For a one parameter family of Calabi-Yau threefolds, Green et al. (2009) expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV (Bershadsky-Cecotti-Ooguri-Vafa) invariants, studied by Fang et al. (2008). On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.

Due to the orbifold singularities, the intersection numbers on the moduli space of curves Mg, n are in general rational numbers rather than integers. We study the properties of the denominators of these intersection numbers and their relationship with the orders of automorphism groups of stable curves. We also present a conjecture about a multinomial type numerical property for a general class of Hodge integrals.

We consider a measurable matrix-valued cocycle A: Z+ X Rd d, driven by a measurepreserving transformation T of a probability space (X,, ), with the integrability condition log+ A (1,) L1 (). We show that for -ae x X, if limn 1 n log A (n, x) v= 0 for all v Rd\{0}, then the trajectory {A (n, x) v} n= 0 is far away from 0 (ie lim supn A (n, x) v> 0) and there is some nonzero v such that lim supn A (n, x) v v. This improves the classical multiplicative ergodic theorem of Oseledec. We here present an application to linear random processes to illustrate the importance.

We give a short proof of Penner-Grushevsky-Schumacher-Trapanis large genus asymptotics of Weil-Petersson volumes of moduli spaces of curves. We also study asymptotic expansions for certain integrals of pure \psi classes and answer a question of Mirzakhani on the asymptotic behavior of one-point volume polynomials of moduli spaces of curves.

We define the quantum correction of the Teichmller space \mathcal {T} of Calabi-Yau manifolds. Under the assumption of no weak quantum correction, we prove that the Teichmller space \mathcal {T} is a locally symmetric space with the Weil-Petersson metric. For Calabi-Yau threefolds, we show that no strong quantum correction is equivalent to that, with the Hodge metric, the image \mathcal {T} of the Teichmller space \mathcal {T} under the period map \mathcal {T} is an open submanifold of a globally Hermitian symmetric space \mathcal {T} of the same dimension as \mathcal {T} . Finally, for Hyperkhler manifold of dimension \mathcal {T} , we find both locally and globally defined families of \mathcal {T} and \mathcal {T} -classes over the Teichmller space of polarized Hyperkhler manifolds.