Kähler-Einstein Metrics on Fano Manifolds

Transcription

1 Kähler-Einstein etrics on Fano anifolds Catherine Cannizzo Contents 1 Introduction Kähler metrics Holomorphic tangent bundle Connections Chern classes Calabi-Yau Theorem 7 3 Calabi-Futaki invariant Formula for the Calabi-Futaki invariant Example Example Fano surfaces Asymptotic Chow Stability Background Chow form and asymptotic Chow stability oment map and symplectic quotient Chow polystability is equivalent to balanced K-stability Background Weak K-stability is an obstruction Conclusion 34 1

2 1 Introduction One way to obtain information about the geometry of a complex manifold is by looking at the metrics it admits. When do complex manifolds admit metrics which give them a special structure? For example, requiring a metric of constant bisectional curvature puts a strong restriction on the manifold; in the case of a complete Kähler manifold, it then has universal cover CP n, C n or the unit ball in C n and its metric pulls back to the canonical metrics on these covers, up to scaling, [Tia00, Theorem 1.12]. So one may look for a slightly weaker condition, such as the existence of Kähler-Einstein metrics. These are Kähler metrics for which the Ricci form is proportional to the Kähler form. Kähler-Einstein KE metrics provide a special solution to the Einstein equation, which describes how space-time curves as a result of gravitation from mass and energy. An Einstein metric solves the Einstein equation in a vacuum, [Bes87]. In order for KE metrics to exist on a compact complex manifold, the first Chern class must be definite. Under this condition, the first Chern class separates complex manifolds into three cases. always admits a KE metric when c 1 0. In the case of positive first Chern class, is called a Fano manifold and does not always admit a Kähler-Einstein metric. Obstructions include the vanishing of the Calabi-Futaki invariant, asymptotic Chow stability and K-stability. The answer to when KE metrics exist is known for Fano surfaces. These surfaces have been classified by N. Hitchin in [Hit75]. CP 1 CP 1 and CP 2 admit Kähler-Einstein metrics. If = CP 2 p 1,..., p k, the blow up of CP 2 in k points in general position, then does not admit a Kähler-Einstein metric for k {1, 2} and it does for k {3,..., 8} [Tia00, pg 87]. The layout of this essay is as follows: in Section 1 I will give background for Kähler manifolds and Chern classes. In Section 2 I will state the Calabi-Yau theorem without proof and note some corollaries. In Section 3 I will introduce the Calabi-Futaki invariant, show CP 2 p and CP 1 CP 1 p do not admit Kähler-Einstein metrics, and discuss the other Fano surfaces. In Section 4 I will introduce asymptotic Chow stability and describe an equivalence between Chow polystability and balanced varieties as given in Wang [Wan04], and state the theorem of Donaldson [Don01] that the existence of a constant scalar curvature Kähler csck metric implies there is a sequence of balanced metrics converging to the csck metric, when the automorphism group is discrete. In Section 5 I will define Tian s K-stability and describe his proof [Tia97] that Kähler-Einstein implies weakly K-stable. Acknowledgments. I would like to thank Dr Julius Ross for advising me and for discussions about the material, as well as Emile Bouaziz and Ruadhaí Dervan for discussions on algebraic geometry; I attended a talk given by the latter in the Part III seminar series. 2

3 1.1 Kähler metrics The background material of this section is similar to that in [Tia00]. Let, g be a smooth Riemannian manifold. An almost complex structure J : T T is an endomorphism of the tangent bundle such that J 2 = id. The Nijenhuis tensor NJ is NJ : T T T NJu, v = [u, v] + J[Ju, v] + J[u, Jv] [Ju, Jv], u, v V ect is a complex manifold if it admits an almost complex structure J which is also integrable, meaning J is induced from multiplication by i on the holomorphic tangent bundle T 1,0 T C := T C, defined below. By a theorem of Newlander and Nirenberg, J is integrable if and only if NJ = 0. Definition 1. J is compatible with g if gju, Jv = gu, v, Definition 2. The Kähler form of g is defined as ω g u, v := gu, Jv u, v V ect Remark 3. The Kähler form is alternating by compatibility of J: ω g v, u = gv, Ju = gjv, u = gu, Jv = ω g u, v Let denote the Levi-Civita connection on. This is the unique torsion-free connection on such that g = 0. Definition 4. A Riemannian manifold, g with a compatible almost complex structure J is a Kähler manifold if J = 0. Then g is a Kähler metric. Throughout this essay, Kähler metric ω g means Kähler form ω g corresponding to Kähler metric g. Remark 5. Note that J = 0 = NJ = 0. This can be seen as follows: is symmetric so [u, v] = u v v u. Further, J Γ, EndT means X JY := X JY J X Y 1 for all vector fields X, Y on. Thus taking X = Ju, Y = v and X = v, Y = Ju respectively, and so J Ju v = Ju Jv Ju Jv J v Ju = v u v JJu NJu, v = [u, v] + J[Ju, v] + J[u, Jv] [Ju, Jv] = u v v u + J Ju v v Ju + J u Jv Jv u Ju Jv Jv Ju = u v v u J v Ju u Jv Ju Jv + Jv Ju = v JJu u JJv Ju Jv + Jv Ju = 0 if J = 0. So a Kähler manifold is a complex manifold. Equivalently, a Kähler manifold could be defined as a complex manifold with metric g such that the induced Kähler form ω g is d-closed. Then g is a Kähler metric. See [Tia00, Prop 1.5] for this equivalence. 3

4 1.1.1 Holomorphic tangent bundle Definition 6. Assume is a Kähler manifold. J induces a splitting of T C into eigenspaces T 1,0 T 0,1 corresponding to eigenvalues +i, i respectively, called the holomorphic and antiholomorphic tangent bundles. We can extend g C-linearly to g C on T C. Note that T = T 1,0 as real vector bundles; if has local real coordinates x 1,..., x 2n and local complex coordinates z j := x j +ix n+j, 1 j n, then this isomorphism is given by := 1 i x i z i 2 x i x n+i i x n+i z i for 1 i n. T 0,1 is generated by z i := 1 2 x i + i x n+i for 1 i n. Define a hermitian inner product on T 1,0 by hu, v = gu, v. In local complex coordinates, h = g ij dz i dz j, where g ij = g C z i, z j. Then from the definition of the Kähler form, ω g i,j is locally The metric h is hermitian since g is symmetric and Also ω g = ω g so the Kähler form is real Connections We can extend C-linearly to T C. Then ω g = i g 2 ij dz i dz j 2 i,j z i h ij = g ij = g ij = g ji = h ji z i z i = Γ k ij + Γ k ij z j z k z k = Γ k + Γ k z ij j z ij k z k = z i, therefore where Γ k ij denote the Christoffel symbols. Since J = 0, by Equation 1 we have z i J = J z j z i z j 3 So putting the Christoffel symbols in 3 and using that of J respectively, J = i z i z j z i z j = i Γ k ij z i, z i + Γ k ij z k z k are in the i and i eigenspaces 4

5 and J z i = J z j = i Γ k ij Γ k ij = Γ k ij = 0 + Γ k ij z k z k Γ k ij z k z k By similar calculations, all Christoffel symbols are zero except Γ k ij and Γk. Then the connection ij matrix for the induced connection on T 1,0 is θ given by As is Kähler, dω g = 0. Note that z j = θ k j dω g = 0 i,j,k g ij z k z k = Γ k ijdz i z k 4 [ gij dz k + g ] ij dz k dz i dz j = 0 5 z k z k = g kj z i, g ij z k = g ik z j 6 Since g ij = g ij = 0 by compatibility of J, we can write out the Christoffel symbols as Γ l ij = 1 gir 2 glr + g jr g ij = g lr g ir 7 z j z i z r z j 1.2 Chern classes Set n := dim C. It is a property of the Chern connection on that its induced curvature form on K 1 := n T 1,0 is equal to the trace of the curvature form on T 1,0, that is Θ K 1 = trθ T 1,0 Let e 1,..., e n be a local frame for T 1,0 so e 1... e n is a local frame for K 1. Since K 1 is a line bundle, the curvature form induced by the Chern connection is Θ K 1 where h = he 1... e n, e 1... e n := deth ij = detg ij. = log h 8 is also a Riemannian manifold. The Riemannian curvature induced by g is R ijkl = R,,, = g R,, z i z j z k z l z i z j z l z k and the Ricci curvature R kl = g ij R ijkl. Using Equation 7 we can obtain R ijkl. Recall Thus taking u = z i, v = Ru, vw = u v w v u w [u,v] w z j, w = z l R z i, we have = R r 9 z j z ijl l z r 10 z l = z i z j 5

6 Using the product rule for differentiation of g pr g lr = δ p l we find Also from 7 above Thus 10 is z i Γ r jl g pr z i = z r z i = = g ps g nr g ns z i Γ r jl = gpr g pj z l g pr g pj z l z r g ps g nr g ns z i g pj z l + g pr 2 g pj z i z l z r The Ricci curvature is = R ijkl = R r ijl g rk = g ps g ks z i g pj z l g kj z i z l R ij = log det g z i z kl j The Ricci form is defined to be Ricg = i R 2 ij dz i dz j = i 2 log det g ij i,j Note that the Ricci form is, up to a factor of i 2, the same as Θ K 1 in 8. On a Kähler manifold, the Chern connection and Levi-Civita connection are equivalent on T 1,0 = T [Huy05, Prop 4.A.9]. Let Ω j i = gjp R ipkl dz k dz l. By the equivalence between Chern and Levi-Civita connections, we can define the Chern classes in terms of Ω. Let c = det I + t i 2π Ω. It is a fact that the coefficients on t k are real closed k, k forms and their cohomology classes in H k,k, C H 2k, R are independent of g. [Huy05, pgs , 198]. The kth Chern class c k is defined to be the cohomology class represented by this coefficient. Definition 7. The first Chern class c 1 is i 2π [trω] = 1 π [Ricg]. We write c 1 > 0 if the first Chern class can be represented by a form with coefficients in local coordinates given by 1 φ ij for φ ij positive definite, c 1 < 0 if c 1 > 0 and c 1 = 0 if it can be represented by a form cohomologous to zero. Definition 8. [ω] H 1,1, C H 2, R is a Kähler class if it can be represented by a form corresponding to a Kähler metric, i.e. we can choose ω to be i 2 i,j g ij dz i dz j for some g ij positive definite. Remark 9. In particular if c 1 > 0 for a compact complex manifold, then c 1 is a Kähler class since it can be represented by a positive definite closed 1, 1 form, so is a Kähler manifold. In general if L is a positive line bundle over, i.e. a holomorphic line bundle with c 1 L > 0, then is Kähler. The case c 1 > 0 is the special case L = K 1. 6

7 2 Calabi-Yau Theorem Definition 10. A Kähler metric g on a Kähler manifold is said to be Kähler-Einstein if Ricg = λω g for some λ R. In this case, is called a Kähler-Einstein manifold. The question of when a complex manifold admits a Kähler-Einstein metric has been answered in the cases c 1 0. The answer makes use of the following theorem first conjectured by Calabi and later proved by Yau. Theorem 11 Calabi-Yau. Let be a compact Kähler manifold. Let Ω be a representative form for πc 1 and [ω] H 1,1, C H 2, R a Kähler class. Then there exists a unique Kähler metric g with ω g [ω] such that Ricg = Ω. Thus when c 1 = 0, the Calabi-Yau theorem implies has a Ricci flat metric g, i.e. Ricg = 0, so g is a Kähler-Einstein metric. Aubin and Yau independently proved that when c 1 < 0, there exists a unique Kähler-Einstein metric g such that Ricg = ω g. The full answer is in progress when c 1 > 0. In this case, is called a Fano manifold. Some ideas in the proof of Theorem 11. I will not give a proof of the Calabi-Yau theorem but will note a few points from the proof in [Tia00, Theorem 5.1]. The proof makes use of the -lemma which will also be used later. The part of the lemma needed is the following. Lemma 12 -Lemma. Let, g be a compact Kähler manifold. Suppose α H 1,1, C is d-exact. Then there exists a smooth function β such that α = β. Remark 13. If ω 1 and ω 2 are two cohomologous Kähler forms associated to Kähler metrics in particular they are real, then by the -lemma ω 1 ω 2 = β is real so β = β = β = β which implies we can choose β such that β = i f for some f C, R. Proof of Lemma, [Huy05, Cor ]. Since α is d-exact we can write α = dη for some η H 1, C. is Kähler so the notions of, and d harmonicity are equivalent. If, denotes the inner product on p, q forms given by ψ, η g C ψ, ηωg n = hψ, ηωg n then d is the formal adjoint of d with respect to this inner product. ψ is d-harmonic if and only if dψ = d ψ = 0. Thus for all ψ H 1,1, C, the space of harmonic 1, 1 forms, α, ψ = dη, ψ = η, d ψ = 0 i.e. α H 1,1, C. We know dα = 0, thus α = α = 0. Since α ker, by the Hodge decomposition for, α is in the direct sum of the harmonic 1, 1 forms and the image of on 0, 1 forms. We know α / H 1,1, C hence α = γ for some 0, 1 form γ. Again by Hodge decomposition, now for γ = β + β + β = α = β + β = β β 7

8 for some β H 0,1. Note = by the Hodge identities. So α = 0 = β = 0 = 0 = β, β = β, β = β 2 = β = 0 = α = β Returning to the Calabi-Yau theorem, in local coordinates ω g = i g 2 ij dz i dz j i,j Ricg = i 2 log detg ij Since Ricg and Ω are cohomologous, the -lemma says there exists a real smooth function f such that Ω Ricg = i 2 f We normalize f so that e f ωg n = ωn g, and then such an f is unique. We want to find some metric ω [ω g ] such that Ricω = Ω. Again by the -lemma, ω must be of the form ω g + i 2 φ, so ω corresponds to the metric with coefficients g ij + 2 φ z i z j. Then Ricω g + i 2 φ = Ω = Ricg + i 2 f In local coordinates this is i 2 log det g ij + 2 φ = i z i z j 2 logdetg ij + i 2 f = f = log det g ij + 2 φ z i z j detg ij = f + c = log det g ij + 2 φ z i z j detg ij for some constant c, since harmonic functions on a compact complex manifold are constant. The left hand side is defined globally so the right hand side is as well. Exponentiating both sides gives det g ij + 2 φ = e f+c detg z i z ij 11 j Equation 11 is equivalent to ω g + i 2 φn = e f+c ω n g 12 8

9 Note that using Stokes theorem for 0 < m n ωg n m φ m = = = 0 ω n m g dω n m g φ m 1 φ φ m 1 φ since ω g is closed and we can replace with d since the form is a n 1, n form. So integrating both sides of Equation 12 over implies ωg n = e f+c ωg n so by the normalization condition above, c = 0. Thus to find an ω as in the Calabi-Yau theorem, we need to solve the complex onge-ampère equation, which is ω g + i 2 φn = e f ω n g 13 Yau s proof of this conjecture is given in [Tia00, Theorem 5.1] and involves a continuity argument on solutions to ω g + i 2 φn = e fs ω n g 14 where f s := sf + c s, s [0, 1], and c s are constants uniquely determined by the normalization condition e fs 1ω n g = 0. It is shown that the set S = {s [0, 1] there is a solution to 14 for all t s} is open and closed and non-empty, since it contains zero by setting φ = constant hence S = [0, 1] and there is a solution at f. 3 Calabi-Futaki invariant There are obstructions to the existence of Kähler-Einstein metrics on Fano manifolds. The vanishing of the Calabi-Futaki invariant is necessary and is an obstruction related to holomorphic vector fields. Assume is a compact Fano manifold. Let Ka denote the set of Kähler classes on and η the space of holomorphic vector fields on, i.e. in local coords z 1,..., z n, vector fields of the form X i z i with X i holomorphic. Choose [ω] Ka and Kähler metric ω g [ω]. Let sg denote the complex scalar curvature of g, i.e. sg = g ij R ij locally. Define a function h g on by sg 1 sgωg n = h g V where V = ωn g is the volume. Definition 14. The Calabi-Futaki invariant f is f : Ka η C f [ω g ], X = Xh g ωg n 9

10 Calabi and Futaki proved Theorem 15. f [ω], X is a holomorphic invariant independent of the representative g chosen in [ω]. In particular, if there exists a constant scalar curvature metric ω g [ω], then f [ω], = 0. We can restrict the first argument to πc 1 > 0. Let πc 1 = [ω g ] = [Ricω g ]. So Ricω g ω g = i 2 h g for some function h g, by the -Lemma. A Kähler manifold locally admits normal coordinates about any x [Tia00, Prop 1.6], where g ij x = δ ij. We can assume ω g = i 2 dzi dz i, Ricg = i 2 Rii dz i dz i at x, and Ricg ω n 1 g = n 1! i R ii ω n g n! at x. Thus 1 V sgω n g = n V = n V = n V = 1 n sgωn g Ricg ω n 1 g ω g + i 2 h g ω n g = n ω n 1 g where the last step follows since h g ωg n 1 = h g ωg n 1 as ω g is closed, which equals dh g ωg n 1 as h g ωg n 1 is an n 1, n form, so this integral vanishes by Stokes theorem. So in the special case [ω g ] = πc 1, f [ω g ], gives Futaki s invariant [Fut88, 3.1] where h g is equivalently defined as Ricω g ω g = i 2 h g 15 This is equivalent since contracting the coefficients in 15 with g ij gives sg n = h g and we already saw that n = 1 V sgωn g. Remark 16. In πc 1 > 0, the notions of constant scalar curvature Kähler csck metrics and KE metrics are equivalent. We have πc 1 = [ω g ] = [Ricg] and Ricg ω g = i 2 h g. KE implies csck since Ricg = λω g implies locally R ij = λg ij therefore sg = nλ by taking the trace of both sides, where n = dim C. Conversely, suppose ω g πc 1 has constant scalar curvature. We know n = 1 V sgωn g. Thus if s is constant, s = n so h g is harmonic on a compact manifold hence constant. Therefore Ricg = ω g and, g is Kähler-Einstein. Proof of Theorem 15, [Fut83],[TD92]. The following proof is for the Kähler-Einstein case using the Futaki invariant and follows [Fut83] and [TD92]. I will give the idea behind the proof for the general case, given in [Tia00, Theorem 3.3]. The second statement of the theorem is clear in the more general case; if sg is constant then h g = 0 so h g is constant. Hence Xh g = 0 X η and f [ω], = 0. Let f X := f πc 1, X. To show f X is independent of the representative g chosen, it suffices to show f t X := Xh g t ω n t is locally constant for an arbitrary differentiable family 10

11 of Kähler metrics ω t := ω gt in πc 1, that is d dt f tx = 0. This will suffice since the set of all Kähler metrics in πc 1 is a cone, hence contractible. Let πc 1 be represented by ω. By the -lemma, there exists smooth real functions ψ t such that ω t ω = i 2 ψ t Then differentiating with respect to t gives ω t t = i 2 ψt t Set φ t := ψt t. Here ωn t denotes detg t ij. Then using Ricω t ω t = i 2 h t t ωn t = n ω t t ωn 1 t = ni 2 φ t ωt n 1 t Ricω t = i 2 t log ωn t = i 2 φt ωt n ωt n = φ t ω n t = i 2 φ t t h g t = 2i t Ricω t ω t So choose h gt h t such that Then = φ t φ t d dt f tx = = = h t t = φ t φ t The following argument is from [TD92]. Note that t Xh tωt n ht X + Xh t φ t ωt n t X φ t φ t + Xh t φ t ω n t X φ t = X i jk g 2 φ t t z i z j z k = X i jk g 2 φ t X i t φ t z i z j z k z i The first term vanishes when integrating over, by the divergence theorem, and the second term is divx φ t. Similarly we can replace X φ t with divx φ t. Also in local coordinates, i X ω g = i 2 Xj g jk dz k and ω g is closed so g ij z k = g ik z j. Since the X j are holomorphic 11

12 2ii X ω t = X j g t jk dz k g t jk [ = X j dz i dz k + z i i<k i>k g t jk [ = X j dz i dz k z i i<k i<k g t jk z i g t jk z i dz i dz k dz i dz k ] ] = 0 So by the Hodge theorem ι X ω t = α t + η t for some harmonic 0, 1 form α t and smooth function η t. The α t will vanish in the integral n.b. α t, φ t = α t, φ t = 0 so we can assume it is zero. Then L X ω t = dι X ω t = ι X ω t by Cartan s formula and divxωt n = L X ωt n, so ι X ω t = η t = divx = η t Since φ t, η t = φ t, η t with respect to the inner product, induced by integrating over d dt f tx = [divx + Xh t φ t + divxφ t ] ωt n = η t + Xh t + η t φ t ωt n We show η t + Xh t + η t = 0. Since 2 h t z i z j = R ij g t ij where R ij depends on t Xh t = X i h t z i = X i 2 h t z i z j dz j = ι X Ricg t ω t η t = ι X ω t and from the definition of Ricg t we show η t = ι X Ricg t from [Tia00, pg 25] ι X Ricg t = i 2 2 Xi log detg t z i z kl dz j j = i 2 X i log detg t z kl i = i X 2 i kl g g tkl t z i = i X 2 i kl g g til t z k = i 2 kl g t X i g t z il g t kl X i g t il k z k = i 2 kl g t X i g t z il k kl = g t η t z k z l = η t 12

13 The fact that X is holomorphic is used in the second line and sixth lines, the definition of the 1 inverse of a matrix A as det AadjA, where adja is the adjugate matrix, is used in the third line, the fourth line uses that ω g is closed, the fifth line is the Chain rule, and the penultimate line uses ι X ω t = η t. So η t + η t + Xh t = 0 and by the chain rule and divergence theorem d dt f tx = 0. In the more general case, one can show that f [ω], X = θ X g h g ω n g, for a specified function θ X. By the Hodge theorem, i X ω g = i 2 α + θ X for some harmonic 1-form α and smooth function θ X. Thus X j = g jk α k + θ X z k. Let g denote the -Laplacian on functions, which is. Note that the inner product g C ψ, η on forms is the dual of the inner product on vector fields, so has coefficients g jk. Then f [ω], X = Xh g ωg n = X j h g ωg n z j = g jk α k + θ X hg ωg n z k z j = h g, α + h g, θ X = α, h g + θ X, h g = α, h g + θ X, h g = θ X g h g ωg n The final line follows since α is harmonic, so α = 0. Then defining F g, X = n + 12 n+1 h g θ X ω n g, one takes a family of metrics {g t } in the given Kähler class [ω] and shows d dt F g t, X t=0 = 0, done in [Tia00, pg 24 27]. 3.1 Formula for the Calabi-Futaki invariant With the additional condition that X V ect be non-degenerate, Futaki gives a formula for f πc 1,, by looking at the zero set of X, [Fut88]. Here is the set-up. Suppose Z is a smooth complex submanifold and let N Z denote the normal bundle T /T Z to Z in. A metric g on induces an orthogonal decomposition T Z = T Z N Z. If is the Levi-Civita connection on, X induces a section DX of EndN Z, given by restricting to vectors in N Z, taking the covariant derivative of X in the given direction, and projecting the result to N Z. With the orthogonal decomposition above, DX = X N Z where X denotes the component in T perpendicular to T Z. Definition 17. We say X V ect is non-degenerate if zerox = λ Λ Z λ where the Z λ are smooth complex connected submanifolds, and D z X : T z /T z Z λ T z /T z Z λ 13

14 is a non-degenerate linear map of vector spaces, i.e. has nonzero determinant, for all z Z λ, λ Λ. Then Theorem 18 [Fut88, Theorem 5.2.8]. For X a non-degenerate vector field on f πc 1, X = πn n + 1 [trl λ X + c 1 ] Zλ n+1 λ Λ detl λ X + i 2π K λ 16 where n = dim C, L λ X = X N Zλ and K λ is the induced curvature form on N Zλ. Consider the special case where is a complex surface and Λ = Λ 0 Λ 1, where Λ i consists of i dimensional submanifolds. Since only forms of degree 2 dim Z λ contribute to the integral in 16, the sum over λ Λ 0 becomes π 2 3 trl λ X3 detl λ X λ Λ 0 For one-dimensional submanifolds Z λ, T Z λ and the normal bundle are line bundles, i.e. rank one vector bundles, so L λ X and K λ are both one-by-one matrices hence we can omit the trace and determinant. Note that X is non-degenerate so L λ X 0. Then using an expansion for the denominator and omitting terms not of degree 2 we find Zλ L λ X + c 1 3 = = = Z λ Z λ Z λ L λ X + i 2π K λ L λ X 3 + 3L λ X 2 c 1 L λ X1 + i 2π K λl λ X 1 L λ X 2 + 3L λ Xc 1 1 i 3L λ Xc 1 i 2π L λxk λ = L λ X2c 1 Z λ + 2 2gZ λ 2π K λl λ X 1 The final step was obtained as follows. K λ is the induced curvature form on the line bundle N Zλ thus [ i 2π K λ] = c 1 N Zλ. Note that T Zλ = T Z λ N Zλ = c 1 T Zλ = c 1 T Z λ + c 1 N Zλ 17 Then using the pairing of cohomology on homology, given by integrating a form over a submanifold, we find by the Gauss-Bonnet theorem C λ c 1 Z λ = c 1 T Z λ = Z λ Z λ Z λ 2π Φ = χz λ where Φ is the volume form and C λ is the Gaussian curvature of Z λ, using the fact that C λ Φ is i times the curvature form on the line bundle T Z λ, see [GH94, pg 77]. Further c 1 T Zλ Z λ = c 1 T Zλ = c 1 T Z λ Z λ = c 1 Z λ = c 1 Z λ 14

15 So by Equation 17 c 1 N Zλ Z λ = c 1 T Zλ Z λ c 1 T Z λ Z λ i 2π K λ = c 1 N Zλ Z λ Z λ and we have the result above. So f πc 1, X = π Example 1 = c 1 Z λ χz λ = c 1 Z λ 2 2gZ λ trl λ X3 detl λ X + π2 L λ X2c 1 Z λ + 2 2gZ λ 3 λ Λ 0 λ Λ 1 This example was done in [Tia00, pg 32 33]. Consider CP 2 p, the blow up of CP 2 in a point p. Using the automorphism group of CP 2, SL3, C/ where A λa, λ C, we may assume p = [1 : 0 : 0]. The blow up of a general complex manifold in a point p is given by taking a local coordinate chart centred about p, blowing up at the origin in C n and then glueing the resulting B 0 C n back onto the manifold. The following theory is from [GH94, pg ]. Here we consider a neighbourhood U = {[x : y : z] x 0} = C 2 in CP 2. We have a parametrization given by f : C 2 U x, y [1 : x : y] and B 0 C 2 = {x, y, [ξ : η] xη = yξ} C 2 CP 1. Thus there is a commutative diagram B 0 C 2 C 2 π 1 > f > U where π 1 is the projection onto the first factor. Note that fπ 1 restricts to an isomorphism B 0 C 2 {0, 0 CP 1 } U {p}. Then the blow-up of CP 2 at p is obtained by glueing along this restriction = CP 2 \{p} fπ1 B 0 C 2 = CP 2 \{p} Ũ where Ũ = {[1 : x : y], [ξ : η] xη = yξ} and E = {[1 : 0 : 0], [ξ : η]} = CP 1 is the exceptional divisor which has replaced p. Points [1 : x : y], [ξ : η] Ũ\E are identified one-to-one with points [1 : x : y] CP 2 \{p}. Away from E, is isomorphic to CP 2, which is a complex manifold. We define local coordinates about E as follows. Take a cover V 0, V 1 of Ũ given by V 0 = {[1 : x : y], [ξ : η] Ũ ξ 0} V 1 = {[1 : x : y], [ξ : η] Ũ η 0} 15

16 On these open sets we have coordinates z1 0, z0 2 = x, η/ξ and z1 1, z1 2 = ξ/η, y since y and x are then respectively determined by xη = yξ. In particular E V 0 = {z1 0 = 0}, E V 1 = {z2 1 = 0}. Thus the transition functions for [E] E = N E are the inverse of those for the hyperplane bundle. Any line bundle L over CP 1 is a multiple of the hyperplane bundle. This multiple is denoted degl and corresponds to the image of L under the isomorphism {line bundles} = H 1 CP 1, O CP 1 = H 2 CP 1, Z = Z L degl N.B. This isomorphism arises from taking the long exact sequence of cohomology from the exponential sequence 0 Z O CP 1 O CP 1 0 and observing that H 1 CP 1, O CP 1 = 0 and H 2 CP 1, O CP 1 = H 0,2 CP1 = 0. degl is also the integer obtained by pairing cohomology and homology, c 1 LCP 1. Further, in the case of the normal bundle to a submanifold Z, this degree coincides with the self-intersection number of Z, denoted Z Z. Since the transition functions for [E] E are inverse those for the hyperplane bundle, we have E E = 1. Any other curve on which is not E can be considered a curve on CP 2, and these always have self-intersection number +1. To obtain a holomorphic vector field X, we first define a flow φ t on CP 2 and then lift it to. Note that φ t must fix p, so that when p is blown up to E, φ t lifts to where it fixes E. On U, set φ t [1 : x : y] = [1 : e t x : e t y] so φ t p = p. Extend this to CP 1 := {[0 : x : y]} by taking a limit φ t [0 : x : y] = lim λ φ t[1 : λx : λy] Note that φ t fixes CP 1 ; we have [1 : e t λx : e t λy] = [e t λ 1 : x : y] for λ 0, so as λ becomes large, we see taking the limit gives [0 : x : y] again. φ t lifts to by φ t [1 : x : y], [ξ : η] [1 : e t x : e t y], [ξ : η] and is defined on CP 1 as above. The fixed points of φ t on Ũ are precisely E, and everything outside Ũ, namely CP1, is fixed. So if X is the vector field induced by φ t F ixφt = E CP 1 = zerox = E CP 1 Both E, CP 1 are isomorphic to CP 1 so have genus 0 and Euler characteristic 2. As noted earlier E E = 1, CP 1 CP 1 = +1. It remins to compute L λ X. In coordinates about E, e.g. in V 0, the flow sends x 0 1, x 0 2 = x, η/ξ e t x, η/ξ Since d dt et x, η/ξ = e t x, 0, X is locally given by X = x 0 1. Since E V x 0 0 = {x 0 1 = 0}, we have 1 a local frame for T E V0 given by therefore we can choose as a local frame for N E. So with respect to the basis x 0 1 x 0 2 for N E, X = 1 x 0 1 x x 0 1 x 0 1 x 0 1 x 0 1

17 On E, x 0 1 = 0 so we get L λx = 1 on E. Next we find local coordinates about CP 1. CP 1 is contained in U 1 U 2 where U i := {[s 0 : s 1 : s 2 ] s i 0}. U 1 has coordinates u, v = s 0 /s 1, s 2 /s 1 on which φ t is u, v e t u, v so X = u u locally. CP1 U 1 = {u = 0} so we can choose N CP 1 As in the calculation above, u So putting everything together X = u on CP1 and L λ X = 1. to be generated by u. 1 π 2 f πc 1, X = 1 [2c1 E + 2 2gE] [2c 1 CP gCP 1 ] = 2 [χe + E E] [χcp CP 1 CP 1 ] = = therefore CP 2 p never admits a Kähler-Einstein metric. 3.3 Example 2 Let be the blow-up of CP 1 CP 1 at the point [1 : 0], [1 : 0]. That is, is the blow up of the image of the Segre embedding φ : CP 1 CP 1 CP 3 at p := [1 : 0 : 0 : 0]. Let X = φcp 1 CP 1 and w 0,..., w 3 be coordinates on CP 3. So X is the zero set of w 0 w 3 w 1 w 2. Note that if we blow up CP 1 CP 1 in any point p, using the automorphism group P SL2, C P SL2, C CP 1 CP 1 we may assume p = [1 : 0], [1 : 0]. On X, p lies in the open set U := U 0 X on X, where U i is the open set in CP 3 consisting of points with nonzero ith coordinate. We have local parametrization f : C 2 U x, y [1 : x : y : xy] The projection onto the first factor is π 1 : B 0 C 2 C 2 as earlier. Thus = X\{p} fπ1 B 0 C 2 = X\{p} Ũ where Ũ = {[1 : x : y : xy], [ξ : η] xη = yξ}. The exceptional divisor is E = {[1 : 0 : 0 : 0], [ξ : η]}. Take V 0 = {[1 : x : y : xy], [ξ : η] Ũ ξ 0} V 1 = {[1 : x : y : xy], [ξ : η] Ũ η 0} Then V 0 has coordinates x 0 1, x0 2 = x, η/ξ and V 1 has coordinates x 1 1, x1 2 = ξ/η, y, where E is given by {x = 0} and {y = 0} on V 0 and V 1 respectively. So as in Example 1, E E = 1. Define a flow on CP 3 given by φ t [w 0 : w 1 : w 2 : w 3 ] = [w 0 : e t w 1 : e t w 2 : e 2t w 3 ] 17

18 This restricts to a flow on X and fixes p, so lifts to a flow on fixing E. Also [0 : e t w 1 : e t w 2 : e 2t w 3 ] = [0 : w 1 : w 2 : e t w 3 ] = [0 : w 1 : w 2 : w 3 ] t = w 3 = 0 Thus {[0 : w 1 : w 2 : 0]} X = [0 : 1 : 0 : 0] [0 : 0 : 1 : 0] = F ixφ t = E q 1 q 2 where q 1, q 2 are the two fixed points. The flow locally on V 0 is x 0 1, x 0 2 e t x 0 1, x 0 2 thus X := d dt φ t = x 0 1 x 0 1 on V 0 gives X E = E x 0 1 x 0 1 = dx 0 1 x 0 1 = L λ X = 1 on E x 0 1 where we used that x 0 1 = 0 on E so is a generator for N E. About q 1 we have open set U 1 and coordinates u, v = w 0 /w 1, w 3 /w 1 and then w 2 is determined by w 0 w 3 = w 1 w 2. Here φ t u, v = e t u, e t v = X U1 = u u + v v Then X at q 1 is X = u u + v v = du u + dv v since X vanishes at q 1. As q 1 is a 1 0 point, its normal bundle is all of T q1 so L λ X is a 2 2 matrix given by L λ X = 0 1 therefore trl λ X = 0. The calculation for q 2 is identical, with coordinates u, v = w 0 /w 2, w 3 /w 2, φ t u, v = e t u, e t v, X U2 and L λ X are the same, so trl λ X = 0. So the points do not contribute to f. Then f πc 1, X = π2 3 2c 1E + 2 2gE = 2π2 χe + E E = 2π2 4π = So CP 1 CP 1 p never admits a Kähler-Einstein metric. 18

19 3.4 Fano surfaces We ve shown above that CP 1 CP 1 p and CP 2 p do not admit Kähler-Einstein metrics. CP n admits the Fubini-Study metric, given by where z 2 = n z i 2 for local coordinates z 1,..., z n on CP n ω F S = i 2 log1 + z 2 = i 2 i ωf n S = n! 2 i=1 i dz i dz i 1 + z 2 i z idz i j z jdz j 1 + z n dz 1 dz 1... dz n dz n 1 + z 2 n+1 19 The latter equality can be seen as follows: z i dz i 2 z j dz j = z i z j z k z l dz i dz j dz k dz l i j = i<k = i<k Thus in ω n F S, the terms Thus i,j,k,l z i z j z k z l dz i dz j dz k dz l + i>k z i z j z k z l dz i dz j dz k dz l z i z j z k z l dz i dz j dz k dz l dz i dz j dz k dz l = 0 i z idz i j z jdz j 1+ z 2 2 k for k 2 are zero. Then 2iω F S n i = dz i dz n i i 1 + z 2 n dz i dz n 1 i i 1 + z 2 z idz i j z jdz j 1 + z 2 2 = i dz i dz i n 1 + z z 2 n+1 z 2 i dz i dz i n 1 + z 2 n+1 So CP n is Kähler-Einstein. = i dz i dz i n 1 + z 2 n+1 Ricω F S = i 2 logωn F S = i 2 log z 2 n+1 = n + 1 i 2 log1 + z 2 = n + 1ω F S := CP 1 CP 1 is also Kähler-Einstein. Let π i : CP 1 be the projection onto the ith factor. admits a Kähler structure as a product of two Kähler manifolds, induced by the metric g := g 1 F S + g2 F S where g 1 F S + g 2 F Su, v = g 1 F Sπ 1 u, π 1 v + g 2 F Sπ 2 u, π 2 v So in local coordinates z 1, z 2 on CP 1 CP 1 ω = i dz1 dz z dz 2 dz z = ωf 1 S + ωf 2 S 19

20 Then Ricω = i log ω2 2 = i 2 logω1 F S ωf 2 S = i [log z = 2ω 1 F S + ω 2 F S = 2ω + log z Theorem 19 [Hit75]. The Fano surfaces are CP 1 CP 1 and CP 2 p 1,..., p k blown up at k points in general position for 0 k 8. Note that CP 2 p 1, p 2 = CP 1 CP 1 p, see [GH94, pg ]. Ideas in the proof of 19. Hitchin s proof showed that c 1 > 0 implies is birational to CP 2, and further blowing down does not change the sign of c 1 so we may look at the minimal models for rational surfaces, which are CP 2 and F n := PH n 1, n 1, where H is the hyperplane bundle over CP 1. Using the Riemann-Roch theorem for surfaces for a non-singular rational curve D on gives c 1 [D] + [D] 2 = 2 so using c 1 > 0 one can show this implies [D] 2 > 2. The manifold F n has a rational curve with self-intersection n so we can exclude all F n except F 0 = CP 1 CP 1. As CP 1 CP 1 p = CP 2 p 1, p 2, we can assume we blow up in points on CP 2. The first Chern classes of a blow up π : ˆ of in a point are related by c 1 ˆ = π c 1 [E] where E is the exceptional divisor. If we blow up CP 2 in k points then this implies c 1 CP 2 p 1,..., p k 2 = 9 k > 0 using the result that c 1 CP 2 corresponds to the integer 3. So we require k 8. We must blow up in distinct points since blowing up in a point gives an exceptional divisor of self-intersection 1 and blowing up again in the same point gives a curve of self-intersection 2, but [C] 2 > 2 for all non-singular rational curves C. The following theorem and the previous examples answer the Kähler-Einstein question for Fano surfaces. Theorem 20 [Tia00, pg 87]. The Fano surfaces CP 2 p 1,..., p k for 3 k 8 and p i in general position all admit Kähler-Einstein metrics. ] 20

21 4 Asymptotic Chow Stability The existence of KE metrics is also related to stability. We can generalize to the case of a compact complex manifold with positive line bundle L. Then the pair, L is called a polarized manifold. c 1 L is represented by a positive closed 1, 1 form, locally given by i 2 i,j g ij dz i dz j, where g ij is a positive definite hermitian matrix. Then g := i,j g ij dz i dz j is a hermitian Kähler metric and is a Kähler manifold. Note that since L is positive, for sufficiently large k we can embed ι k : CP N k via sections of L k. This is the statement of the Kodaira Embedding theorem, [GH94, pg 181]. We seek to find a constant scalar curvature Kähler csck metric in c 1 L, which is a generalization of finding a KE metric in c 1. I will describe some results related to this question and then give the background behind them. Let Aut, L denote the subgroup of AutL consisting of automorphisms of L which commute with the C -action on fibers. In particular, these descend to automorphisms of, so Aut, L can be identified with a subgroup of Aut. When Aut, L is discrete its Lie algebra is trivial. Donaldson proved in [Don01], for the sequence of metrics ω k := 2π k ι k ω F S, Theorem 21 [Don01]. Suppose, L is a polarized manifold and Aut, L is discrete. If ω is a csck metric in 2πc 1 L, then, L k is balanced for all sufficiently large k and the sequence of metrics ω k converges in C to ω, as k. Corollary 22. When Aut, L is discrete, if 2πc 1 L admits a csck metric it is unique. Wang showed using moment maps and symplectic reduction Theorem 23 [Wan04]. Let, L be polarized by a very ample line bundle L, with embedding P N via L. Then, L is Chow polystable if and only if it can be balanced. Theorem 23 was originally due to Zhang, and there is a proof by Paul as well. Corollary 24. Asymptotic Chow stability of a polarized manifold, L is an obstruction to the existence of csck metrics in c 1 L when Aut, L is discrete. 4.1 Background Chow form and asymptotic Chow stability The Chow form gives a way of parametrizing polarized manifolds. Chow polystability is defined in terms of stability of the Chow form. There are two equivalent ways of defining the Chow form. Let, L be a polarized manifold as above with embedding ι k : CP N k. Set n := dim C and d k is the degree of ι k CP N k. 1 Consider the set of all N k n + 1 dimensional subspaces V of CP N k, i.e. points in GN k n, N k + 1, the Grassmannian of N k n dimensional subspaces of C Nk+1. Given a set of coordinates e 1,..., e n on C n, we have a set of coordinates e i1... e ir, 1 i 1 <... < i r n on r C n, called the Plücker coordinates. Then GN k n, N k + 1 embeds into P N k n C n by sending a space V spanned by e 1,..., e Nk n to the one dimensional space spanned by e 1... e Nk n in N k n C n. This is called the Plücker embedding. V does not generically intersect since dim V + dim = dim CP N k 1. However, the set of V such that V φ forms an irreducible codimension one subvariety of GN k n, N k

22 and hence is defined by the vanishing of some polynomial degree d k in the Plücker coordinates of GN k n, N k + 1. This is proved in [GKZ94, Chapter 3, 2, A and B]. This polynomial is the Chow point or Chow form. It is a point in the Chow space CHOW P N k n, d k PH 0 GN k n, N k + 1, O G d k whose points parametrize polarized varieties, L where L k induces an embedding ι k : P N k, n = dim C and d k = degι k. This is the approach taken in [Wan04]. 2 Following Futaki [Fut11], let V k = H 0, L k so N k +1 = dim V k. Elements in PV k define hyperplanes in PV k so if D := {H 1,..., H n+1 PV k... PV k H 1... H n+1 φ} then D is a divisor in PVk... PV k defined by a polynomial in Symd kv k n+1, also called the Chow form. These two definitions of the Chow form are equivalent as described by umford [um77, pg 16 17]. Let Chow, L k denote the kth Chow form of, L as defined above. Note that G := SLN k + 1 acts on CP N k, which corresponds to changing the chosen basis for H 0, L k. This induces an action on Sym d kv k n+1 or PH 0 GN k n, N k + 1, O G d k so we can consider the orbit of Chow, L k. Chow stability of corresponds with the geometric invariant theory GIT stability of Chow, L k with respect to this G-action. Definition 25. Let, L be a polarized manifold. 1. is Chow polystable w.r.t. L k if the orbit of Chow, L k under G is closed. 2. is Chow stable w.r.t L k if it is polystable and the stabilizer of Chow, L k is finite. 3. is asymptotically Chow stable w.r.t. L if there exists a k 0 > 0 such that for all k k 0, is Chow stable w.r.t L k oment map and symplectic quotient I learned the following background from [DK90], [Tho06], [Wan04] and Wikipedia. Definition 26. A symplectic manifold is a smooth manifold equipped with a closed, nondegenerate global 2-form ω. Definition 27. We say an automorphism g of is symplectic or a symplectomorphism if it preserves ω, i.e. g ω = ω. For example a Kähler manifold is symplectic, where ω is its Kähler metric. If = CP N is equipped with the Fubini-Study metric ω F S = i 2 log z 2 then g GLN + 1, C preserves ω F S if and only if g UN + 1 since gz 2 = z T g T gz = z T z T = z 2 for all z = z 1,..., z N+1 if and only if g T g = I i.e. g UN

23 Let G be a compact Lie group acting on by symplectomorphisms. Let g denote the Lie algebra of G. Each element ξ g defines a vector field σξ on by considering its infinitesimal action. On T z the vector σ z ξ is defined to be σ z ξ := d dt exptξ z t=0 where exptξ is the 1-parameter subgroup 1-PS of G induced by ξ via the exponential map. Since G acts by symplectomorphisms the Lie derivative L σξ ω = 0. Thus using Cartan s formula and the fact that ω is closed we obtain 0 = L σξ ω = dι σξ ω + ι σξ dω = dι σξ ω So if H 1, R = 0, then ι σξ ω is exact. For example, when is Kähler and c 1 > 0, the Calabi-Yau theorem states that admits a metric with positive Ricci curvature, which implies is simply connected [Tia00, Remark 2.14]. So H 1, R = 0 in that case. Then we may write ι σξ ω = dm ξ for some function m ξ on. Note that in general if df = 0, then f lies in the kernel of d, which consists of locally constant functions on, so for simply connected f is constant. That is, m ξ is unique up to a constant. Definition 28. A moment map for the action of G on is a map where mz, ξ = m ξ z, such that m : g d m, ξ = ι σξ ω 20 The final equation is an expression in terms of z, where m, ξ z = mz, ξ, and ι σξ ωz = ι σzξω. That is, a moment map for the action of G on is obtained by combining all the components m ξ into a map. Here, denotes the evaluation pairing g g R. If we have a bi-invariant non-degenerate pairing on g g, such as taking the trace of the product of two matrices in the Lie algebra of the special linear group, then we can identify g with its dual via this pairing and consider m as a map g. Definition 29. G acts on itself by conjugation, which induces the adjoint action on g. ψg : G G h ghg 1 This induces Adg := dψg e : T e G T e G, the adjoint action of G on g. Then m is G-equivariant if mg z, ξ = Adg mz, ξ, z, ξ g where the right hand side is mz, Adg 1 ξ. Recall the m ξ are unique up to constants. It is possible to choose these constants such that m is a G-equivariant moment map. Then m is uniquely defined up to addition of a central element in g, [DK90],[Tho06]. 23

24 The moment map generalizes the notion of angular and linear momentum, hence its name. The possible locations of a particle in 3-space form its configuration space, R 3. The phase space corresponding to this configuration space is := T R 3 ; each point of corresponds to a unique state of the particle, describing its position and momentum. The group G of translations R 3 and rotations SO3 acts on and the components of the moment map m : g are the components of angular and linear momentum in each of the three directions. Assuming m is G-equivariant, G acts on m 1 0. Definition 30. The symplectic quotient of by G is //G := m 1 0/G When is a compact Kähler manifold we consider the action of a compact Lie group G and extend the action to its complexification G C, where g C = g + ig, via the almost complex structure J. So σiξ = Jσξ. For example, SUN + 1 SLN + 1, C, sln + 1, C = trace 0 matrices over C and sun + 1 = trace 0 skew-hermitian matrices, as a Lie algebra over R. Since any trace 0 matrix A can be written as a sum of trace 0 hermitian and skewhermitian matrices A = 1 2 A+AT A AT, we have sln +1, C = sun +1+i sun +1. As an example P N, ω F S has a canonical moment map µ F S µ F S : z 0 :... : z N 1 zi z j 2i z 2 δ ij sun + 1 N + 1 Note that the trace of the image is 1 z 2 2i δ z 2 ii N+1 = 1 1 = 0 as required. Proof follows [FK94, Example 8.1ii] and [Kir84, Lemma 2.5]. As noted above UN + 1 acts on P N by symplectomorphisms. Note that 1 z 2 2i δ z 2 ii N+1 can be identified with its dual element in sun +1 via the Killing form tra b for a, b sun +1. This is symmetric and bilinear. So µ F S z, a su = i 2 tr zi z j z 2 δ ij a jk N + 1 = 1 zi z j a ji 2i z 2 a ii N + 1 A G-equivariant moment map is unique up to addition of an element which is central in the Lie algebra, and in this case the central elements of un + 1 are constant scalar multiples of the identity which are skew-hermitian, i.e. elements of the form i ri N+1 for r R. Thus we ve chosen to add the constant i δ ij 2 N+1 above so the image of the moment map is in sun + 1, i.e. it is trace free. First note that µ F S is independent of the non-zero representative chosen for z in C N+1 ; this is because if we chose λz instead then the λ 2 cancel in the numerator and denominator. µ F S is SUN + 1-equivariant since for g SUN + 1 and z P N, µ F S g z, a = 1 z T g T agz 2i z 2 tra N + 1 = 1 2i z T g 1 agz z 2 = Adg µ F S z, a trg 1 ag N

25 Since UN + 1 is transitive on P N, it suffices to prove that Equation 20 holds at the point p = 1 : 0 :... : 0 which corresponds to the origin in the coordinate chart on U 0, the set of points with nonzero first coordinate. The vector field induced by a sun + 1 takes the value at p in coordinates z 1,..., z n on U 0 given by d dt e ta p = d t=0 dt I + ta + Ot 2 p t=0 From Equation 18, at p = a p = a 10,..., a n0 ω F S = i 2 Thus in coordinates on U 0 n.b. p 2 = 1 using that a is skew-hermitian. d p µ F S, a = d p [ 1 2i n dz i dz i i=1 = 1 2i d p z i z j a ji = 1 2i n zi z j a ji z 2 a ] ii N + 1 i,j=0 k=1 n a ji [δ ik z j dz k + δ jk z i dz k ] p = 1 a 0k dz k + a k0 dz k 2i k = i a k0 dz k a k0 dz k 2 k = ι σa ω F S p 4.2 Chow polystability is equivalent to balanced The space of Chow points admits the structure of a Kähler manifold, with Kähler form Ω as follows. Each f CHOW P N k n, d k is a symmetric degree d k polynomial in the Plücker coordinates of GN k n, N k + 1, and parameterizes a polarized variety, L. The tangent space T f CHOW P N k n, d k can be identified with Γ, T P N k. In order to see how CHOW P N k n, d k varies infinitesimally at f, we can look at the velocity of each point on, since corresponds to the point f. That is, a tangent vector at f in the Chow space corresponds to assigning a direction everywhere on, i.e. a global section of T P N k, because we know P N k [GKZ94, 4.3]. Define Ω f on u, v Γ, T P N k by Ω f u, v = where ι u denotes contraction with u. ι v ι u ω n+1 F S n + 1! 21 25

26 Lemma 31 [Wan04, Proposition 17]. The map given by µ Ω : CHOW P N k n, d k sun k + 1 Chow, L k is a moment map for CHOW P N k n, d k, Ω. We can check that µ F S ω n F S n! d µ Ω, ξ = ι σξ Ω, ξ sun k holds pointwise at each Chow point f. The right side is computed by evaluating at some Y T f CHOW P N k n, d k, which can be identified as an element in Γ, T P N k. The left side of 22 is computed by taking a path f t in the Chow space with f 0 = f such that its velocity at t = 0 is Y cf [Wan04]. So µ Ω is the required moment map. In particular, µ Ω inherits G-equivariance from µ F S so we can form the symplectic quotient CHOW P N k n, d k //SUN + 1 = µ 1 Ω 0/SUN + 1 Definition 32. We say that, L k can be balanced if there exists a choice of basis for embedding in CP N k such that Chow, L k is a zero of the moment map µ Ω. Sketch proof of Theorem 23. There are several intermediate results in the following which I state without proof, hence I ve labelled this as a sketch proof. Since L is very ample we can assume k = 1 and drop the k s. By the Hilbert-umford Criterion [FK94, 2.1], to check polystability of Chow, L with respect to the SUN + 1-action we need only check it for all 1-parameter subgroups outside its stabilizer. In order to define an action of a one-parameter subgroup, we need something which is invariant. Define := lim t e itξ for ξ sun + 1 aut, L. Here aut, L is the Lie algebra of Aut, L. Then is invariant under the action of the 1-PS {e tξ } t C, since SUN + 1 acts by symplectomorphisms and we ve made invariant under isun + 1 as well. Definition 33. The λ-weight ρx of a C -action on an element x in an invariant one-dimensional space is the exponent of the eigenvalue, where λ C acts by λ x = λ ρx x In general, given a C -action on a space V, there are eigenvectors v 1,..., v m with eigenspaces of dimension d i with respect to the action such that λ C acts by λ v i = λ ai v i. Then the λ-weight of the action on this space is i a id i. Equivalently, C V = C top V and we can compute the weight on top V as in the one-dimensional case. Definition 34. The ξ-weight of Chow, L is defined to be the weight of the action induced by ξ on the one dimensional space Z := O CHOWP N n,d1 at the point Chow, L. Chow polystability was defined earlier by looking at orbits of Chow, L. There is an equivalent notion of polystability by [FK94]: is Chow polystable if the ξ-weight of Chow, L is negative for all ξ sun + 1 aut, L. We use the following two results: 26

27 Proposition 35 [um77, Prop 2.11]. Suppose is an n-dimensional manifold embedded in P N via very ample line bundle L and is fixed by a 1-PS ξ of SLN + 1. Let R N be the degree N part of the projective homogenous coordinate ring for. Let a be the ξ-weight of Chow, L and rn the ξ-weight of R N. Then for large N, rn is represented by a polynomial in N of degree at most n + 1, with normalized leading coefficient a, i.e. the leading coefficient is a /n!. In the notation of [Wan04], the leading coefficient of r N Theorem 36 [Wan04, Theorem 26]. w,0ξ n + 1 is denoted w,0ξ. = lim µ Ω e itξ Chow, L, ξ t su Thus the ξ-weight of Chow, L has the same sign as lim µω e itξ Chow, L, ξ t su. Hence Chow polystability of is equivalent to lim µ Ω e itξ Chow, L, ξ > 0 t su for all ξ sun + 1 aut, L. We then use the following results from [DK90, Section 6.5.2], also described in [Tho06]. Let G = SUN + 1. We can lift the G-action on CHOW P N n, d to one on Z, which extends to G C. We have a projection map π : Z CHOW P N n, d π : T Z T CHOW P N n, d Then T Z is the direct sum of the vertical subspace, which is ker π, and a horizontal subspace orthogonal to the vertical subspace, with respect to the induced inner product on Z. Tangent vectors on the Chow space have a horizontal lift to the horizontal subspace. Let σξ be the lift to this horizontal subspace, which projects to σξ via π. Recall µ Ω, ξ is the function corresponding to the vector field σξ on CHOW P N n, d, i.e. d µ Ω, ξ = ι σξ Ω. Then the infinitesimal action of ξ on Z is defined at a point γ over z where z corresponds to the point Chow, L σ z ξ + i µ Ω z, ξ γ 23 Z carries a G-invariant hermitian metric induced by Ω so g γ = γ for all g G. So when considering how g γ changes along a 1-PS, we need only look at iξ ig g C. Let H ξ t = log e itξ γ 2 defined by exponentiating the infinitesimal action above. Then defining z t := e itξ z H ξ t = 2 d dt eitξ γ, e itξ γ e itξ γ 2 = 2 i µ Ω z t, iξ e itξ γ, e itξ γ e itξ γ 2 = 2 µ Ω z t, ξ 27

28 using Equation 23 and orthogonality of horizontal and vertical subspaces. If γ is a point in some G C -orbit G C γ 0, then it is a critical point of log g γ 0 2 if and only if µ Ω πγ = 0. Critical points of H ξ t are minima; note that H ξ t = 2d µ Ω, ξ zt σ zt iξ = 2ι σzt ξωjσ zt ξ = 2Ωσ zt ξ, Jσ zt ξ = 2 σ zt ξ 2 Thus H ξ t is convex and has at most one minimum. This minimum is either attained on G C /G or at infinity; the former occurs if and only if lim H t ξ t > 0. So lim t [H ξ t = 2 µ Ωz t, ξ ] > 0! minimum of H ξ t ξ g g z ξ g g z g SLN + 1, µ Ω g z = 0 where g z is the Lie algebra of the stabilizer of z = Chow, L, which is aut, L. This final expression is the condition that, L can be balanced, i.e. µ Ω Chowg, L = 0. Thus by Donaldson s theorem, Theorem 21 above, asymptotic Chow stability of, L is an obstruction to the existence of a csck metric on c 1 L when the stabilizer Aut, L is discrete. 5 K-stability It has been conjectured by Yau, Tian and Donaldson that another type of stability, K-stability, is equivalent to the existence of csck metrics. I will describe Tian s proof that the existence of a Kähler-Einstein metric implies weak K-stability, [Tia97]. 5.1 Background The following background is from [Tia00]. K-stability is defined by looking at special degenerations of a Kähler manifold into normal varieties. Assume dim C = n 3. Definition 37. A fibration is a map π : A B between two topological spaces which satisfies the homotopy lifting property. Definition 38. A special degeneration of is a fibration π : W n+1, where is the unit disc in C, such that π 1 s is smooth s 0, π 1 1/2 is biholomorphic to and there exists v W ηw such that π v W = s s, generating a 1-PS e s z on. W is trivial if W =. Tian assumes the central fiber π 1 0 and all other fibers are smooth in [Tia00], and defines special degenerations for π 1 0 a normal variety in [Tia97]. This more general version of special degeneration is that for which K-stability is defined. Donaldson re-defined K-stability for polarized varieties. Let W t denote the fiber π 1 t where t = e s. Since π v W vanishes at t = 0, v W restricts to a vector field on W 0, as it has no component in the /t direction on W 0. Thus the Calabi- Futaki invariant f W0 v W = f W0 v W W0 makes sense, using a generalized version of the Futaki invariant when W 0 is normal. 28