Date

2014

Document Type

Dissertation

Degree

Doctor of Philosophy

Department

Mathematics

First Adviser

Cao, Huai-Dong

Other advisers/committee members

Napier, Terrence; Corvino, Justin

Abstract

In this thesis, two topics will be studied. In the first part, we investigate the geometric quantization of the Weil-Petersson metric on the moduli space of Fano Kaehler-Einstein manifolds. In the second part, we investigate the (weak) pseudo-convexity of the Teichmuller space of Kaehler-Einstein manifolds of general type. In Chapter 1, we review the (infinitesimal) deformation theory of complex structures on compact complex manifolds. Based on Hodge theory, the existence of (infinitesimal) deformations will be discussed in detail. In Chapter 2, we explore the deformation theory of complex structures on compact Fano Kaehler-Einstein manifolds with respect to the Kuranishi-divergence gauge. We also give the construction of local canonical sections of the relative tangent bundle. Based on these works, we show that the Weil-Petersson metric can be approximated by the curvatures of the natural L^2 metrics on the direct image of the tensor powers of relative anti-canonical bundles after normalization. In Chapter 3, we look at the Teichmuller space T of Kaehler-Einstein manifolds of general type whose complex structure is unobstructed. Let N be a Riemannian manifold with nonpositive sectional curvature. We prove that the harmonic energy from T to N is pluri-subharmonic.

Included in

Mathematics Commons

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