Geometry of the CalabiYau Moduli


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1 Geometry of the CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau Moduli 3/51
8 Theorem (L,1999) We have ρ 3 n 3 ρ2 Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau Moduli 5/51
10 CY moduli has the similar Kähler metric structure. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau 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 hyperkähler. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 6/51
13 We consider a polarized CalabiYau manifold (X, L), where 1 X is a CalabiYau manifold, Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 7/51
14 We consider a polarized CalabiYau manifold (X, L), where 1 X is a CalabiYau manifold, a simply connected compact Kähler manifold with zero first Chern class. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 7/51
15 We consider a polarized CalabiYau manifold (X, L), where 1 X is a CalabiYau 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 CalabiYau Moduli 7/51
16 We consider a polarized CalabiYau manifold (X, L), where 1 X is a CalabiYau 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 CalabiYau Moduli 7/51
17 The following theorem of Yau is classical in understanding the socalled CalabiYau manifold: Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 8/51
18 The following theorem of Yau is classical in understanding the socalled CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau Moduli 10/51
21 We will study the differential geometry of CY moduli. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 11/51
22 We will study the differential geometry of CY moduli. 1 local properties Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 11/51
23 We will study the differential geometry of CY moduli. 1 local properties 2 semilocal (semiglobal) properties Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 11/51
24 We will study the differential geometry of CY moduli. 1 local properties 2 semilocal (semiglobal) properties 3 global properties Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 11/51
25 A very quick review of KodairaSpencer 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 CalabiYau Moduli 12/51
26 A very quick review of KodairaSpencer 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 CalabiYau Moduli 12/51
27 A very quick review of KodairaSpencer 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 CalabiYau Moduli 12/51
28 A very quick review of KodairaSpencer theory2 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 CalabiYau Moduli 13/51
29 A very quick review of KodairaSpencer theory2 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 KodairaSpencer map is defined as ε ϕ Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 13/51
30 The integrability condition can be written as ( ϕ(t)) 2 = 0. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 14/51
31 The integrability condition can be written as ( ϕ(t)) 2 = 0. or using the notation of superlie 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 CalabiYau Moduli 14/51
32 Sufficient conditions for the existence of smooth Kuranishi spaces Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau manifold Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau 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 quasiprojective. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 quasiprojective. 2 Unlike the moduli space of curves, the compactification of CY moduli is not modular. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 quasiprojective. 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 CalabiYau Moduli 16/51
40 WeilPetersson metric The metric is an L 2 metric, defined by the following Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 17/51
41 WeilPetersson metric The metric is an L 2 metric, defined by the following Definition Let Z be a polarized CalabiYau 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 CalabiYau manifold, via the KodairaSpencer 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 WeilPetersson metric of M. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 17/51 Z
42 The differential geometry of the WeilPetersson metric on CalabiYau moduli is called the WeilPetersson geometry. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 18/51
43 The differential geometry of the WeilPetersson metric on CalabiYau moduli is called the WeilPetersson 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 CalabiYau 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 CalabiYau 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 WeilPetersson metric can be expressed explicitly in terms of the variation of Hodge structures. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 (BryantGriffiths) 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 CalabiYau 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 (BryantGriffiths) 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 CalabiYau 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 (BryantGriffiths) 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 skewsymmetric and can be written as ( ) 1 Q = 1 Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau Moduli 24/51
56 Local WeilPetersson geometry Good properties Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 25/51
57 Local WeilPetersson geometry Good properties Kählerian Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 25/51
58 Local WeilPetersson geometry Good properties Kählerian Formula of Strominger Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 25/51
59 Local WeilPetersson geometry Good properties Kählerian Formula of Strominger Many others Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 25/51
60 Local WeilPetersson geometry Good properties Kählerian Formula of Strominger Many others Bad property Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 25/51
61 Local WeilPetersson 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 CalabiYau 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 3fold 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 CalabiYau 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 3fold 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 CalabiYau 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 3fold 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 (nonpositive bisectional curvature, negative Ricci and holomorphic sectional curvature, etc). Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 26/51
65 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
66 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Semiglobal theory Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
67 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Semiglobal theory The BCOV conjecture (settled by Zinger07 Hao FangLYoshikawa 06) Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
68 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Semiglobal theory The BCOV conjecture (settled by Zinger07 Hao FangLYoshikawa 06) Incompleteness of the WeilPetersson metric (96 for 1d, L12 in preparation) Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
69 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Semiglobal theory The BCOV conjecture (settled by Zinger07 Hao FangLYoshikawa 06) Incompleteness of the WeilPetersson metric (96 for 1d, L12 in preparation) A lot of others Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
70 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Semiglobal theory The BCOV conjecture (settled by Zinger07 Hao FangLYoshikawa 06) Incompleteness of the WeilPetersson metric (96 for 1d, L12 in preparation) A lot of others Global theory Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
71 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Semiglobal theory The BCOV conjecture (settled by Zinger07 Hao FangLYoshikawa 06) Incompleteness of the WeilPetersson metric (96 for 1d, L12 in preparation) A lot of others Global theory Freed Conjecture (L99) Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
72 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Semiglobal theory The BCOV conjecture (settled by Zinger07 Hao FangLYoshikawa 06) Incompleteness of the WeilPetersson metric (96 for 1d, L12 in preparation) A lot of others Global theory Freed Conjecture (L99) GaussBonnet Theorem (Joint/w M. R. Douglas) Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
73 We want to study the differential geometric properties of the moduli space with respect to the WeilPetersson metric. Local theory Semiglobal theory The BCOV conjecture (settled by Zinger07 Hao FangLYoshikawa 06) Incompleteness of the WeilPetersson metric (96 for 1d, L12 in preparation) A lot of others Global theory Freed Conjecture (L99) GaussBonnet Theorem (Joint/w M. R. Douglas) Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 27/51
74 The BCOV Conjecture BCOV=BershadskyCecottiOoguriVafa Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 28/51
75 The BCOV Conjecture BCOV=BershadskyCecottiOoguriVafa 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 CalabiYau Moduli 28/51
76 Define the multivalued 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 CalabiYau Moduli 29/51
77 Conjecture (A) Conjecture (A) Let n g (d) be the genusg degreed 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 CalabiYau Moduli 30/51
78 Conjecture (A) was proved by Aleksey Zinger. Aleksey Zinger The Reduced GenusOne GromovWitten Invariants of CalabiYau Hypersurfaces ArXiv: v2, JAMS 08 Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 31/51
79 Setup of Conjecture (B) Let X be a compact Kähler manifold. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 welldefined? Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 welldefined? ζ function regularization (for example: Riemann ζfunction) Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 32/51
85 Setup of Conjecture B BershadskyCeccottiOoguriVafa defined T def = (det p,q ) ( 1)p+qpq. p,q Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 33/51
86 Setup of Conjecture B BershadskyCeccottiOoguriVafa 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 CalabiYau 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 WeilPetersson 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 CalabiYau Moduli 34/51
88 Conjecture B was proved by FangLYoshikawa. Hao FangLYoshikawa Asymptotic behavior of the BCOV torsion of CalabiYau moduli ArXiv: JDG (80), 2008, , Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau Moduli 36/51
90 Reduction of the problem Theorem (FangL) On the moduli space of a primitive CY 3fold, 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 CalabiYau Moduli 37/51
91 Reduction of the problem Theorem (FangL) On the moduli space of a primitive CY 3fold, 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 CalabiYau Moduli 37/51
92 Corollary If X is a primitive CalabiYau, N M is a kdimensional 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 CalabiYau Moduli 38/51
93 Corollary Assume that a polarized CalabiYau 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 CalabiYau Moduli 39/51
94 Corollary Assume that a polarized CalabiYau 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 CalabiYau Moduli 39/51
95 Corollary Assume that a polarized CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 3fold 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau Moduli 41/51
102 (FangL) Using Song s result, a direct method of computing the BCOV torsion? Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 CalabiYau 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 ChernWeil form with respect to the polynomial f is defined by f(r). Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 ChernWeil form; Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 ChernWeil form; 2 [f(r)] H (M, Z), and is independent of the choice of the connection. GaussBonnetChern Theorem Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 45/51
110 This is joint with Michael R. Douglas Zhiqin Lu and Michael R. Douglas GaussBonnetChern theorem on moduli spaces arxiv: Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 46/51
111 This is joint with Michael R. Douglas Zhiqin Lu and Michael R. Douglas GaussBonnetChern theorem on moduli spaces arxiv: Theorem Let f be an invariant polynomial with rational coefficients. Let R be the curvature tensor with respect the WeilPetersson metric. Then f(r) is a rational number. M Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau 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 CalabiYau Moduli 47/51
114 Some remarks The moduli space is often noncompact. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 48/51
115 Some remarks The moduli space is often noncompact. If the moduli space were compact, then the ChernWeil forms define Chern classes, and the theorem follows. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 48/51
116 Some remarks The moduli space is often noncompact. If the moduli space were compact, then the ChernWeil forms define Chern classes, and the theorem follows. If the growth of the ChernWeil forms and the connection were mild at infinity, the theorem follows from a theorem of Mumford. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau 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 CalabiYau Moduli 49/51
118 The above result implies that the ChernWeil forms are integrable. But it is not that hard to prove the integrability condition directly. Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 50/51
119 The above result implies that the ChernWeil forms are integrable. But it is not that hard to prove the integrability condition directly. We first establish the following wellknown 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 CalabiYau Moduli 50/51
120 The above result implies that the ChernWeil forms are integrable. But it is not that hard to prove the integrability condition directly. We first establish the following wellknown 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 CalabiYau Moduli 50/51
121 The above result implies that the ChernWeil forms are integrable. But it is not that hard to prove the integrability condition directly. We first establish the following wellknown 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 CalabiYau Moduli 50/51
122 The above result implies that the ChernWeil forms are integrable. But it is not that hard to prove the integrability condition directly. We first establish the following wellknown 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 CalabiYau Moduli 50/51
123 Thank you! Zhiqin Lu, Dept. Math, UCI Geometry of the CalabiYau Moduli 51/51
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