Two problems in Kahler manifolds of non-positive curvature

Yu-Lin Chang, Purdue University

Abstract

This thesis contains three parts. They are related under the same assumption that all manifolds we consider have a metric of non-positive curvature. In fact two conjectures are studied. The role of locally hermitian symmetric spaces is important. They are used to test whether the conjectures hold or not. ^ In the first part of this thesis we consider a non-vanishing conjecture of Kawamata. We treat this problem in an analytic way. By using Atiyah's L² Index Theorem, we can pass the difficulty to the universal covering X of M. We show that the conjecture holds for any compact Kähler manifold of non-positive sectional curvature. ^ The second part studies the behavior of p(m ) = dimC H⁰(M, K + mL) when m is small. We show that p( m) is increasing for m ≥ 1 provided M is a compact quotient of an irreducible bounded symmetric domain of rank ≥ 2. ^ In the third part we consider a dual problem of Frankel's conjecture. By imposing a topology condition, we settle the problem for dimension 2.^

Degree

Ph.D.

Advisors

Sai Kee Yeung, Purdue University.

Subject Area

Engineering, Electronics and Electrical

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