Hilbert series and obstructions to asymptotic Chow semistability

Transcription

1 Hilbert series and obstructions to asymptotic Chow semistability Akito Futaki Tokyo Institute of Technology Kähler and Sasakian Geometry in Rome Rome, June 16th-19th, 2009 In memory of Krzysztof Galicki 1

2 1. Obstructions to Asymptotic Chow semistability (background) Def : (M, L) is a polarized manifold if M is a comapct complex manifold and L a positive line bundle, i.e. c 1 (L) > 0. c 1 (L) is regarded as a Kähler class (or the space of Kähler forms). We seek a constant scalar curvature Kähler (csck) metric in c 1 (L). 2

3 1. Obstructions to Asymptotic Chow semistability (background) Def : (M, L) is a polarized manifold if M is a comapct complex manifold and L a positive line bundle, i.e. c 1 (L) > 0. c 1 (L) is regarded as a Kähler class (or the space of Kähler forms). We seek a constant scalar curvature Kähler (csck) metric in c 1 (L). 2-a

4 1. Obstructions to Asymptotic Chow semistability (background) Def : (M, L) is a polarized manifold if M is a comapct complex manifold and L a positive line bundle, i.e. c 1 (L) > 0. c 1 (L) is regarded as a Kähler class (or the space of Kähler forms). We seek a constant scalar curvature Kähler (csck) metric in c 1 (L). 2-b

5 1. Obstructions to Asymptotic Chow semistability (background) Def : (M, L) is a polarized manifold if M is a comapct complex manifold and L a positive line bundle, i.e. c 1 (L) > 0. c 1 (L) is regarded as a Kähler class (or the space of Kähler forms). We seek a constant scalar curvature Kähler (csck) metric in c 1 (L). 2-c

6 Besides the obstructions due to Matsushima and myself, there are obstructions related to GIT stability (Yau, Tian, Donaldson). Theorem A. (Donaldson, JDG 2001) Let (M, L) be a polarized manifold with Aut(M, L) discrete. If a csck metric then (M, L) is asymptotically Chow stable. 3

7 Besides the obstructions due to Matsushima and myself, there are obstructions related to GIT stability (Yau, Tian, Donaldson). Theorem A. (Donaldson, JDG 2001) Let (M, L) be a polarized manifold with Aut(M, L) discrete. If a csck metric then (M, L) is asymptotically Chow stable. 3-a

8 Claim of this talk: This is not the case if Aut(M, L) is not discrete. Theorem (Ono-Sano-Yotsutani) There are toric Fano Kähler-Einstein manifolds which are not asymptotically Chow-semistable. 4

9 Claim of this talk: This is not the case if Aut(M, L) is not discrete. Theorem (Ono-Sano-Yotsutani) There are toric Fano Kähler-Einstein manifolds which are not asymptotically Chow-semistable (polystable). 4-a

10 What is (asymptotic) Chow stability? V k := H 0 (M, O(L k )) M k P(V k ) the image of Kodaira embedding by L k d k = the degree of M k in P(V k ) 5

11 What is (asymptotic) Chow stability? V k := H 0 (M, O(L k )) M k P(V k ) the image of Kodaira embedding by L k d k = the degree of M k in P(V k ) 5-a

12 What is (asymptotic) Chow stability? V k := H 0 (M, O(L k )) M k P(V k ) the image of Kodaira embedding by L k d k = the degree of M k in P(V k ) 5-b

13 Let m = dim C M. An element of P(Vk ) P(V k ) (m + 1 times) defines m + 1 hyperplanes H 1,, H m+1 in P(V k ). {(H 1,, H m+1 ) P(V k ) P(V k ) H 1 H m+1 M k } defines a divisor in P(V k ) P(V k ) and this divisor is defined by ˆM k (Sym d k(v k )) (m+1). 6

14 Let m = dim C M. An element of P(Vk ) P(V k ) (m + 1 times) defines m + 1 hyperplanes H 1,, H m+1 in P(V k ). {(H 1,, H m+1 ) P(V k ) P(V k ) H 1 H m+1 M k } defines a divisor in P(V k ) P(V k ) and this divisor is defined by ˆM k (Sym d k(v k )) (m+1). 6-a

15 Let m = dim C M. An element of P(Vk ) P(V k ) (m + 1 times) defines m + 1 hyperplanes H 1,, H m+1 in P(V k ). {(H 1,, H m+1 ) P(V k ) P(V k ) H 1 H m+1 M k } defines a divisor in P(V k ) P(V k ) and this divisor is defined by ˆM k (Sym d k(v k )) (m+1). 6-b

16 Let m = dim C M. An element of P(Vk ) P(V k ) (m + 1 times) defines m + 1 hyperplanes H 1,, H m+1 in P(V k ). {(H 1,, H m+1 ) P(V k ) P(V k ) H 1 H m+1 M k } defines a divisor in P(V k ) P(V k ) and this divisor is defined by ˆM k (Sym d k(v k )) (m+1). 6-c

17 The point [ ˆM k ] P((Sym d k(v k )) (m+1) ) is called the Chow point. The Chow point determines M k P(V k ). Stabilizer of ˆM k under SL(V k )-action is Aut(M, L). In Theorem A (Donaldson), Aut(M, L) is discrete means the stabilizer is finite. 7

18 The point [ ˆM k ] P((Sym d k(v k )) (m+1) ) is called the Chow point. The Chow point determines M k P(V k ). Stabilizer of ˆM k under SL(V k )-action is Aut(M, L). In Theorem A (Donaldson), Aut(M, L) is discrete means the stabilizer is finite. 7-a

19 The point [ ˆM k ] P((Sym d k(v k )) (m+1) ) is called the Chow point. The Chow point determines M k P(V k ). Stabilizer of ˆM k under SL(V k )-action is Aut(M, L). In Theorem A (Donaldson), Aut(M, L) is discrete means the stabilizer is finite. 7-b

20 The point [ ˆM k ] P((Sym d k(v k )) (m+1) ) is called the Chow point. The Chow point determines M k P(V k ). Stabilizer of ˆM k under SL(V k )-action is Aut(M, L). In Theorem A (Donaldson), Aut(M, L) is discrete means the stabilizer is finite. 7-c

21 M is said to be Chow polystable w.r.t. L k if the orbit of ˆM k in (Sym d k(v k )) (m+1) under the action of SL(V k ) is closed. M is Chow stable w.r.t L k if M is polystable and the stabilizer at ˆM k of the action of SL(V k ) is finite. M is Chow semistable w.r.t. L k if the closure of the orbit of ˆM k in (Sym d k(v k )) (m+1) under the action of SL(V k ) does not contain o (Sym d k(v k )) (m+1). M is asymptotically Chow polystable (resp. stable or semistable) w.r.t. L if there exists a k 0 > 0 such that M is Chow polystable (resp. stable or semistable) w.r.t. L k for all k k 0. 8

22 M is said to be Chow polystable w.r.t. L k if the orbit of ˆM k in (Sym d k(v k )) (m+1) under the action of SL(V k ) is closed. M is Chow stable w.r.t L k if M is polystable and the stabilizer at ˆM k of the action of SL(V k ) is finite. M is Chow semistable w.r.t. L k if the closure of the orbit of ˆM k in (Sym d k(v k )) (m+1) under the action of SL(V k ) does not contain o (Sym d k(v k )) (m+1). M is asymptotically Chow polystable (resp. stable or semistable) w.r.t. L if there exists a k 0 > 0 such that M is Chow polystable (resp. stable or semistable) w.r.t. L k for all k k 0. 8-a

23 M is said to be Chow polystable w.r.t. L k if the orbit of ˆM k in (Sym d k(v k )) (m+1) under the action of SL(V k ) is closed. M is Chow stable w.r.t L k if M is polystable and the stabilizer at ˆM k of the action of SL(V k ) is finite. M is Chow semistable w.r.t. L k if the closure of the orbit of ˆM k in (Sym d k(v k )) (m+1) under the action of SL(V k ) does not contain o (Sym d k(v k )) (m+1). M is asymptotically Chow polystable (resp. stable or semistable) w.r.t. L if there exists a k 0 > 0 such that M is Chow polystable (resp. stable or semistable) w.r.t. L k for all k k 0. 8-b

24 M is said to be Chow polystable w.r.t. L k if the orbit of ˆM k in (Sym d k(v k )) (m+1) under the action of SL(V k ) is closed. M is Chow stable w.r.t L k if M is polystable and the stabilizer at ˆM k of the action of SL(V k ) is finite. M is Chow semistable w.r.t. L k if the closure of the orbit of ˆM k in (Sym d k(v k )) (m+1) under the action of SL(V k ) does not contain o (Sym d k(v k )) (m+1). M is asymptotically Chow polystable (resp. stable or semistable) w.r.t. L if there exists a k 0 > 0 such that M is Chow polystable (resp. stable or semistable) w.r.t. L k for all k k 0. 8-c

25 Theorem B (Mabuchi, Osaka J. Math. 2004) Let (M, L) be a polarized manifold. If Aut(M, L) is not discrete then there is an obstruction to asymptotic Chow semistability. Theorem C (Mabuchi, Invent. Math. 2005) Let (M, L) be a polarized manifold, and suppose Aut(M, L) is not discrete. If a csck metric in c 1 (L) and if the obstruction vanishes then (M, L) is asymptotically Chow polystable. 9

26 Theorem B (Mabuchi, Osaka J. Math. 2004) Let (M, L) be a polarized manifold. If Aut(M, L) is not discrete then there is an obstruction to asymptotic Chow semistability. Theorem C (Mabuchi, Invent. Math. 2005) Let (M, L) be a polarized manifold, and suppose Aut(M, L) is not discrete. If a csck metric in c 1 (L) and if the obstruction vanishes then (M, L) is asymptotically Chow polystable. 9-a

27 h 0 = {X holo vector field zero(x) } = {X holo vector field u C (M) C s.t. X = grad u = g i j u z j z i} = {X holo vector field lift to an infinitesimal action on L} = Lie(Aut(M, L)) Theorem D (F, Internat. J. Math. 2004) Let (M, L) be a polarized manifold with dim C M = m. (1) The vanishing of Mabuchi s obstruction is equivalent to the vanishing of Lie algebra characters F Td i : h 0 C, i = 1,, m. (2) F Td 1 = obstruction to of csck metric (Futaki invariant). 10

28 h 0 = {X holo vector field zero(x) } = {X holo vector field u C (M) C s.t. X = grad u = g i j u z j z i} = {X holo vector field lift to an infinitesimal action on L} = Lie(Aut(M, L)) Theorem D (F, Internat. J. Math. 2004) Let (M, L) be a polarized manifold with dim C M = m. (1) The vanishing of Mabuchi s obstruction is equivalent to the vanishing of Lie algebra characters F Td i : h 0 C, i = 1,, m. (2) F Td 1 = obstruction to of csck metric (Futaki invariant). 10-a

29 h 0 = {X holo vector field zero(x) } = {X holo vector field u C (M) C s.t. X = grad u = g i j u z j z i} = {X holo vector field lift to an infinitesimal action on L} = Lie(Aut(M, L)) Theorem D (F, Internat. J. Math. 2004) Let (M, L) be a polarized manifold with dim C M = m. (1) The vanishing of Mabuchi s obstruction is equivalent to the vanishing of Lie algebra characters F Td i : h 0 C, i = 1,, m. (2) F Td 1 = obstruction to of csck metric (Futaki invariant). 10-b

30 Question (a) In Theorem C, can t we omit the assumption of the vanishing of the obstruction? That is to say, if a csck metric then doesn t the obstruction necessarily vanish? Question (b) In Theorem D, if F Td 1 = 0 then F Td 2 = = F Td m = 0? Question (c) dim span{f Td 1,, F Td m} = 1? 11

31 Question (a) In Theorem C, can t we omit the assumption of the vanishing of the obstruction? That is to say, if a csck metric then doesn t the obstruction necessarily vanish? Question (b) In Theorem D, if F Td 1 = 0 then F Td 2 = = F Td m = 0? Question (c) dim span{f Td 1,, F Td m} = 1? 11-a

32 Question (a) In Theorem C, can t we omit the assumption of the vanishing of the obstruction? That is to say, if a csck metric then doesn t the obstruction necessarily vanish? Question (b) In Theorem D, if F Td 1 = 0 then F Td 2 = = F Td m = 0? Question (c) dim span{f Td 1,, F Td m} = 1? 11-b

33 Futaki-Ono-Sano, 2008 (1) Question (c) is not true in general. (2) F Td 1,, F Td m are obtained as derivatives of the Hilbert series. (3) The derivatives of the Hilbert series are computed by imputting toric data into a computer. Ono-Sano-Yotsutani recently showed the answers to Questions (a) and (b) are No. 12

34 Futaki-Ono-Sano, 2008 (1) Question (c) is not true in general. (2) F Td 1,, F Td m are obtained as derivatives of the Hilbert series. (3) The derivatives of the Hilbert series are computed by imputting toric data into a computer. Ono-Sano-Yotsutani recently showed the answers to Questions (a) and (b) are No. 12-a

35 Futaki-Ono-Sano, 2008 (1) Question (c) is not true in general. (2) F Td 1,, F Td m are obtained as derivatives of the Hilbert series. (3) The derivatives of the Hilbert series are computed by imputting toric data into a computer. Ono-Sano-Yotsutani recently showed the answers to Questions (a) and (b) are No. 12-b

36 Futaki-Ono-Sano, 2008 (1) Question (c) is not true in general. (2) F Td 1,, F Td m are obtained as derivatives of the Hilbert series. (3) The derivatives of the Hilbert series are computed by imputting toric data into a computer. Ono-Sano-Yotsutani recently showed the answers to Questions (a) and (b) are No. 12-c

37 2. Lie algebra characters F Td i θ : connection form on L zero section X : horizontal lift of X to L z : fiber coordinate of L Then the lift X to L of X is written as X = θ( X)iz z + X. The ambiguity of X is const iz z. 13

38 2. Lie algebra characters F Td i θ : connection form on L zero section X : horizontal lift of X to L z : fiber coordinate of L Then the lift X to L of X is written as X = θ( X)iz z + X. The ambiguity of X is const iz z. 13-a

39 X + ciz z = θ( X + ciz z )iz z + X = (θ( X) + c)iz z + X. 14

40 Given a Kähler form ω c 1 (L), suppose the connection θ is so chosen that Then If we put then 1 θ = ω. 2π i(x)ω = 1 θ( X). 2π u X = 1 θ( X) 2π i(x)ω = u X. Conclusion: Ambiguity of Hamiltonian function ambiguity of lifting of X to L. 15

41 Given a Kähler form ω c 1 (L), suppose the connection θ is so chosen that Then If we put then 1 θ = ω. 2π i(x)ω = 1 θ( X). 2π u X = 1 θ( X) 2π i(x)ω = u X. Conclusion: Ambiguity of Hamiltonian function ambiguity of lifting of X to L. 15-a

42 Given a Kähler form ω c 1 (L), suppose the connection θ is so chosen that Then If we put then 1 θ = ω. 2π i(x)ω = 1 θ( X). 2π u X = 1 θ( X) 2π i(x)ω = u X. Conclusion: Ambiguity of Hamiltonian function ambiguity of lifting of X to L. 15-b

43 Given a Kähler form ω c 1 (L), suppose the connection θ is so chosen that Then If we put then 1 θ = ω. 2π i(x)ω = 1 θ( X). 2π u X = 1 θ( X) 2π i(x)ω = u X. Conclusion: Ambiguity of Hamiltonian function ambiguity of lifting of X to L. 15-c

44 Assume the normalization M u Xω m = 0. Choose a type (1, 0)-connection in T M. Put L(X) = X L X Γ(End(T M)) and let Θ Γ(Ω 1,1 (M) End(T M)) be the (1,1)-part of the curvature form of. 16

45 Assume the normalization M u Xω m = 0. Choose a type (1, 0)-connection in T M. Put L(X) = X L X Γ(End(T M)) and let Θ Γ(Ω 1,1 (M) End(T M)) be the (1,1)-part of the curvature form of. 16-a

46 Assume the normalization M u Xω m = 0. Choose a type (1, 0)-connection in T M. Put L(X) = X L X Γ(End(T M)) and let Θ Γ(Ω 1,1 (M) End(T M)) be the (1,1)-part of the curvature form of. 16-b

47 Def: For φ I p (GL(m, C)), we define F φ (X) = (m p + 1) + M φ(θ) u X ω m p (1) M φ(l(x) + Θ) ωm p+1. Theorem D : Vanishing of Mabuchi s obstruction is equivalent to F Td 1(X) = = F Td m(x) = 0 for all X h 0. 17

48 Def: For φ I p (GL(m, C)), we define F φ (X) = (m p + 1) + M φ(θ) u X ω m p (2) M φ(l(x) + Θ) ωm p+1. Theorem D : Vanishing of Mabuchi s obstruction is equivalent to F Td 1(X) = = F Td m(x) = 0 for all X h a

49 3. Hilbert series M: toric Fano manifold of dim M = m. L = K 1 M T m acts on M. T m+1 acts on K 1 M. For g T m+1, L(g) := k=0 Tr(g H 0 (M,K k M )) the formal sum of the Lefchetz numbers. 18

50 3. Hilbert series M: toric Fano manifold of dim M = m. L = K 1 M T m acts on M. T m+1 acts on K 1 M. For g T m+1, L(g) := k=0 Tr(g H 0 (M,K k M )) the formal sum of the Lefchetz numbers. 18-a

51 3. Hilbert series M: toric Fano manifold of dim M = m. L = K 1 M T m acts on M. T m+1 acts on K 1 M. For g T m+1, L(g) := k=0 Tr(g H 0 (M,K k M )) the formal sum of the Lefchetz numbers. 18-b

52 For x T m+1 C we may analytically continue to L(x). Let {v j Z m } j be the generators of 1-dimensional faces of the fan of M. λ j := (v j, 1) Z m+1, C := {y R m+1 λ j y 0, j} g = (Lie(T m+1 )), P := {w R m v j w 1, j} C is a cone over P. Integral points in C k=1 basis of H 0 (M, K k M ) a = (w, k) Z m+1 C 19

53 For x T m+1 C we may analytically continue to L(x). Let {v j Z m } j be the generators of 1-dimensional faces of the fan of M. λ j := (v j, 1) Z m+1, C := {y R m+1 λ j y 0, j} g = (Lie(T m+1 )), P := {w R m v j w 1, j} C is a cone over P. Integral points in C k=1 basis of H 0 (M, K k M ) a = (w, k) Z m+1 C 19-a

54 For x T m+1 C we may analytically continue to L(x). Let {v j Z m } j be the generators of 1-dimensional faces of the fan of M. λ j := (v j, 1) Z m+1, C := {y R m+1 λ j y 0, j} g = (Lie(T m+1 )), P := {w R m v j w 1, j} C is a cone over P. Integral points in C k=1 basis of H 0 (M, K k M ) a = (w, k) Z m+1 C 19-b

55 For x T m+1 C we may analytically continue to L(x). Let {v j Z m } j be the generators of 1-dimensional faces of the fan of M. λ j := (v j, 1) Z m+1, C := {y R m+1 λ j y 0, j} g = (Lie(T m+1 )), P := {w R m v j w 1, j} C is a cone over P. Integral points in C k=1 basis of H 0 (M, K k M ) a = (w, k) Z m+1 C 19-c

56 For x T m+1 C we may analytically continue to L(x). Let {v j Z m } j be the generators of 1-dimensional faces of the fan of M. λ j := (v j, 1) Z m+1, C := {y R m+1 λ j y 0, j} g = (Lie(T m+1 )), P := {w R m v j w 1, j} C is a cone over P. Integral points in C k=1 basis of H 0 (M, K k M ) a = (w, k) Z m+1 C 19-d

57 For x TC m+1, we put x a = x a 1 1 xa m+1 m+1. Def: C(x, C ) := a C Z m+1 x a Hilbert series Fact: C(x, C ) is a rational function of x. Lemma : C(x, C ) = L(x). For b R m+1 = g = Lie(T m+1 ), e tb := (e b 1t,, e b m+1t ) C(e tb, C ) = a C Z m+1 e ta b : rational function in t 20

58 For x TC m+1, we put x a = x a 1 1 xa m+1 m+1. Def: C(x, C ) := a C Z m+1 x a Hilbert series Fact: C(x, C ) is a rational function of x. Lemma : C(x, C ) = L(x). For b R m+1 = g = Lie(T m+1 ), e tb := (e b 1t,, e b m+1t ) C(e tb, C ) = a C Z m+1 e ta b : rational function in t 20-a

59 For x TC m+1, we put x a = x a 1 1 xa m+1 m+1. Def: C(x, C ) := a C Z m+1 x a Hilbert series Fact: C(x, C ) is a rational function of x. Lemma : C(x, C ) = L(x). For b R m+1 = g = Lie(T m+1 ), e tb := (e b 1t,, e b m+1t ) C(e tb, C ) = a C Z m+1 e ta b : rational function in t 20-b

60 For x TC m+1, we put x a = x a 1 1 xa m+1 m+1. Def: C(x, C ) := a C Z m+1 x a Hilbert series Fact: C(x, C ) is a rational function of x. Lemma : C(x, C ) = L(x). For b R m+1 = g = Lie(T m+1 ), e tb := (e b 1t,, e b m+1t ) C(e tb, C ) = a C Z m+1 e ta b : rational function in t 20-c

61 C R := {(b 1,, b m, m + 1) (b 1,, b m ) (m + 1)P } g where P is the dual polytope of P. C R is the space of Reeb vector fields of Sasakian structures on S, the total space of the associated U(1)-bundle of K M. The tangent space of C R at (0,, 0, m + 1) is T (0,,0,m+1) C R = {c = (c 1,, c m, 0)} g. This defines another way of lifting of T m -action to L. 21

62 C R := {(b 1,, b m, m + 1) (b 1,, b m ) (m + 1)P } g where P is the dual polytope of P. C R is the space of Reeb vector fields of Sasakian structures on S, the total space of the associated U(1)-bundle of K M. The tangent space of C R at (0,, 0, m + 1) is T (0,,0,m+1) C R = {c = (c 1,, c m, 0)} g. This defines another way of lifting of T m -action to L. 21-a

63 C R := {(b 1,, b m, m + 1) (b 1,, b m ) (m + 1)P } g where P is the dual polytope of P. C R is the space of Reeb vector fields of Sasakian structures on S, the total space of the associated U(1)-bundle of K M. The tangent space of C R at (0,, 0, m + 1) is T (0,,0,m+1) C R = {c = (c 1,, c m, 0)} g. This defines another way of lifting of T m -action to L. 21-b

64 Put b = (0,, 0, m + 1). Comparing the two liftings of T m -action to L we can show Theorem: (1) The coeffocients of the Laurant series of the rational function d ds s=0c(e t(b+sc), C ) in t span the linear space spanned by F Td 1,, F Td m. (2) In dimension 2, the linear spans are 1-dimensional. In dimension 3 the linear spans are at most 2-dimensional, and there are examples in which the linear spans are 2- dimensional. Remark Martelli-Sparks-Yau: Futaki invariant. From t m term we get the 22

65 Put b = (0,, 0, m + 1). Comparing the two liftings of T m -action to L we can show Theorem: (1) The coeffocients of the Laurant series of the rational function d ds s=0c(e t(b+sc), C ) in t span the linear space spanned by F Td 1,, F Td m. (2) In dimension 2, the linear spans are 1-dimensional. In dimension 3 the linear spans are at most 2-dimensional, and there are examples in which the linear spans are 2- dimensional. Remark Martelli-Sparks-Yau: Futaki invariant. From t m term we get the 22-a

66 Our computations show that the question is closely related to a question raised by Batyrev and Selivanova: Is a toric Fano manifold with vanishing f(= F Td 1) for the anticanonical class necessarily symmetric? 23

67 Our computations show that the question is closely related to a question raised by Batyrev and Selivanova: Is a toric Fano manifold with vanishing f(= F Td 1) for the anticanonical class necessarily symmetric? 23-a

68 Recall that a toric Fano manifold M is said to be symmetric if the trivial character is the only fixed point of the action of the Weyl group on the space of all algebraic characters of the maximal torus in Aut(M). Nill and Paffenholz gave a counterexample to Batyrev- Selyvanova. 24

69 Recall that a toric Fano manifold M is said to be symmetric if the trivial character is the only fixed point of the action of the Weyl group on the space of all algebraic characters of the maximal torus in Aut(M). Nill and Paffenholz gave a counterexample to Batyrev- Selyvanova. 24-a