Positivity and vanishing theorems in complex and algebraic geometry

In this thesis, we consider geometric properties of vector bundlesarising from algebraic and Hermitian geometry. On vector bundles in algebraic geometry, such as ample, nef and globally generated vectorbundles, we are able to construct positive Hermitian metrics indifferent senses(e.g. Griffiths-positive, Nakano-positive anddual-Nakano-positive) by $L^2$-method and deduce many new vanishingtheorems for them by analytic method instead of the Le Potier-Lerayspectral sequence method.On Hermitian manifolds, we find that the second Ricci curvaturetensors of various metric connections are closely related to thegeometry of Hermitian manifolds. We can derive various vanishingtheorems for Hermitian manifolds and also for complex vector bundlesover Hermitian manifolds by their second Ricci curvature tensors. Wealso introduce a natural geometric flow on Hermitian manifolds byusing the second Ricci curvature tensor