moment mapping. Invent. Math., 67(3):491 513, 1982. [15] Yuji Sano Hajime Ono and Naoto Yotsutani. An example of asymptotically chow unstable manifolds with constant scalar curvature. arxiv.0906.3836 (to appear in Annales de L Institut Fourier). [16] Hongnian Huang. Convergence of the calabi flow on toric varieties and related kaehler manifolds. arxiv:1207.5969.

[17] Hongnian Huang. On the extension of the calabi flow on toric varieties. arxiv:1101.0638. [18] Eveline Legendre. Toric geometry of convex quadrilaterals. J. Symplectic Geom., 9(3):343 385, 2011. [19] Eugene Lerman and Susan Tolman. Hamiltonian torus actions on symplectic orbifolds and toric varieties. Trans. Amer. Math. Soc., 349(10):4201 4230, 1997. [20] Benjamin Nill and Andreas Paffenholz. Examples of Kähler-Einstein toric Fano manifolds associated to non-symmetric reflexive polytopes. Beitr. Algebra Geom., 52(2):297 304, 2011. [21] Julius Ross and Richard Thomas. Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics. J. Differential Geom., 88(1):109 159, 2011. [22] Gábor Székelyhidi. Extremal metrics and K-stability. Bull. Lond. Math. Soc., 39(1):76 84, 2007. [23] Gábor Székelyhidi. Optimal test-configurations for toric varieties. J. Differential Geom., 80(3):501 523, 2008. [24] Xu-jia Wang and Bin Zhou. On the existence and nonexistence of extremal metrics on toric Kähler surfaces. Adv. Math., 226(5):4429 4455, 2011. [25] Xu-Jia Wang and Xiaohua Zhu. Kähler-Ricci solitons on toric manifolds with positive first Chern class. Adv. Math., 188(1):87 103, 2004. [26] Bin Zhou and Xiaohua Zhu. K-stability on toric manifolds. Proc. Amer. Math. Soc., 136(9):3301 3307, 2008. [27] Bin Zhou and Xiaohua Zhu. Relative K-stability and modified K-energy on toric manifolds. Adv. Math., 219(4):1327 1362, 2008. [28] Xiaohua Zhu. Khler-ricci flow on a toric manifold with positive first chern class. arxiv:math/0703486. 20