Positive vector bundles in complex and convex geometry

This thesis concerns various aspects of the geometry of holomorphic vector bundles and their analytical theory which all, vaguely speaking, are related to the notion of positive curvature in general, and L^2-methods for the dbar-equation in particular. The thesis contains four papers.In Paper I we introduce and study the notion of singular hermitian metrics on holomorphic vector bundles. We define what it means for such metrics to be positively curved in the sense of Griffiths, and investigate the assumptions needed in order to define the curvature tensor of such metrics as currents with measure coefficients. We also investigate the regularisation of such metrics.In Paper II we prove the Nakano vanishing theorem with H\uf6rmander L^2-estimates on a compact K\ue4hler manifold using Siu\u27s d-dbar-Bochner-Kodaira method. We then introduce the singular hermitian metrics and regularisation results of Paper I, and use these to prove a Demailly-Nadel type of vanishing theorem for vector bundles over Riemann surfaces.A fundamental tool in complex geometry closely related to the notion of positivity is the Ohsawa-Takegoshi extension theorem. In Paper III the d-dbar-Bochner-Kodaira method is applied to extend this theorem from line bundles to vector bundles over compact K\ue4hler manifolds. Another way of obtaining a vector bundle version of this theorem is to reduce it to the line bundle setting through the useful algebraic geometric procedure of studying the projective bundle associated with the vector bundle. In Paper III we also investigate the relationship between these two different approaches.On a trivial line bundle, a positively curved metric is the complex-analytic counterpart of a log concave function in the real-variable setting. In Paper IV we extend this link between complex and convex geometry to trivial vector bundles. We define two new notions of log concavity for real, matrix-valued functions, corresponding to Griffiths and Nakano positivity, and we prove a matrix-valued Prekopa theorem