Gauss-Bonnet theorem on moduli spaces

In this talk, I will first give a survey of the Weil-Petersson geometry on Calabi-Yau moduli. Then I will show the recent result of the Gauss-Bonnet-Chern theorem on moduli space of polarized Kahler manifolds. 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. I will show that all these integrals are finite (and also rational). The second part of the talk is based on joint work with Michael R. Douglas.