Existence and implications of positively curved metrics on holomorphic vector bundles

Abstract

This thesis is divided into two parts. In the first part, we study interpolating and uniformly flat hypersurfaces in complex Euclidean space. The study of interpolation and sampling in the Bargmann-Fock spaces on the complex plane started with the work of K. Seip in 1992. In several papers, Seip and his collaborators have entirely characterised the interpolating and sampling sequences for the Bargmann-Fock spaces on the complex plane. This problem has been studied for the Bargmann-Fock spaces on the higher dimensional complex Euclidean spaces. Very few results on interpolating and sampling hypersurfaces in higher dimensions are known. We have proven certain hypersurfaces are not interpolating. Ortega-Cerd\'{a}, Schuster and Varolin have defined uniformly flat smooth hypersurfaces and proved that uniform flatness is one of the sufficient conditions for smooth hypersurfaces to be interpolating and sampling in higher dimensions. We have studied uniformly flat hypersurfaces in dimensions greater than or equal to two and proved a complete characterization of them. In dimension two, we gave a sufficient condition for a smooth algebraic hypersurface to be interpolating in terms of its projectivization. The second part deals with the existence of a Griffiths positively curved metric on the Vortex bundle. Given a Hermitian holomorphic vector bundle of arbitrary rank on a projective manifold, we have the notions of Nakano positivity, Griffiths positivity, and ampleness. All these notions of positivity are equivalent for line bundles. In general, Griffiths positivity implies ampleness. A conjecture due to Griffiths says that ampleness implies Griffiths positivity. To prove the equivalence between Griffiths positivity and ampleness, Demailly designed several systems of equations of Hermitian-Yang-Mills type for the curvature tensor. We have studied these systems on the Vortex bundle.