# Geometry of the Calabi-Yau Moduli

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

2 An Example (Strominger and Freed) Let X be a Kähler manifold. That is, X is a complex manifold with a closed positive (1, 1) form ω ω = 1 2π g i jdz i d z j, Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 2/51

3 An Example (Strominger and Freed) Let X be a Kähler manifold. That is, X is a complex manifold with a closed positive (1, 1) form ω ω = 1 2π g i jdz i d z j, g i j = 2 f z i z j Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 2/51

4 An Example (Strominger and Freed) Let X be a Kähler manifold. That is, X is a complex manifold with a closed positive (1, 1) form ω ω = 1 2π g i jdz i d z j, g i j = 2 f z i z j Let U = {U α } be a cover of X. Assume that on each U α, there exists a holomorphic function u = u α such that the Kähler metric ω can be represented by ω = ( ) 2 u 1 Im dz i d z j = 1 g z i z i jdz i d z j. j Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 2/51

5 An Example (Strominger and Freed) Let X be a Kähler manifold. That is, X is a complex manifold with a closed positive (1, 1) form ω ω = 1 2π g i jdz i d z j, g i j = 2 f z i z j Let U = {U α } be a cover of X. Assume that on each U α, there exists a holomorphic function u = u α such that the Kähler metric ω can be represented by ω = ( ) 2 u 1 Im dz i d z j = 1 g z i z i jdz i d z j. j Then X is called a special Kähler manifold (in the sense of Stronminger and Freed). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 2/51

6 Theorem The curvature of the metric is R i jk l = 1 4 gm n u ikm u jln In particular, the scalar curvature of the metric is ρ = 1 4 gm n g i j g k lu ikm u jln, where u ijk = 3 u z i z j z k. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 3/51

7 Theorem The curvature of the metric is R i jk l = 1 4 gm n u ikm u jln In particular, the scalar curvature of the metric is ρ = 1 4 gm n g i j g k lu ikm u jln, where u ijk = 3 u z i z j z k. Thus X is Ricci nonnegative and the scalar curvature is nonnegative. Moreover, if the scalar curvature is zero, then so is the curvature tensor. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 3/51

8 Theorem (L,1999) We have ρ 3 n 3 ρ2 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 4/51

9 From the above argument, we proved that Theorem (L,1999, a conjecture of Freed) If X is a complete special Kähler manifold, then X has to be a flat space. Proof. (Assume that X is compact) By the maximum principle, we have ρ 0, which implies that the sectional curvatures are zero. Therefore the manifold has to be flat. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 5/51

10 CY moduli has the similar Kähler metric structure. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 6/51

11 CY moduli has the similar Kähler metric structure. 1 If M is special Kähler, then T M is hyper Kähler. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 6/51

12 CY moduli has the similar Kähler metric structure. 1 If M is special Kähler, then T M is hyper Kähler. 2 (roughly speaking) If M is CY moduli, then T M is pseudo hyper-kähler. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 6/51

13 We consider a polarized Calabi-Yau manifold (X, L), where 1 X is a Calabi-Yau manifold, Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 7/51

14 We consider a polarized Calabi-Yau manifold (X, L), where 1 X is a Calabi-Yau manifold, a simply connected compact Kähler manifold with zero first Chern class. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 7/51

15 We consider a polarized Calabi-Yau manifold (X, L), where 1 X is a Calabi-Yau manifold, a simply connected compact Kähler manifold with zero first Chern class. 2 L is an ample line bundle over X. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 7/51

16 We consider a polarized Calabi-Yau manifold (X, L), where 1 X is a Calabi-Yau manifold, a simply connected compact Kähler manifold with zero first Chern class. 2 L is an ample line bundle over X. 3 Why polarization? Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 7/51

17 The following theorem of Yau is classical in understanding the so-called Calabi-Yau manifold: Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 8/51

18 The following theorem of Yau is classical in understanding the so-called Calabi-Yau manifold: Theorem (Yau) There exists a unique Ricci flat Kähler metric ω in the cohomology class defined by [ω]. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 8/51

19 Why polarization? The Ricci flat metric is unique only when fixing the cohomology class of a Kähler metric. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 9/51

20 Then why moduli space (CY moduli)? Moduli space gives a way to compare CY manifolds that are close to each other. So it gives a platform of linearization. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 10/51

21 We will study the differential geometry of CY moduli. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 11/51

22 We will study the differential geometry of CY moduli. 1 local properties Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 11/51

23 We will study the differential geometry of CY moduli. 1 local properties 2 semi-local (semi-global) properties Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 11/51

24 We will study the differential geometry of CY moduli. 1 local properties 2 semi-local (semi-global) properties 3 global properties Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 11/51

25 A very quick review of Kodaira-Spencer theory Let Z be a compact complex manifold. A complex structure J is a real operator J : T C Z T C Z such that J 2 = I (plus the integrability conditions). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 12/51

26 A very quick review of Kodaira-Spencer theory Let Z be a compact complex manifold. A complex structure J is a real operator J : T C Z T C Z such that J 2 = I (plus the integrability conditions). For fixed frames, we have ( ) 1I J = 1I Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 12/51

27 A very quick review of Kodaira-Spencer theory Let Z be a compact complex manifold. A complex structure J is a real operator J : T C Z T C Z such that J 2 = I (plus the integrability conditions). For fixed frames, we have ( ) 1I J = 1I A variation of the complex structure is a real matrix A such that (J + εa) 2 = I Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 12/51

28 A very quick review of Kodaira-Spencer theory-2 Or we have Thus we have AJ + JA = 0 ( A = 2Ā1 ) 2A 1 for some A 1 : T 1,0 Z T 0,1 Z, or equivalently A 1 can be represented by ϕ Λ 0,1 (T 1,0 Z) (equivalent to Beltrami differential for Riemann Surface) Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 13/51

29 A very quick review of Kodaira-Spencer theory-2 Or we have Thus we have AJ + JA = 0 ( A = 2Ā1 ) 2A 1 for some A 1 : T 1,0 Z T 0,1 Z, or equivalently A 1 can be represented by ϕ Λ 0,1 (T 1,0 Z) (equivalent to Beltrami differential for Riemann Surface) The Kodaira-Spencer map is defined as ε ϕ Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 13/51

30 The integrability condition can be written as ( ϕ(t)) 2 = 0. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 14/51

31 The integrability condition can be written as ( ϕ(t)) 2 = 0. or using the notation of super-lie bracket ϕ(t) 1 ϕ [ϕ(t), ϕ(t)] = 0, 2 with ϕ(0) = 0 and ϕ (0) = θ given. t = 0 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 14/51

32 Sufficient conditions for the existence of smooth Kuranishi spaces Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 15/51

33 Sufficient conditions for the existence of smooth Kuranishi spaces 1 if Z is a Riemann surface (Riemann) Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 15/51

34 Sufficient conditions for the existence of smooth Kuranishi spaces 1 if Z is a Riemann surface (Riemann) 2 if H 2 (Z, T Z) = 0 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 15/51

35 Sufficient conditions for the existence of smooth Kuranishi spaces 1 if Z is a Riemann surface (Riemann) 2 if H 2 (Z, T Z) = 0 3 if Z is a Calabi-Yau manifold Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 15/51

36 Very roughly speaking, the moduli space is the parameter space, which can be identified with the space of complex structures. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 16/51

37 Very roughly speaking, the moduli space is the parameter space, which can be identified with the space of complex structures. 1 Moduli spaces are quasi-projective. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 16/51

38 Very roughly speaking, the moduli space is the parameter space, which can be identified with the space of complex structures. 1 Moduli spaces are quasi-projective. 2 Unlike the moduli space of curves, the compactification of CY moduli is not modular. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 16/51

39 Very roughly speaking, the moduli space is the parameter space, which can be identified with the space of complex structures. 1 Moduli spaces are quasi-projective. 2 Unlike the moduli space of curves, the compactification of CY moduli is not modular. 3 Differential Geometry? Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 16/51

40 Weil-Petersson metric The metric is an L 2 metric, defined by the following Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 17/51

41 Weil-Petersson metric The metric is an L 2 metric, defined by the following Definition Let Z be a polarized Calabi-Yau manifold with the Ricci flat Kähler metric µ whose Kähler form defines the polarization. Let X, Y H 1 (Z, T (1,0) Z). Define the L 2 inner product by (X, Y ) = 1 X, Y µ n. n! For a Calabi-Yau manifold, via the Kodaira-Spencer map: T Z M H 1 (Z, T (1,0) Z), which is an isomorphism, the above inner product defines a metric on the smooth part of M. The metric happens to be Kählerian, and is called the Weil-Petersson metric of M. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 17/51 Z

42 The differential geometry of the Weil-Petersson metric on Calabi-Yau moduli is called the Weil-Petersson geometry. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 18/51

43 The differential geometry of the Weil-Petersson metric on Calabi-Yau moduli is called the Weil-Petersson geometry. The Aim: We want to be able to tell the properties of the moduli space through geometric analysis. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 18/51

44 Extrinsic characterization of the WP metric Theorem (Tian) Let M be the CY moduli of Z. Let Ω be a holomorphic family of nonzero (n, 0) forms. Then the WP metric can be expressed as 1 ω = 2π log Ω Ω Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 19/51

45 Extrinsic characterization of the WP metric Theorem (Tian) Let M be the CY moduli of Z. Let Ω be a holomorphic family of nonzero (n, 0) forms. Then the WP metric can be expressed as 1 ω = 2π log Ω Ω This is quite unique! The Weil-Petersson metric can be expressed explicitly in terms of the variation of Hodge structures. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 19/51

46 Hodge theory! Recall that for any compact complex manifold, we can define the cohomology groups H p,q. By Hodge theorem, they are made from harmonic forms. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 20/51

47 If we deform the CY manifolds, we deform the Hodge flags. This is called the variation of Hodge structure. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 21/51

48 Let s take the example of moduli space of a CY 3 fold. On the moduli space, we can define the following Hodge bundles: F 3 = H 3,0 F 2 = H 3,0 H 2,1 F 1 = H 3,0 H 2,1 H 1,2 F 0 = H 3,0 H 2,1 H 1,2 H 0,3 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 22/51

49 Let s take the example of moduli space of a CY 3 fold. On the moduli space, we can define the following Hodge bundles: Serre Duality F 3 = H 3,0 F 2 = H 3,0 H 2,1 F 1 = H 3,0 H 2,1 H 1,2 F 0 = H 3,0 H 2,1 H 1,2 H 0,3 H 1,2 = H 2 (X, Ω 1 ) = H 1 (X, T X) K S T M. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 22/51

50 Since F 3 is a subbundle of a locally flat bundle, we can write a holomorphic section Ω of F 3 as (Bryant-Griffiths) Ω = (1, 1 1 z 1,, z n, u z i u i, u 1,, u n ), 2 2 where n = dim H 2,1 = dim M, (z 1,, z n ) can be used as holomorphic coordinates. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 23/51

51 Since F 3 is a subbundle of a locally flat bundle, we can write a holomorphic section Ω of F 3 as (Bryant-Griffiths) Ω = (1, 1 1 z 1,, z n, u z i u i, u 1,, u n ), 2 2 where n = dim H 2,1 = dim M, (z 1,, z n ) can be used as holomorphic coordinates. We define Q(ϕ, ψ) = ϕ ψ Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 23/51

52 Since F 3 is a subbundle of a locally flat bundle, we can write a holomorphic section Ω of F 3 as (Bryant-Griffiths) Ω = (1, 1 1 z 1,, z n, u z i u i, u 1,, u n ), 2 2 where n = dim H 2,1 = dim M, (z 1,, z n ) can be used as holomorphic coordinates. We define Q(ϕ, ψ) = ϕ ψ Then Q is skew-symmetric and can be written as ( ) 1 Q = 1 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 23/51

53 With the above theorem, we have Ω Ω = Im (u (Re z i ) u i ). (In special Kähler case, g i j = Im u ij ). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 24/51

54 With the above theorem, we have Ω Ω = Im (u (Re z i ) u i ). (In special Kähler case, g i j = Im u ij ). The curvature of the metric is (for the CY moduli of CY 3 folds) R i jk l = g i jg k l + g i lg k j g m n F ikm Fjln, where F ijk is the Yukawa coupling. (In special Kähler case, R i jk l = g m n F ikm Fjln ). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 24/51

55 With the above theorem, we have Ω Ω = Im (u (Re z i ) u i ). (In special Kähler case, g i j = Im u ij ). The curvature of the metric is (for the CY moduli of CY 3 folds) R i jk l = g i jg k l + g i lg k j g m n F ikm Fjln, where F ijk is the Yukawa coupling. (In special Kähler case, R i jk l = g m n F ikm Fjln ). The curvature is similar to that of special Kähler manifolds. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 24/51

56 Local Weil-Petersson geometry Good properties Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 25/51

57 Local Weil-Petersson geometry Good properties Kählerian Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 25/51

58 Local Weil-Petersson geometry Good properties Kählerian Formula of Strominger Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 25/51

59 Local Weil-Petersson geometry Good properties Kählerian Formula of Strominger Many others Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 25/51

60 Local Weil-Petersson geometry Good properties Kählerian Formula of Strominger Many others Bad property Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 25/51

61 Local Weil-Petersson geometry Good properties Kählerian Formula of Strominger Many others Bad property The curvature is neither positive nor negative. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 25/51

62 Rebuilding the local geometry The following result was proved Theorem (L,1997) Let M be the moduli space of a CY 3-fold with dimension m. Define the following metric ω H = (m + 3)ω W P + Ric (ω W P ), Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 26/51

63 Rebuilding the local geometry The following result was proved Theorem (L,1997) Let M be the moduli space of a CY 3-fold with dimension m. Define the following metric ω H = (m + 3)ω W P + Ric (ω W P ), which I called it Hodge metric. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 26/51

64 Rebuilding the local geometry The following result was proved Theorem (L,1997) Let M be the moduli space of a CY 3-fold with dimension m. Define the following metric ω H = (m + 3)ω W P + Ric (ω W P ), which I called it Hodge metric. The curvature has good properties (non-positive bisectional curvature, negative Ricci and holomorphic sectional curvature, etc). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 26/51

65 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

66 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Semi-global theory Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

67 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Semi-global theory The BCOV conjecture (settled by Zinger-07 Hao Fang-L-Yoshikawa -06) Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

68 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Semi-global theory The BCOV conjecture (settled by Zinger-07 Hao Fang-L-Yoshikawa -06) Incompleteness of the Weil-Petersson metric (-96 for 1d, L-12 in preparation) Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

69 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Semi-global theory The BCOV conjecture (settled by Zinger-07 Hao Fang-L-Yoshikawa -06) Incompleteness of the Weil-Petersson metric (-96 for 1d, L-12 in preparation) A lot of others Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

70 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Semi-global theory The BCOV conjecture (settled by Zinger-07 Hao Fang-L-Yoshikawa -06) Incompleteness of the Weil-Petersson metric (-96 for 1d, L-12 in preparation) A lot of others Global theory Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

71 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Semi-global theory The BCOV conjecture (settled by Zinger-07 Hao Fang-L-Yoshikawa -06) Incompleteness of the Weil-Petersson metric (-96 for 1d, L-12 in preparation) A lot of others Global theory Freed Conjecture (L-99) Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

72 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Semi-global theory The BCOV conjecture (settled by Zinger-07 Hao Fang-L-Yoshikawa -06) Incompleteness of the Weil-Petersson metric (-96 for 1d, L-12 in preparation) A lot of others Global theory Freed Conjecture (L-99) Gauss-Bonnet Theorem (Joint/w M. R. Douglas) Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

73 We want to study the differential geometric properties of the moduli space with respect to the Weil-Petersson metric. Local theory Semi-global theory The BCOV conjecture (settled by Zinger-07 Hao Fang-L-Yoshikawa -06) Incompleteness of the Weil-Petersson metric (-96 for 1d, L-12 in preparation) A lot of others Global theory Freed Conjecture (L-99) Gauss-Bonnet Theorem (Joint/w M. R. Douglas) Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 27/51

74 The BCOV Conjecture BCOV=Bershadsky-Cecotti-Ooguri-Vafa Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 28/51

75 The BCOV Conjecture BCOV=Bershadsky-Cecotti-Ooguri-Vafa Definition The mirror map is the holomorphic map from a neighborhood of P 1 to a neighborhood of 0 defined by the following formula ( q := (5ψ) 5 5 exp y 0 (ψ) n=1 { 5n (5n)! (n!) 5 j=n+1 } 1 j ) 1, (5ψ) 5n where ψ 1, and y 0 (ψ) := n=0 (5n)!, ψ > 1. (n!) 5 (5ψ) 5n The inverse of the mirror map is denoted by ψ(q). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 28/51

76 Define the multi-valued function F top 1,B (ψ) as ( ) 62 ψ F top 1,B (ψ) := 3 (ψ 5 1) 1 dψ 6 q y 0 (ψ) dq, and F top top 1,A (q) := F1,B (ψ(q)). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 29/51

77 Conjecture (A) Conjecture (A) Let n g (d) be the genus-g degree-d instanton number of a quintic in CP 4 for g = 0, 1. Then the following identity holds: q d top log F1,A dq (q) = nd qnd n 1 (d) 1 q nd n,d=1 d=1 n 0 (d) 2d q d 12(1 q d ). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 30/51

78 Conjecture (A) was proved by Aleksey Zinger. Aleksey Zinger The Reduced Genus-One Gromov-Witten Invariants of Calabi-Yau Hypersurfaces ArXiv: v2, JAMS 08 Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 31/51

79 Setup of Conjecture (B) Let X be a compact Kähler manifold. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 32/51

80 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 32/51

81 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 32/51

82 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 32/51

83 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Not well-defined? Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 32/51

84 Setup of Conjecture (B) Let X be a compact Kähler manifold. Let = p,q be the Laplacian on (p, q) forms; By compactness, the spectrum of are eigenvalues: 0 λ 0 λ 1 λ n +. Define det = λ i 0 λ i. Not well-defined? ζ function regularization (for example: Riemann ζ-function) Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 32/51

85 Setup of Conjecture B Bershadsky-Ceccotti-Ooguri-Vafa defined T def = (det p,q ) ( 1)p+qpq. p,q Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 33/51

86 Setup of Conjecture B Bershadsky-Ceccotti-Ooguri-Vafa defined T def = (det p,q ) ( 1)p+qpq. p,q Why define such a strange quantity? Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 33/51

87 Conjecture (B) Let be the Hermitian metric on the line bundle (π K W/CP 1) 62 (T (CP 1 )) 3 CP 1 \D induced from the L 2 -metric on π K W/CP 1 and from the Weil-Petersson metric on T (CP 1 ). Then the following identity holds: ( ) 62 ( 1 Ωψ τ BCOV (W ψ ) = Const. F top q d ) 2 3 3, 1,B (ψ)3 y 0 (ψ) dq where Ω is the local holomorphic section of the (3, 0) forms. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 34/51

88 Conjecture B was proved by Fang-L-Yoshikawa. Hao Fang-L-Yoshikawa Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli ArXiv: JDG (80), 2008, , Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 35/51

89 Donaldson s Functional A(X) = Vol(X, γ) χ(x) 12 [ exp 1 ( 1η η log 12 γ 3 /3! X ) Vol(X, γ) η 2 L 2 c 3 (X, γ)] Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 36/51

90 Reduction of the problem Theorem (Fang-L) On the moduli space of a primitive CY 3-fold, we have Then 1 ( 1) n ω H 2π log τ BCOV = χ X 12 ω W P, where χ top (X) is the Euler characteristic number of X. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 37/51

91 Reduction of the problem Theorem (Fang-L) On the moduli space of a primitive CY 3-fold, we have Then 1 ( 1) n ω H 2π log τ BCOV = χ X 12 ω W P, where χ top (X) is the Euler characteristic number of X. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 37/51

92 Corollary If X is a primitive Calabi-Yau, N M is a k-dimensional complete subvariety of M where M is the moduli space of X, then the following volume identity holds: [ ] ( 1) n k Vol H n(n) = 12 χ(x) Vol WP(N). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 38/51

93 Corollary Assume that a polarized Calabi-Yau manifold X is primitive, and that ( 1) n+1 χ X > 24. Let M be the moduli space of X. Then there exists no complete curve in M; hence, there exists no projective subvariety of M (of positive dimensions). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 39/51

94 Corollary Assume that a polarized Calabi-Yau manifold X is primitive, and that ( 1) n+1 χ X > 24. Let M be the moduli space of X. Then there exists no complete curve in M; hence, there exists no projective subvariety of M (of positive dimensions). In particular, M is not compact. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 39/51

95 Corollary Assume that a polarized Calabi-Yau manifold X is primitive, and that ( 1) n+1 χ X > 24. Let M be the moduli space of X. Then there exists no complete curve in M; hence, there exists no projective subvariety of M (of positive dimensions). In particular, M is not compact. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 39/51

96 Using the curvature formula, the BCOV torsion is determined up to a pluriharmonic function. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 40/51

97 Using the curvature formula, the BCOV torsion is determined up to a pluriharmonic function. Thus we only need to determine the asymptotic behavior of the BCOV torsion at infinity to complete the proof. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 40/51

98 Using the curvature formula, the BCOV torsion is determined up to a pluriharmonic function. Thus we only need to determine the asymptotic behavior of the BCOV torsion at infinity to complete the proof. In fact, we proved that, near a CY 3-fold with one ordinary rational double point, log τ BCOV (X t ) = 1 6 log t 2 + O(log( log t 2 )). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 40/51

99 Recent Results Jian Song On a conjecture of Candelas and de la Ossa arxiv: Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 41/51

100 Recent Results Jian Song On a conjecture of Candelas and de la Ossa arxiv: Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 41/51

101 Recent Results Jian Song On a conjecture of Candelas and de la Ossa arxiv: It is possible to write out the approximation Ricci flat metric near ODP. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 41/51

102 (Fang-L) Using Song s result, a direct method of computing the BCOV torsion? Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 42/51

103 Chern classes A quick review We assume that E M is a holomorphic vector bundle over a compact complex manifold M. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 43/51

104 Chern classes A quick review We assume that E M is a holomorphic vector bundle over a compact complex manifold M. Let h be a Hermitian metric on E. Let Γ = h h 1. be the matrix of the connection. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 43/51

105 Chern classes A quick review We assume that E M is a holomorphic vector bundle over a compact complex manifold M. Let h be a Hermitian metric on E. Let Γ = h h 1. be the matrix of the connection. Let R = Γ be the curvature matrix. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 43/51

106 Let f be an invariant polynomial. That is, f is a polynomial on C r2 such that f(a) = f(t 1 AT ). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 44/51

107 Let f be an invariant polynomial. That is, f is a polynomial on C r2 such that f(a) = f(t 1 AT ). The Chern-Weil form with respect to the polynomial f is defined by f(r). Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 44/51

108 If in addition, we assume that f has integer coefficients, then we have the following results: 1 f(r) is closed. This is the Chern-Weil form; Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 45/51

109 If in addition, we assume that f has integer coefficients, then we have the following results: 1 f(r) is closed. This is the Chern-Weil form; 2 [f(r)] H (M, Z), and is independent of the choice of the connection. Gauss-Bonnet-Chern Theorem Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 45/51

110 This is joint with Michael R. Douglas Zhiqin Lu and Michael R. Douglas Gauss-Bonnet-Chern theorem on moduli spaces arxiv: Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 46/51

111 This is joint with Michael R. Douglas Zhiqin Lu and Michael R. Douglas Gauss-Bonnet-Chern theorem on moduli spaces arxiv: Theorem Let f be an invariant polynomial with rational coefficients. Let R be the curvature tensor with respect the Weil-Petersson metric. Then f(r) is a rational number. M Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 46/51

112 For example, (Ric(M)) m M M c m (ω W P ) V ol ωw P (M) are all rational numbers. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 47/51

113 For example, are all rational numbers. M M (Ric(M)) m c m (ω W P ) V ol ωw P (M) If string theory is correct, then the number of parallel Universes is finite. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 47/51

114 Some remarks The moduli space is often non-compact. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 48/51

115 Some remarks The moduli space is often non-compact. If the moduli space were compact, then the Chern-Weil forms define Chern classes, and the theorem follows. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 48/51

116 Some remarks The moduli space is often non-compact. If the moduli space were compact, then the Chern-Weil forms define Chern classes, and the theorem follows. If the growth of the Chern-Weil forms and the connection were mild at infinity, the theorem follows from a theorem of Mumford. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 48/51

117 Important! We realized that Theorem Let E M be a Hodge bundle over the moduli space of a polarized Kähler manifold. Let R be the curvature tensor with respect to the Hodge bundle. Then f(r) is a rational number. M Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 49/51

118 The above result implies that the Chern-Weil forms are integrable. But it is not that hard to prove the integrability condition directly. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 50/51

119 The above result implies that the Chern-Weil forms are integrable. But it is not that hard to prove the integrability condition directly. We first establish the following well-known Proposition Let M any quasiprojective manifold. Then there is a Kähler metric on M such that 1 It is complete; Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 50/51

120 The above result implies that the Chern-Weil forms are integrable. But it is not that hard to prove the integrability condition directly. We first establish the following well-known Proposition Let M any quasiprojective manifold. Then there is a Kähler metric on M such that 1 It is complete; 2 Its Ricci curvature has a lower bound; and Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 50/51

121 The above result implies that the Chern-Weil forms are integrable. But it is not that hard to prove the integrability condition directly. We first establish the following well-known Proposition Let M any quasiprojective manifold. Then there is a Kähler metric on M such that 1 It is complete; 2 Its Ricci curvature has a lower bound; and 3 it is of finite volume. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 50/51

122 The above result implies that the Chern-Weil forms are integrable. But it is not that hard to prove the integrability condition directly. We first establish the following well-known Proposition Let M any quasiprojective manifold. Then there is a Kähler metric on M such that 1 It is complete; 2 Its Ricci curvature has a lower bound; and 3 it is of finite volume. The metric is called the Poincaré metric. Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 50/51

123 Thank you! Zhiqin Lu, Dept. Math, UCI Geometry of the Calabi-Yau Moduli 51/51

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