UCLA

In this thesis, we consider geometric properties of vector bundles

arising from algebraic and Hermitian geometry.

On vector bundles in algebraic geometry, such as ample, nef and globally generated vector

bundles, we are able to construct positive Hermitian metrics in

different senses(e.g. Griffiths-positive, Nakano-positive and

dual-Nakano-positive) by $L^2$-method and deduce many new vanishing

theorems for them by analytic method instead of the Le Potier-Leray

spectral sequence method.

On Hermitian manifolds, we find that the second Ricci curvature

tensors of various metric connections are closely related to the

geometry of Hermitian manifolds. We can derive various vanishing

theorems for Hermitian manifolds and also for complex vector bundles

over Hermitian manifolds by their second Ricci curvature tensors. We

also introduce a natural geometric flow on Hermitian manifolds by

using the second Ricci curvature tensor.