Gauss-bonnet-chern Theorem on Moduli Space Zhiqin Lu and Michael R. Douglas

The moduli space of complex structures of a polarized Kähler manifold is of fundamental interest to algebraic geometers and to string theorists. The study of its geometry is greatly enriched by considering the Hodge structure of the underlying manifolds. The moduli space at infinity is particularly interesting because it is related to the degeneration of Kähler manifolds. Even when we know very little about the degeneration itself, by the works of Schmid [27], Steenbrink [28], Cattani-Kaplan-Schmid [4], and many others, we have a good understanding of the degeneration of the corresponding Hodge structures. The purposes of this paper are twofold: first, based on the work of [27, 28, 4], we prove a Gauss-Bonnet-Chern type theorem in full generality for the Chern-Weil forms of Hodge bundles. That is, the Chern-Weil forms compute the corresponding Chern classes. This settles a long standing problem. Second, we apply the result to Calabi-Yau moduli, and proved the corresponding Gauss-Bonnet-Chern type theorem in the setting of Weil-Petersson geometry. As an application of our results in string theory, we prove that the number of flux vacua of type II string compactified on a Calabi-Yau