Positivity and vanishing theorems in complex and algebraic geometry

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 L2-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.