Article

# On the Curvature Tensor of the Hodge Metric of Moduli Space of Polarized Calabi-Yau Threefolds

Journal of Geometric Analysis (Impact Factor: 0.86). 06/2005; DOI: 10.1007/BF02930760

Source: arXiv

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**ABSTRACT:**This is the first of a series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi Yau metrics acting on (0,q) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in case of the elliptic curves the Kronecker limit formula gives an explicit formula for the regularized determinants of the flat metrics with fixed volume on the elliptic curves. The following formula holds in this case; the regularized determinant is the product of the imaginary part of the complex number in the Siegel upper half plane with the Dedekind eta function. It is well known fact that the Dedekind eta function in power 24 is a cusp automorphic form of weight 12 related to the discriminant of the elliptic curve. Thus we can view that the regularized determinant is the norm of a section of some power of the line bundle of the classes of cohomologies of (1,0) forms of the elliptic curves over its moduli space. Our purpose is to generalize this fact in the case of CY manifolds. In this paper we will establish the local analogue of the Kronecker limit formula for CY manifolds.04/2005; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we proved the Gauss-Bonnet-Chern theorem on moduli space of polarized Kahler manifolds. Using our results, we proved the rationality of the Chern-Weil forms (with respect to the Weil-Petersson metric) on CY moduli. As an application in physics, by the Ashok-Douglas theory, counting the number of flux compactifications of the type IIb string on a Calabi-Yau threefold is related to the integrations of various Chern-Weil forms. We proved that all these integrals are finite (and also rational).03/2009; - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, we study the Chern classes on the moduli space of polarized Calabi-Yau manifolds. We prove that the integrations of the invariants of the curvature of the Weil-Petersson metric are finite. In some special cases, they are even rational numbers.04/2006;

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