Remarks on hypersurface K-stability. Complex Geometry: A Conference Honoring Simon Donaldson

Size: px
Start display at page:

Download "Remarks on hypersurface K-stability. Complex Geometry: A Conference Honoring Simon Donaldson"

Transcription

1 Remarks on hypersurface K-stability Zhiqin Lu, UC Irvine Complex Geometry: A Conference Honoring Simon Donaldson October 26, 2009 Zhiqin Lu, UC. Irvine Hypersurface K-stability 1/42

2 The Result Theorem (L-Phong) If M is smooth hypersurface of CP n of degree 3 d n, then M is K-stable. Zhiqin Lu, UC. Irvine Hypersurface K-stability 2/42

3 The Result Theorem (L-Phong) If M is smooth hypersurface of CP n of degree 3 d n, then M is K-stable. Our K-stability is related to the K-stability in the sense of Donaldson, but they are not the same. Note that we don t assume the existence of csck metrics. Zhiqin Lu, UC. Irvine Hypersurface K-stability 2/42

4 Background Futaki (1983) defined an invariant (Futaki invariant): Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42

5 Background Futaki (1983) defined an invariant (Futaki invariant): Let M be a Fano manifold, and let ω c 1 (M) be a Kähler metric. Then there is a real function f such that Ric(ω) ω = 1 f Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42

6 Background Futaki (1983) defined an invariant (Futaki invariant): Let M be a Fano manifold, and let ω c 1 (M) be a Kähler metric. Then there is a real function f such that Ric(ω) ω = 1 f Let v be a holomorphic vector field of M. Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42

7 Background Futaki (1983) defined an invariant (Futaki invariant): Let M be a Fano manifold, and let ω c 1 (M) be a Kähler metric. Then there is a real function f such that Ric(ω) ω = 1 f Let v be a holomorphic vector field of M. Then the Futaki invariant can be defined as Fut(v) = v(f )ω n. M Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42

8 Background Futaki (1983) defined an invariant (Futaki invariant): Let M be a Fano manifold, and let ω c 1 (M) be a Kähler metric. Then there is a real function f such that Ric(ω) ω = 1 f Let v be a holomorphic vector field of M. Then the Futaki invariant can be defined as Fut(v) = v(f )ω n. If Fut(v) 0, then the Kähler-Einstein metric doesn t exist. M Zhiqin Lu, UC. Irvine Hypersurface K-stability 3/42

9 Donaldson s point of view Let (M, L) be a polarized Kähler manifold. Zhiqin Lu, UC. Irvine Hypersurface K-stability 4/42

10 Donaldson s point of view Let (M, L) be a polarized Kähler manifold. Let v be a holomorphic vector field of M which induces an action to the line bundle L (We say v linearizes on L). Zhiqin Lu, UC. Irvine Hypersurface K-stability 4/42

11 Donaldson s point of view Let (M, L) be a polarized Kähler manifold. Let v be a holomorphic vector field of M which induces an action to the line bundle L (We say v linearizes on L). Consider the space H 0 (M, L k ). Zhiqin Lu, UC. Irvine Hypersurface K-stability 4/42

12 Donaldson s point of view Let Tr v (H 0 (M, L k )) dim H 0 (M, L k ) = a 0k + a 1 + O( 1 k ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 5/42

13 Donaldson s point of view Let Tr v (H 0 (M, L k )) dim H 0 (M, L k ) = a 0k + a 1 + O( 1 k ) The Donaldson s Futaki invariant is defined to be Don Fut(v) = a 1 Zhiqin Lu, UC. Irvine Hypersurface K-stability 5/42

14 Donaldson s point of view Let Tr v (H 0 (M, L k )) dim H 0 (M, L k ) = a 0k + a 1 + O( 1 k ) The Donaldson s Futaki invariant is defined to be We can prove Don Fut(v) = a 1 Theorem Let L = K 1 M, then Don Fut(v) = 1 2 Fut(v). Zhiqin Lu, UC. Irvine Hypersurface K-stability 5/42

15 The Futaki invariant depends only on the cohomology class c 1 (M), not on the particular Kähler metric in the class. Zhiqin Lu, UC. Irvine Hypersurface K-stability 6/42

16 The Futaki invariant depends only on the cohomology class c 1 (M), not on the particular Kähler metric in the class. In Donaldson s definition, the metric doesn t even appear. Zhiqin Lu, UC. Irvine Hypersurface K-stability 6/42

17 Test Configuration Definition (Donaldson) Let L M be an ample line bundle over a compact Kähler manifold. A test configuration consists of the following data Zhiqin Lu, UC. Irvine Hypersurface K-stability 7/42

18 Test Configuration Definition (Donaldson) Let L M be an ample line bundle over a compact Kähler manifold. A test configuration consists of the following data 1. A scheme X with a C action ρ Zhiqin Lu, UC. Irvine Hypersurface K-stability 7/42

19 Test Configuration Definition (Donaldson) Let L M be an ample line bundle over a compact Kähler manifold. A test configuration consists of the following data 1. A scheme X with a C action ρ 2. A C equivariant line bundle L X, where C is ample on all fibers Zhiqin Lu, UC. Irvine Hypersurface K-stability 7/42

20 Test Configuration Definition (Donaldson) Let L M be an ample line bundle over a compact Kähler manifold. A test configuration consists of the following data 1. A scheme X with a C action ρ 2. A C equivariant line bundle L X, where C is ample on all fibers 3. A flat C equivariant map π : X C where C acts on C by multiplication. Zhiqin Lu, UC. Irvine Hypersurface K-stability 7/42

21 1. Let H be a finite dimensional space. Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42

22 1. Let H be a finite dimensional space. 2. Let A be an endomorphism of H. Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42

23 1. Let H be a finite dimensional space. 2. Let A be an endomorphism of H. 3. Let V be a subspace of H which is A invariant. Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42

24 1. Let H be a finite dimensional space. 2. Let A be an endomorphism of H. 3. Let V be a subspace of H which is A invariant. 4. Tr A (V ) is well-defined. Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42

25 1. Let H be a finite dimensional space. 2. Let A be an endomorphism of H. 3. Let V be a subspace of H which is A invariant. 4. Tr A (V ) is well-defined. 5. How to define Tr A (V ), if V is not A invariant? Zhiqin Lu, UC. Irvine Hypersurface K-stability 8/42

26 Nonlinear Trace We assume that, under some basis, that we can write A as where λ j are integers. A = λ 0... λ n Zhiqin Lu, UC. Irvine Hypersurface K-stability 9/42

27 Nonlinear Trace We assume that, under some basis, that we can write A as A = λ 0... where λ j are integers. Then there is a C action ρ(t), acts on H. Let λ n V 0 = lim t 0 ρ(t)v Zhiqin Lu, UC. Irvine Hypersurface K-stability 9/42

28 Nonlinear Trace We assume that, under some basis, that we can write A as A = λ 0... where λ j are integers. Then there is a C action ρ(t), acts on H. Let λ n V 0 = lim t 0 ρ(t)v Then V 0 is A invariant. Thus Tr A (V ) = Tr A (V 0 ) is well defined. Zhiqin Lu, UC. Irvine Hypersurface K-stability 9/42

29 Example Let M CP N. For any k, define H = H k = H 0 (CP N, H k ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 10/42

30 Example Let M CP N. For any k, define H = H k = H 0 (CP N, H k ) Define W = {S H S M = 0}, V = H/W Zhiqin Lu, UC. Irvine Hypersurface K-stability 10/42

31 Example Let M CP N. For any k, define H = H k = H 0 (CP N, H k ) Define W = {S H S M = 0}, V = H/W Using Tr A (V )/ dim V to define the Donaldson s Futaki invariant. Zhiqin Lu, UC. Irvine Hypersurface K-stability 10/42

32 Example (Stoppa) Let v be a holomorphic vector field on a (X, L). Let p X. Let W = {S H 0 (M, L k ) S(p) = 0}, V = H 0 (M, L k )/W Zhiqin Lu, UC. Irvine Hypersurface K-stability 11/42

33 Mabuchi s K-energy Nonlinearizing the Futaki s invariant, Mabuchi was able to define the following K-energy functional: Zhiqin Lu, UC. Irvine Hypersurface K-stability 12/42

34 Mabuchi s K-energy Nonlinearizing the Futaki s invariant, Mabuchi was able to define the following K-energy functional: Definition Let ω 0, ω 1 be two Kähler metrics of M such that Then we define K(ω 0, ω 1 ) = where ω s = ω 0 + s φ. ω 1 = ω φ. 1 0 M φ(ric(ω s ) ω s ) ωs n 1 ds, Zhiqin Lu, UC. Irvine Hypersurface K-stability 12/42

35 Mabuchi s K-energy satisfies the co-cycle condition K(ω 0, ω 1 ) + K(ω 1, ω 2 ) = K(ω 0, ω 2 ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 13/42

36 Mabuchi s K-energy satisfies the co-cycle condition K(ω 0, ω 1 ) + K(ω 1, ω 2 ) = K(ω 0, ω 2 ) If ω 1 = ω φ, and when φ is the Hamiltonian function of a holomorphic vector field, the Mabuchi s functional is the same as the Futaki invariant. Zhiqin Lu, UC. Irvine Hypersurface K-stability 13/42

37 Mabuchi s K-energy satisfies the co-cycle condition K(ω 0, ω 1 ) + K(ω 1, ω 2 ) = K(ω 0, ω 2 ) If ω 1 = ω φ, and when φ is the Hamiltonian function of a holomorphic vector field, the Mabuchi s functional is the same as the Futaki invariant. It is a very important tool in Kähler-Einstein geometry, in particular in the uniquesness of Kähler-Einstein metrics and the csck metrics. Zhiqin Lu, UC. Irvine Hypersurface K-stability 13/42

38 Let M CP N for some N. Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42

39 Let M CP N for some N. Let ω be the Kähler metric of M obtained by the restriction of the Fubini-Study metric to M. Define ρ(t) ω be the pull-back of the FS metric by the action ρ(t), where t C. Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42

40 Let M CP N for some N. Let ω be the Kähler metric of M obtained by the restriction of the Fubini-Study metric to M. Define ρ(t) ω be the pull-back of the FS metric by the action ρ(t), where t C. Define Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42

41 Let M CP N for some N. Let ω be the Kähler metric of M obtained by the restriction of the Fubini-Study metric to M. Define ρ(t) ω be the pull-back of the FS metric by the action ρ(t), where t C. Define Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Is this non-linear Futaki invariant depends only on the cohomology class of ω? Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42

42 Let M CP N for some N. Let ω be the Kähler metric of M obtained by the restriction of the Fubini-Study metric to M. Define ρ(t) ω be the pull-back of the FS metric by the action ρ(t), where t C. Define Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Is this non-linear Futaki invariant depends only on the cohomology class of ω? What are we interested in this question? Zhiqin Lu, UC. Irvine Hypersurface K-stability 14/42

43 Zhiqin Lu, UC. Irvine Hypersurface K-stability 15/42

44 Analysis/ Geometry Side Algebraic Sides Zhiqin Lu, UC. Irvine Hypersurface K-stability 15/42

45 Analysis/ Geometry Side Making nonlinear Futaki more algebraic, Algebraic Sides Zhiqin Lu, UC. Irvine Hypersurface K-stability 15/42

46 Analysis/ Geometry Side Algebraic Sides Making nonlinear Futaki more algebraic, which was done by Sean Paul. Zhiqin Lu, UC. Irvine Hypersurface K-stability 15/42

47 Sean Paul s result Nonlinear Fut(ω) = A + B where both A, B are nonnegative. A is related to the Chow weight, and B is related to the X-hyperdiscriminant and the Chow form. Both of which are algebraic. Zhiqin Lu, UC. Irvine Hypersurface K-stability 16/42

48 Zhiqin Lu K energy and K stability on hypersurfaces CAG, 2004, 12(3), Zhiqin Lu, UC. Irvine Hypersurface K-stability 17/42

49 Zhiqin Lu K energy and K stability on hypersurfaces CAG, 2004, 12(3), Zhiqin Lu, UC. Irvine Hypersurface K-stability 17/42

50 For given λ, set Λ = max λ, α. α I Zhiqin Lu, UC. Irvine Hypersurface K-stability 18/42

51 For given λ, set Λ = max λ, α. α I Chow weight? Zhiqin Lu, UC. Irvine Hypersurface K-stability 18/42

52 For given λ, set Λ = max λ, α. α I Chow weight? and, for u = (u 0,, u n ) an (n + 1)-vector, φ(u) = max u + λ, α. α I Finally, we define φ j (t) = φ(0,, t j,, 0) for 0 j n. Then the following theorem was proved: Zhiqin Lu, UC. Irvine Hypersurface K-stability 18/42

53 Theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) a generic rational vector with n i=0 λ i = 0, then the non-linear Futaki invariant is equal to lim t d t 0 dt K(ω FS, σ(t) (ω FS )) Λ(d 1)(n + 1) = + n n j=0 0 φ j(t)(φ j(t) 1)dt. Zhiqin Lu, UC. Irvine Hypersurface K-stability 19/42

54 Theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) a generic rational vector with n i=0 λ i = 0, then the non-linear Futaki invariant is equal to lim t d t 0 dt K(ω FS, σ(t) (ω FS )) Λ(d 1)(n + 1) = + n n j=0 0 φ j(t)(φ j(t) 1)dt. Here generic means that the set of λ such that the above equality holds is dense. Zhiqin Lu, UC. Irvine Hypersurface K-stability 19/42

55 Nonlinear Futaki = ( 2 n ( ) XFt d F t 2 M t k=0 k ) (F t ) k ω n 1 + (n d + 1) θω n 1. M t Zhiqin Lu, UC. Irvine Hypersurface K-stability 20/42

56 Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Zhiqin Lu, UC. Irvine Hypersurface K-stability 21/42

57 Nonlinear Fut(ω) = lim t 0 t d dt K(ω, ρ(t) ω) Theorem (Ding-Tian) If the center fiber of the test configuration is a normal variety, then Nonlinear Fut(ω) = Fut(v), where v is the vector field induced by ρ on the center fiber, and Fut(v) is the Futaki invariant of the center fiber. Zhiqin Lu, UC. Irvine Hypersurface K-stability 21/42

58 If the center fiber is normal, then obviously the nonlinear Futaki depends only on the cohomology class. Zhiqin Lu, UC. Irvine Hypersurface K-stability 22/42

59 If the center fiber is normal, then obviously the nonlinear Futaki depends only on the cohomology class. We can manage to generalize the above result when the center fiber is reduced. However, for generic test configuration, the center fiber is non-reduced. Zhiqin Lu, UC. Irvine Hypersurface K-stability 22/42

60 An example of test configuration Let M be defined by the zeroes of the homogeneous polynomial F (Z) = α I a α Z α (1) of degree d, where a α 0 for α in a set of indices I, and we have used the multi-index notation α = (α 0,, α n ), Z α = Z α0 0 Zn αn. Zhiqin Lu, UC. Irvine Hypersurface K-stability 23/42

61 An example of test configuration Let M be defined by the zeroes of the homogeneous polynomial F (Z) = α I a α Z α (1) of degree d, where a α 0 for α in a set of indices I, and we have used the multi-index notation α = (α 0,, α n ), Z α = Z α0 0 Zn αn. Let λ = (λ 0,, λ n ) be such that λ j = 0 and all λ j are integers. Zhiqin Lu, UC. Irvine Hypersurface K-stability 23/42

62 An example of test configuration Let M be defined by the zeroes of the homogeneous polynomial F (Z) = α I a α Z α (1) of degree d, where a α 0 for α in a set of indices I, and we have used the multi-index notation α = (α 0,, α n ), Z α = Z α0 0 Zn αn. Let λ = (λ 0,, λ n ) be such that λ j = 0 and all λ j are integers. Then λ induced an action ρ(t) = (t λ0,, t λn ). The scheme X is defined by a sub-scheme of CP n C defined by the equation F (t λ0 Z 0,, t λn Z n ) = 0 Zhiqin Lu, UC. Irvine Hypersurface K-stability 23/42

63 Generically, the center fiber is defined by Z β = 0, where β I is the unique element in I such that λ, β = max λ, α. Zhiqin Lu, UC. Irvine Hypersurface K-stability 24/42

64 Generically, the center fiber is defined by Z β = 0, where β I is the unique element in I such that λ, β = max λ, α. Thus unless β j = 0, 1 for all j, the center fiber will be non-reduced (with multiplicity). Zhiqin Lu, UC. Irvine Hypersurface K-stability 24/42

65 Theorem (S. Paul) The Mabuchi energy is bounded from below along all degenerations if and only if the Hyperdiscriminant polytope contains the Chow polytope. In particular, the asymptotic behavior of the Mabuchi energy along any degeneration is logarithmic. Zhiqin Lu, UC. Irvine Hypersurface K-stability 25/42

66 Arezzo-della Vedova-la Nave In order to check K-stability, one only needs to check those test configurations where the center fibers are of normal crossings. Zhiqin Lu, UC. Irvine Hypersurface K-stability 26/42

67 A theorem of Kodaira and Spencer Theorem Let M be a smooth hypersurface of CP n of degree at least 3, then M has no holomorphic vector field. Zhiqin Lu, UC. Irvine Hypersurface K-stability 27/42

68 By the above result, we can prove the following Theorem (Donaldson, Stoppa) If M admits a csck metric, then M is K-stable (in the sense of Donaldson). Zhiqin Lu, UC. Irvine Hypersurface K-stability 28/42

69 Definition of K-stability Definition Let M be the hypersurface defined by a homogeneous polynomial F. The hypersurface M is said to be K-stable if for any vector λ = (λ 0,, λ n ) with λ integers and n i=0 λ i = 0, we have lim t 0 t d dt K(ω FS, σ(t) (ω FS )) 0 for the K-energy on M. Moreover, the equality holds if and only if after a permutation of (0,, n), 1. λ = a(n, 1,, 1) for a positive rational number a > 0, and 2. F (1, 0,, 0) 0. Zhiqin Lu, UC. Irvine Hypersurface K-stability 29/42

70 The semi-continuity theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) an arbitrary rational vector with n i=0 λ i = 0. Define ξ(λ) = lim t 0 t d dt K(ω FS, σ(t) (ω FS )) Zhiqin Lu, UC. Irvine Hypersurface K-stability 30/42

71 The semi-continuity theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) an arbitrary rational vector with n i=0 λ i = 0. Define Theorem (L-Phong) Using the above result, we have ξ(λ) = lim t 0 t d dt K(ω FS, σ(t) (ω FS )) ξ(λ) lim λ λ,λ U ξ(λ ). Zhiqin Lu, UC. Irvine Hypersurface K-stability 30/42

72 The semi-continuity theorem If M is a smooth hypersurface, and λ = (λ 0,, λ n ) an arbitrary rational vector with n i=0 λ i = 0. Define Theorem (L-Phong) Using the above result, we have As a corollary, we have ξ(λ) = lim t 0 t d dt K(ω FS, σ(t) (ω FS )) ξ(λ) Λ(d 1)(n + 1) ξ(λ) + n lim λ λ,λ U ξ(λ ). n j=0 0 φ j(t)(φ j(t) 1)dt. Zhiqin Lu, UC. Irvine Hypersurface K-stability 30/42

73 Chow-Mumford stability Definition Let F (Z) = α I a α Z α, where a α 0 for any α I. We say M is Chow-Mumford stable, if Zhiqin Lu, UC. Irvine Hypersurface K-stability 31/42

74 Chow-Mumford stability Definition Let F (Z) = α I a α Z α, where a α 0 for any α I. We say M is Chow-Mumford stable, if 1. for any (λ 0,, λ n ) with λ i all integers and λ i = 0, there is an element α = (α 0,, α n ) I such that λ, α > 0; Zhiqin Lu, UC. Irvine Hypersurface K-stability 31/42

75 Chow-Mumford stability Definition Let F (Z) = α I a α Z α, where a α 0 for any α I. We say M is Chow-Mumford stable, if 1. for any (λ 0,, λ n ) with λ i all integers and λ i = 0, there is an element α = (α 0,, α n ) I such that λ, α > 0; 2. there is no holomorphic vector field on M. Zhiqin Lu, UC. Irvine Hypersurface K-stability 31/42

76 By a theorem of Mumford, smooth hypersurface is Chow-Mumford stable. Zhiqin Lu, UC. Irvine Hypersurface K-stability 32/42

77 The technical heart of the paper is the following result: we define the function Λ(d 1)(n + 1) g(λ) = + n n j=0 0 φ j(t)(φ j(t) 1)dt. Zhiqin Lu, UC. Irvine Hypersurface K-stability 33/42

78 The technical heart of the paper is the following result: we define the function Λ(d 1)(n + 1) g(λ) = + n n j=0 0 φ j(t)(φ j(t) 1)dt. Theorem If M is Chow-Mumford stable, then g(λ) < 0, unless λ a(n, 1,, 1) for any positive number a. Zhiqin Lu, UC. Irvine Hypersurface K-stability 33/42

79 We first prove the following elementary lemma: Lemma Let be a finite set of liner functions. I = {ax + b a 0} Zhiqin Lu, UC. Irvine Hypersurface K-stability 34/42

80 We first prove the following elementary lemma: Lemma Let I = {ax + b a 0} be a finite set of liner functions. Assume that I contains at least one constant function. Define φ I = min f f I Zhiqin Lu, UC. Irvine Hypersurface K-stability 34/42

81 We first prove the following elementary lemma: Lemma Let I = {ax + b a 0} be a finite set of liner functions. Assume that I contains at least one constant function. Define φ I = min f f I Let J I, then we have 0 φ I (t)(φ I (t) 1)dt 0 φ J(t)(φ J(t) 1)dt Zhiqin Lu, UC. Irvine Hypersurface K-stability 34/42

82 Less linear function to minimize, larger value 0 φ (t)(φ (t) 1)dt. Zhiqin Lu, UC. Irvine Hypersurface K-stability 35/42

83 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42

84 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. By the Chow-Mumford stability, λ, β = max λ, α > 0 Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42

85 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. By the Chow-Mumford stability, λ, β = max λ, α > 0 Define κ = (n, 1,, 1) Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42

86 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. By the Chow-Mumford stability, λ, β = max λ, α > 0 Define κ = (n, 1,, 1) Define for a suitable constant c, we have λ = λ cκ Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42

87 Squeezing more from CM stability In terms of stability, it is easier if we identify the polynomial F with the set I where the coefficients of Z α for any α I are not zero. By the Chow-Mumford stability, λ, β = max λ, α > 0 Define κ = (n, 1,, 1) Define λ = λ cκ for a suitable constant c, we have λ, β = max λ, α Zhiqin Lu, UC. Irvine Hypersurface K-stability 36/42

88 In our proof, we pick up finite sequences λ k = λ c k κ, β k Let the set of β k be J. Zhiqin Lu, UC. Irvine Hypersurface K-stability 37/42

89 In our proof, we pick up finite sequences λ k = λ c k κ, β k Let the set of β k be J. By the lemma above, φ (t)(φ (t) 1)dt 0 0 φ J(t)(φ J(t) 1)dt Zhiqin Lu, UC. Irvine Hypersurface K-stability 37/42

90 Define Λ(d 1)(n + 1) g(λ) = n + φ J(t)(φ J(t) 1)dt. 0 Zhiqin Lu, UC. Irvine Hypersurface K-stability 38/42

91 Define Λ(d 1)(n + 1) g(λ) = n + φ J(t)(φ J(t) 1)dt. 0 Then g(λ) g(λ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 38/42

92 Define Λ(d 1)(n + 1) g(λ) = n + φ J(t)(φ J(t) 1)dt. 0 Then g(λ) g(λ) The key observation is that g(λ) g(λ 1 ) g(λ s ) Zhiqin Lu, UC. Irvine Hypersurface K-stability 38/42

93 Using the above inequality, we can prove the semi-k-stability: at least the function g is not positive, Zhiqin Lu, UC. Irvine Hypersurface K-stability 39/42

94 Using the above inequality, we can prove the semi-k-stability: at least the function g is not positive, by reducing an arbitrary text configuration to a trivial one. Zhiqin Lu, UC. Irvine Hypersurface K-stability 39/42

95 Using the above inequality, we can prove the semi-k-stability: at least the function g is not positive, by reducing an arbitrary text configuration to a trivial one. The only exception is that is λ is proportional to κ. By by continuity, we can always perturb it to prove the semi-k-stability. Zhiqin Lu, UC. Irvine Hypersurface K-stability 39/42

96 We can characterize κ in the following Lemma (L-Phong) Let J(t) be the J-functional restricted to the one-parameter group. Then J(t) = O(1) if and only if the test configuration is induced by κ. Here J(t) is Aubin s J-functional. Zhiqin Lu, UC. Irvine Hypersurface K-stability 40/42

97 Example The is a KE metric on the Fermat surface Z n Z n n = 0. However, along the line defined by the test configuration (n, 1,, 1), the Mabuchi functional is of O(1). Zhiqin Lu, UC. Irvine Hypersurface K-stability 41/42

98 Thank you! Zhiqin Lu, UC. Irvine Hypersurface K-stability 42/42

K-stability and Kähler metrics, I

K-stability and Kähler metrics, I K-stability and Kähler metrics, I Gang Tian Beijing University and Princeton University Let M be a Kähler manifold. This means that M be a complex manifold together with a Kähler metric ω. In local coordinates

More information

A Bird Eye s view: recent update to Extremal metrics

A Bird Eye s view: recent update to Extremal metrics A Bird Eye s view: recent update to Extremal metrics Xiuxiong Chen Department of Mathematics University of Wisconsin at Madison January 21, 2009 A Bird Eye s view: recent update to Extremal metrics Xiuxiong

More information

Uniform K-stability of pairs

Uniform K-stability of pairs Uniform K-stability of pairs Gang Tian Peking University Let G = SL(N + 1, C) with two representations V, W over Q. For any v V\{0} and any one-parameter subgroup λ of G, we can associate a weight w λ

More information

CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES

CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES JULIUS ROSS This short survey aims to introduce some of the ideas and conjectures relating stability of projective varieties to the existence of

More information

A Numerical Criterion for Lower bounds on K-energy maps of Algebraic manifolds

A Numerical Criterion for Lower bounds on K-energy maps of Algebraic manifolds A Numerical Criterion for Lower bounds on K-energy maps of Algebraic manifolds Sean Timothy Paul University of Wisconsin, Madison stpaul@math.wisc.edu Outline Formulation of the problem: To bound the Mabuchi

More information

Kähler-Einstein metrics and K-stability

Kähler-Einstein metrics and K-stability May 3, 2012 Table Of Contents 1 Preliminaries 2 Continuity method 3 Review of Tian s program 4 Special degeneration and K-stability 5 Thanks Basic Kähler geometry (X, J, g) (X, J, ω g ) g(, ) = ω g (,

More information

ASYMPTOTIC CHOW SEMI-STABILITY AND INTEGRAL INVARIANTS

ASYMPTOTIC CHOW SEMI-STABILITY AND INTEGRAL INVARIANTS ASYPTOTIC CHOW SEI-STABILITY AND INTEGRAL INVARIANTS AKITO FUTAKI Abstract. We define a family of integral invariants containing those which are closely related to asymptotic Chow semi-stability of polarized

More information

Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy

Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy Li, C. (20) Constant Scalar Curvature Kähler Metric Obtains the Minimum of K-energy, International Mathematics Research Notices, Vol. 20, No. 9, pp. 26 275 Advance Access publication September, 200 doi:0.093/imrn/rnq52

More information

Geometry of the Calabi-Yau Moduli

Geometry of the Calabi-Yau Moduli Geometry of the Calabi-Yau Moduli Zhiqin Lu 2012 AMS Hawaii Meeting Department of Mathematics, UC Irvine, Irvine CA 92697 March 4, 2012 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 1/51

More information

Hilbert series and obstructions to asymptotic Chow semistability

Hilbert series and obstructions to asymptotic Chow semistability 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

More information

A complex geometric proof of Tian-Yau-Zelditch expansion

A complex geometric proof of Tian-Yau-Zelditch expansion A complex geometric proof of Tian-Yau-Zelditch expansion Zhiqin Lu Department of Mathematics, UC Irvine, Irvine CA 92697 October 21, 2010 Zhiqin Lu, Dept. Math, UCI A complex geometric proof of TYZ expansion

More information

Mathematische Annalen

Mathematische Annalen Math. Ann. DOI 10.1007/s00208-017-1592-5 Mathematische Annalen Relative K-stability for Kähler manifolds Ruadhaí Dervan 1,2 Received: 19 April 2017 The Author(s) 2017. This article is an open access publication

More information

The Yau-Tian-Donaldson Conjectuture for general polarizations

The Yau-Tian-Donaldson Conjectuture for general polarizations The Yau-Tian-Donaldson Conjectuture for general polarizations Toshiki Mabuchi, Osaka University 2015 Taipei Conference on Complex Geometry December 22, 2015 1. Introduction 2. Background materials Table

More information

Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang

Mathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS. Shuguang Wang Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERG-WITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free anti-holomorphic involutions

More information

On the Convergence of a Modified Kähler-Ricci Flow. 1 Introduction. Yuan Yuan

On the Convergence of a Modified Kähler-Ricci Flow. 1 Introduction. Yuan Yuan On the Convergence of a Modified Kähler-Ricci Flow Yuan Yuan Abstract We study the convergence of a modified Kähler-Ricci flow defined by Zhou Zhang. We show that the modified Kähler-Ricci flow converges

More information

Moduli space of smoothable Kähler-Einstein Q-Fano varieties

Moduli space of smoothable Kähler-Einstein Q-Fano varieties Moduli space of smoothable Kähler-Einstein Q-Fano varieties Chi Li joint work with Xiaowei Wang and Chenyang Xu Mathematics Department, Stony Brook University University of Tokyo, July 27, 2015 Table of

More information

Balanced Metrics in Kähler Geometry

Balanced Metrics in Kähler Geometry Balanced Metrics in Kähler Geometry by Reza Seyyedali A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy Baltimore, Maryland

More information

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS

LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS LECTURE 6: J-HOLOMORPHIC CURVES AND APPLICATIONS WEIMIN CHEN, UMASS, SPRING 07 1. Basic elements of J-holomorphic curve theory Let (M, ω) be a symplectic manifold of dimension 2n, and let J J (M, ω) be

More information

CHARACTERISTIC CLASSES

CHARACTERISTIC CLASSES 1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact

More information

Trends in Modern Geometry

Trends in Modern Geometry Trends in Modern Geometry July 7th (Mon) 11st (Fri), 2014 Lecture Hall, Graduate School of Mathematical Sciences, the University of Tokyo July 7th (Monday) 14:00 15:00 Xiuxiong Chen (Stony Brook) On the

More information

Stability of algebraic varieties and algebraic geometry. AMS Summer Research Institute in Algebraic Geometry

Stability of algebraic varieties and algebraic geometry. AMS Summer Research Institute in Algebraic Geometry Stability of algebraic varieties and algebraic geometry AMS Summer Research Institute in Algebraic Geometry Table of Contents I Background Kähler metrics Geometric Invariant theory, Kempf-Ness etc. Back

More information

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8

LECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9

More information

Rational Curves On K3 Surfaces

Rational Curves On K3 Surfaces Rational Curves On K3 Surfaces Jun Li Department of Mathematics Stanford University Conference in honor of Peter Li Overview of the talk The problem: existence of rational curves on a K3 surface The conjecture:

More information

arxiv: v1 [math.dg] 8 Jul 2011

arxiv: v1 [math.dg] 8 Jul 2011 b-stability and blow-ups arxiv:1107.1699v1 [math.dg] 8 Jul 2011 1 Introduction S. K. Donaldson August 25, 2018 Dedicated to Professor V. V. Shokurov In [3] the author introduced a notion of b-stability.

More information

arxiv:math/ v2 [math.dg] 19 Jul 2007

arxiv:math/ v2 [math.dg] 19 Jul 2007 DELIGNE PAIRINGS AND THE KNUDSEN-MUMFORD EXPANSION 1 D.H. Phong, Julius Ross and Jacob Sturm arxiv:math/0612555v2 [math.dg] 19 Jul 2007 Department of Mathematics Columbia University, New York, NY 10027

More information

The Kähler-Ricci flow on singular Calabi-Yau varieties 1

The Kähler-Ricci flow on singular Calabi-Yau varieties 1 The Kähler-Ricci flow on singular Calabi-Yau varieties 1 Dedicated to Professor Shing-Tung Yau Jian Song and Yuan Yuan Abstract In this note, we study the Kähler-Ricci flow on projective Calabi-Yau varieties

More information

Radial balanced metrics on the unit disk

Radial balanced metrics on the unit disk Radial balanced metrics on the unit disk Antonio Greco and Andrea Loi Dipartimento di Matematica e Informatica Università di Cagliari Via Ospedale 7, 0914 Cagliari Italy e-mail : greco@unica.it, loi@unica.it

More information

Holomorphic line bundles

Holomorphic line bundles Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

More information

FAKE PROJECTIVE SPACES AND FAKE TORI

FAKE PROJECTIVE SPACES AND FAKE TORI FAKE PROJECTIVE SPACES AND FAKE TORI OLIVIER DEBARRE Abstract. Hirzebruch and Kodaira proved in 1957 that when n is odd, any compact Kähler manifold X which is homeomorphic to P n is isomorphic to P n.

More information

The structure of algebraic varieties

The structure of algebraic varieties The structure of algebraic varieties János Kollár Princeton University ICM, August, 2014, Seoul with the assistance of Jennifer M. Johnson and Sándor J. Kovács (Written comments added for clarity that

More information

arxiv: v4 [math.dg] 7 Nov 2007

arxiv: v4 [math.dg] 7 Nov 2007 The Ricci iteration and its applications arxiv:0706.2777v4 [math.dg] 7 Nov 2007 Yanir A. Rubinstein Abstract. In this Note we introduce and study dynamical systems related to the Ricci operator on the

More information

Lecture VI: Projective varieties

Lecture VI: Projective varieties Lecture VI: Projective varieties Jonathan Evans 28th October 2010 Jonathan Evans () Lecture VI: Projective varieties 28th October 2010 1 / 24 I will begin by proving the adjunction formula which we still

More information

Classifying complex surfaces and symplectic 4-manifolds

Classifying complex surfaces and symplectic 4-manifolds Classifying complex surfaces and symplectic 4-manifolds UT Austin, September 18, 2012 First Cut Seminar Basics Symplectic 4-manifolds Definition A symplectic 4-manifold (X, ω) is an oriented, smooth, 4-dimensional

More information

Conjectures in Kahler geometry

Conjectures in Kahler geometry Conjectures in Kahler geometry S.K. Donaldson Abstract. We state a general conjecture about the existence of Kahler metrics of constant scalar curvature, and discuss the background to the conjecture 1.

More information

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS

BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate

More information

A Joint Adventure in Sasakian and Kähler Geometry

A Joint Adventure in Sasakian and Kähler Geometry A Joint Adventure in Sasakian and Kähler Geometry Charles Boyer and Christina Tønnesen-Friedman Geometry Seminar, University of Bath March, 2015 2 Kähler Geometry Let N be a smooth compact manifold of

More information

arxiv: v1 [math.dg] 24 Nov 2011

arxiv: v1 [math.dg] 24 Nov 2011 BOUNDING SCALAR CURVATURE FOR GLOBAL SOLUTIONS OF THE KÄHLER-RICCI FLOW arxiv:1111.5681v1 [math.dg] 24 Nov 2011 JIAN SONG AND GANG TIAN Abstract. We show that the scalar curvature is uniformly bounded

More information

The geometry of Landau-Ginzburg models

The geometry of Landau-Ginzburg models Motivation Toric degeneration Hodge theory CY3s The Geometry of Landau-Ginzburg Models January 19, 2016 Motivation Toric degeneration Hodge theory CY3s Plan of talk 1. Landau-Ginzburg models and mirror

More information

Introduction to Extremal metrics

Introduction to Extremal metrics Introduction to Extremal metrics Preliminary version Gábor Székelyhidi Contents 1 Kähler geometry 2 1.1 Complex manifolds........................ 3 1.2 Almost complex structures.................... 5 1.3

More information

Cohomology of the Mumford Quotient

Cohomology of the Mumford Quotient Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive G-equivariant line bundle over X. We use a Witten

More information

Gluing problems related to constructing extremal or constant scalar curvature Kähler metrics on blow-ups.

Gluing problems related to constructing extremal or constant scalar curvature Kähler metrics on blow-ups. Ved V. Datar Research Statement My research interests center around geometric analysis, and I am especially interested in the interplay between Riemannian, complex and algebraic geometry. My past and ongoing

More information

Jian Song. Department of Mathematics Columbia University, New York, NY 10027

Jian Song. Department of Mathematics Columbia University, New York, NY 10027 THE α-invariant ON TORIC FANO MANIFOLDS Jian Song Department of Mathematics Columbia University, New York, NY 10027 1 Introduction The global holomorphic invariant α G (X) introduced by Tian [14], Tian

More information

INTRODUCTION TO REAL ANALYTIC GEOMETRY

INTRODUCTION TO REAL ANALYTIC GEOMETRY INTRODUCTION TO REAL ANALYTIC GEOMETRY KRZYSZTOF KURDYKA 1. Analytic functions in several variables 1.1. Summable families. Let (E, ) be a normed space over the field R or C, dim E

More information

On the BCOV Conjecture

On the BCOV Conjecture Department of Mathematics University of California, Irvine December 14, 2007 Mirror Symmetry The objects to study By Mirror Symmetry, for any CY threefold, there should be another CY threefold X, called

More information

FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS

FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS FROM HOLOMORPHIC FUNCTIONS TO HOLOMORPHIC SECTIONS ZHIQIN LU. Introduction It is a pleasure to have the opportunity in the graduate colloquium to introduce my research field. I am a differential geometer.

More information

Algebraic v.s. Analytic Point of View

Algebraic v.s. Analytic Point of View Algebraic v.s. Analytic Point of View Ziwen Zhu September 19, 2015 In this talk, we will compare 3 different yet similar objects of interest in algebraic and complex geometry, namely algebraic variety,

More information

Vanishing theorems and holomorphic forms

Vanishing theorems and holomorphic forms Vanishing theorems and holomorphic forms Mihnea Popa Northwestern AMS Meeting, Lansing March 14, 2015 Holomorphic one-forms and geometry X compact complex manifold, dim C X = n. Holomorphic one-forms and

More information

0.1 Complex Analogues 1

0.1 Complex Analogues 1 0.1 Complex Analogues 1 Abstract In complex geometry Kodaira s theorem tells us that on a Kähler manifold sufficiently high powers of positive line bundles admit global holomorphic sections. Donaldson

More information

Riemannian Curvature Functionals: Lecture III

Riemannian Curvature Functionals: Lecture III Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli

More information

Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016

Topic: First Chern classes of Kähler manifolds Mitchell Faulk Last updated: April 23, 2016 Topic: First Chern classes of Kähler manifolds itchell Faulk Last updated: April 23, 2016 We study the first Chern class of various Kähler manifolds. We only consider two sources of examples: Riemann surfaces

More information

Moduli spaces of log del Pezzo pairs and K-stability

Moduli spaces of log del Pezzo pairs and K-stability Report on Research in Groups Moduli spaces of log del Pezzo pairs and K-stability June 20 - July 20, 2016 Organizers: Patricio Gallardo, Jesus Martinez-Garcia, Cristiano Spotti. In this report we start

More information

Hyperkähler geometry lecture 3

Hyperkähler geometry lecture 3 Hyperkähler geometry lecture 3 Misha Verbitsky Cohomology in Mathematics and Physics Euler Institute, September 25, 2013, St. Petersburg 1 Broom Bridge Here as he walked by on the 16th of October 1843

More information

arxiv: v1 [math.dg] 11 Jan 2009

arxiv: v1 [math.dg] 11 Jan 2009 arxiv:0901.1474v1 [math.dg] 11 Jan 2009 Scalar Curvature Behavior for Finite Time Singularity of Kähler-Ricci Flow Zhou Zhang November 6, 2018 Abstract In this short paper, we show that Kähler-Ricci flows

More information

arxiv: v1 [math.dg] 24 Sep 2009

arxiv: v1 [math.dg] 24 Sep 2009 TORIC GEOMETRY OF CONVEX QUADRILATERALS arxiv:0909.4512v1 [math.dg] 24 Sep 2009 EVELINE LEGENDRE Abstract. We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This

More information

MIYAOKA-YAU-TYPE INEQUALITIES FOR KÄHLER-EINSTEIN MANIFOLDS

MIYAOKA-YAU-TYPE INEQUALITIES FOR KÄHLER-EINSTEIN MANIFOLDS MIYAOKA-YAU-TYPE INEQUALITIES FOR KÄHLER-EINSTEIN MANIFOLDS KWOKWAI CHAN AND NAICHUNG CONAN LEUNG Abstract. We investigate Chern number inequalities on Kähler-Einstein manifolds and their relations to

More information

On the Chow Ring of Certain Algebraic Hyper-Kähler Manifolds

On the Chow Ring of Certain Algebraic Hyper-Kähler Manifolds Pure and Applied Mathematics Quarterly Volume 4, Number 3 (Special Issue: In honor of Fedor Bogomolov, Part 2 of 2 ) 613 649, 2008 On the Chow Ring of Certain Algebraic Hyper-Kähler Manifolds Claire Voisin

More information

THE BEST CONSTANT OF THE MOSER-TRUDINGER INEQUALITY ON S 2

THE BEST CONSTANT OF THE MOSER-TRUDINGER INEQUALITY ON S 2 THE BEST CONSTANT OF THE MOSER-TRUDINGER INEQUALITY ON S 2 YUJI SANO Abstract. We consider the best constant of the Moser-Trudinger inequality on S 2 under a certain orthogonality condition. Applying Moser

More information

Gauged Linear Sigma Model in the Geometric Phase

Gauged Linear Sigma Model in the Geometric Phase Gauged Linear Sigma Model in the Geometric Phase Guangbo Xu joint work with Gang Tian Princeton University International Conference on Differential Geometry An Event In Honour of Professor Gang Tian s

More information

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY

LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in

More information

Degenerate complex Monge-Ampère equations and singular Kähler-Einstein metrics

Degenerate complex Monge-Ampère equations and singular Kähler-Einstein metrics Degenerate complex Monge-Ampère equations and singular Kähler-Einstein metrics Vincent Guedj Institut Universitaire de France & Institut de Mathématiques de Toulouse Université Paul Sabatier (France) Abstract

More information

arxiv: v3 [math.dg] 30 Nov 2016

arxiv: v3 [math.dg] 30 Nov 2016 UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS IN KÄHLER GEOMETRY SÉBASTIEN BOUCKSOM, TOMOYUKI HISAMOTO, AND MATTIAS JONSSON ariv:1603.01026v3 [math.dg] 30 Nov 2016 Abstract. Consider a polarized

More information

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD

INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:

More information

arxiv: v1 [math.ag] 13 Mar 2019

arxiv: v1 [math.ag] 13 Mar 2019 THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show

More information

Essential Spectra of complete manifolds

Essential Spectra of complete manifolds Essential Spectra of complete manifolds Zhiqin Lu Analysis, Complex Geometry, and Mathematical Physics: A Conference in Honor of Duong H. Phong May 7, 2013 Zhiqin Lu, Dept. Math, UCI Essential Spectra

More information

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents

CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS. Contents CHAPTER 1. TOPOLOGY OF ALGEBRAIC VARIETIES, HODGE DECOMPOSITION, AND APPLICATIONS Contents 1. The Lefschetz hyperplane theorem 1 2. The Hodge decomposition 4 3. Hodge numbers in smooth families 6 4. Birationally

More information

Qualifying Exams I, 2014 Spring

Qualifying Exams I, 2014 Spring Qualifying Exams I, 2014 Spring 1. (Algebra) Let k = F q be a finite field with q elements. Count the number of monic irreducible polynomials of degree 12 over k. 2. (Algebraic Geometry) (a) Show that

More information

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current Author, F., and S. Author. (2015) Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current, International Mathematics Research Notices, Vol. 2015, Article ID rnn999, 7 pages. doi:10.1093/imrn/rnn999

More information

HYPERKÄHLER MANIFOLDS

HYPERKÄHLER MANIFOLDS HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly

More information

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,...

Takao Akahori. z i In this paper, if f is a homogeneous polynomial, the correspondence between the Kodaira-Spencer class and C[z 1,... J. Korean Math. Soc. 40 (2003), No. 4, pp. 667 680 HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES Takao Akahori Abstract. The mirror conjecture means originally the deep relation between complex

More information

Notes on canonical Kähler metrics and quantisation

Notes on canonical Kähler metrics and quantisation Notes on canonical Kähler metrics and quantisation Joel Fine Summer school Universtität zu Köln July 2012. Contents 1 Introduction 4 2 Brief review of Kähler basics 5 2.1 Chern connections and Chern classes.............

More information

Algebraic geometry over quaternions

Algebraic geometry over quaternions Algebraic geometry over quaternions Misha Verbitsky November 26, 2007 Durham University 1 History of algebraic geometry. 1. XIX centrury: Riemann, Klein, Poincaré. Study of elliptic integrals and elliptic

More information

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES

MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.

More information

THE VORTEX EQUATION ON AFFINE MANIFOLDS. 1. Introduction

THE VORTEX EQUATION ON AFFINE MANIFOLDS. 1. Introduction THE VORTEX EQUATION ON AFFINE MANIFOLDS INDRANIL BISWAS, JOHN LOFTIN, AND MATTHIAS STEMMLER Abstract. Let M be a compact connected special affine manifold equipped with an affine Gauduchon metric. We show

More information

SEPARABLE RATIONAL CONNECTEDNESS AND STABILITY

SEPARABLE RATIONAL CONNECTEDNESS AND STABILITY SEPARABLE RATIONAL CONNECTEDNESS AND STABILIT ZHIU TIAN Abstract. In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability

More information

SCALAR CURVATURE BEHAVIOR FOR FINITE TIME SINGULARITY OF KÄHLER-RICCI FLOW

SCALAR CURVATURE BEHAVIOR FOR FINITE TIME SINGULARITY OF KÄHLER-RICCI FLOW SALAR URVATURE BEHAVIOR FOR FINITE TIME SINGULARITY OF KÄHLER-RII FLOW ZHOU ZHANG 1. Introduction Ricci flow, since the debut in the famous original work [4] by R. Hamilton, has been one of the major driving

More information

KASS November 23, 10.00

KASS November 23, 10.00 KASS 2011 November 23, 10.00 Jacob Sznajdman (Göteborg): Invariants of analytic curves and the Briancon-Skoda theorem. Abstract: The Briancon-Skoda number of an analytic curve at a point p, is the smallest

More information

Logarithmic geometry and rational curves

Logarithmic geometry and rational curves Logarithmic geometry and rational curves Summer School 2015 of the IRTG Moduli and Automorphic Forms Siena, Italy Dan Abramovich Brown University August 24-28, 2015 Abramovich (Brown) Logarithmic geometry

More information

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS

LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS LECTURE 5: COMPLEX AND KÄHLER MANIFOLDS Contents 1. Almost complex manifolds 1. Complex manifolds 5 3. Kähler manifolds 9 4. Dolbeault cohomology 11 1. Almost complex manifolds Almost complex structures.

More information

ON SINGULAR CUBIC SURFACES

ON SINGULAR CUBIC SURFACES ASIAN J. MATH. c 2009 International Press Vol. 13, No. 2, pp. 191 214, June 2009 003 ON SINGULAR CUBIC SURFACES IVAN CHELTSOV Abstract. We study global log canonical thresholds of singular cubic surfaces.

More information

ON NEARLY SEMIFREE CIRCLE ACTIONS

ON NEARLY SEMIFREE CIRCLE ACTIONS ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)

More information

Scalar curvature and the Thurston norm

Scalar curvature and the Thurston norm Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,

More information

THE QUANTUM CONNECTION

THE QUANTUM CONNECTION THE QUANTUM CONNECTION MICHAEL VISCARDI Review of quantum cohomology Genus 0 Gromov-Witten invariants Let X be a smooth projective variety over C, and H 2 (X, Z) an effective curve class Let M 0,n (X,

More information

This theorem gives us a corollary about the geometric height inequality which is originally due to Vojta [V].

This theorem gives us a corollary about the geometric height inequality which is originally due to Vojta [V]. 694 KEFENG LIU This theorem gives us a corollary about the geometric height inequality which is originally due to Vojta [V]. Corollary 0.3. Given any ε>0, there exists a constant O ε (1) depending on ε,

More information

Geometry of the Moduli Space of Curves and Algebraic Manifolds

Geometry of the Moduli Space of Curves and Algebraic Manifolds Geometry of the Moduli Space of Curves and Algebraic Manifolds Shing-Tung Yau Harvard University 60th Anniversary of the Institute of Mathematics Polish Academy of Sciences April 4, 2009 The first part

More information

Generalized Tian-Todorov theorems

Generalized Tian-Todorov theorems Generalized Tian-Todorov theorems M.Kontsevich 1 The classical Tian-Todorov theorem Recall the classical Tian-Todorov theorem (see [4],[5]) about the smoothness of the moduli spaces of Calabi-Yau manifolds:

More information

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY

RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY RIEMANN SURFACES: TALK V: DOLBEAULT COHOMOLOGY NICK MCCLEEREY 0. Complex Differential Forms Consider a complex manifold X n (of complex dimension n) 1, and consider its complexified tangent bundle T C

More information

Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds

Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds arxiv:0805.3689v1 [hep-th] 23 May 2008 Eigenvalues and Eigenfunctions of the Scalar Laplace Operator on Calabi-Yau Manifolds Volker Braun 1, Tamaz Brelidze 1, Michael R. Douglas 2, and Burt A. Ovrut 1

More information

RICCI SOLITONS ON COMPACT KAHLER SURFACES. Thomas Ivey

RICCI SOLITONS ON COMPACT KAHLER SURFACES. Thomas Ivey RICCI SOLITONS ON COMPACT KAHLER SURFACES Thomas Ivey Abstract. We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming

More information

arxiv: v5 [math.dg] 9 Jul 2018

arxiv: v5 [math.dg] 9 Jul 2018 THE KÄHLER-RICCI FLOW AND OPTIMAL DEGENERATIONS ariv:1612.07299v5 [math.dg] 9 Jul 2018 RUADHAÍ DERVAN AND GÁBOR SZÉKELYHIDI Abstract. We prove that on Fano manifolds, the Kähler-Ricci flow produces a most

More information

Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1

Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1 Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines

More information

Kähler (& hyper-kähler) manifolds

Kähler (& hyper-kähler) manifolds Kähler (& hyper-kähler) manifolds Arithmetic & Algebraic Geometry Seminar, KdVI Reinier Kramer October 23, 2015 Contents Introduction 1 1 Basic definitions 1 2 Metrics and connections on vector bundles

More information

NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD

NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD INDRANIL BISWAS Abstract. Our aim is to review some recent results on holomorphic principal bundles over a compact Kähler manifold.

More information

LECTURE 26: THE CHERN-WEIL THEORY

LECTURE 26: THE CHERN-WEIL THEORY LECTURE 26: THE CHERN-WEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a k-tensor T k V is called symmetric if

More information

The Futaki Invariant of Kähler Blowups with Isolated Zeros via Localization

The Futaki Invariant of Kähler Blowups with Isolated Zeros via Localization The Futaki Invariant of Kähler Blowups with Isolated Zeros via Localization Luke Cherveny October 30, 2017 Abstract We present an analytic proof of the relationship between the Calabi-Futaki invariant

More information

Lecture 4: Harmonic forms

Lecture 4: Harmonic forms Lecture 4: Harmonic forms Jonathan Evans 29th September 2010 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 1 / 15 Jonathan Evans () Lecture 4: Harmonic forms 29th September 2010 2 / 15

More information

Introduction To K3 Surfaces (Part 2)

Introduction To K3 Surfaces (Part 2) Introduction To K3 Surfaces (Part 2) James Smith Calf 26th May 2005 Abstract In this second introductory talk, we shall take a look at moduli spaces for certain families of K3 surfaces. We introduce the

More information

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS

SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS SPACES OF RATIONAL CURVES IN COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e in a general complete intersection of multidegree

More information

Mirror Symmetry: Introduction to the B Model

Mirror Symmetry: Introduction to the B Model Mirror Symmetry: Introduction to the B Model Kyler Siegel February 23, 2014 1 Introduction Recall that mirror symmetry predicts the existence of pairs X, ˇX of Calabi-Yau manifolds whose Hodge diamonds

More information

15 Elliptic curves and Fermat s last theorem

15 Elliptic curves and Fermat s last theorem 15 Elliptic curves and Fermat s last theorem Let q > 3 be a prime (and later p will be a prime which has no relation which q). Suppose that there exists a non-trivial integral solution to the Diophantine

More information

RATIONALLY INEQUIVALENT POINTS ON HYPERSURFACES IN P n

RATIONALLY INEQUIVALENT POINTS ON HYPERSURFACES IN P n RATIONALLY INEQUIVALENT POINTS ON HYPERSURFACES IN P n XI CHEN, JAMES D. LEWIS, AND MAO SHENG Abstract. We prove a conjecture of Voisin that no two distinct points on a very general hypersurface of degree

More information