Monge—Ampère Functionals for the Curvature Tensor of a Holomorphic Vector Bundle

Abstract

Let E be a holomorphic vector bundle on a projective manifold X such that det E is ample. We introduce three functionals ΦP related to Griffiths, Nakano and dual Nakano positivity respectively. They can be used to define new concepts of volume for the vector bundle E, by means of generalized Monge—Ampère integrals of ΦP(ΘE,h), where ΘE,h is the Chern curvature tensor of (E,h). These volumes are shown to satisfy optimal Chern class inequalities. We also prove that the functionals ΦP give rise in a natural way to elliptic differential systems of Hermitian-Yang—Mills type for the curvature, in such a way that the related P-positivity threshold of E ⊗ (det E)t, where t> −1/rank E, can possibly be investigated by studying the infimum of exponents t for which the Yang—Mills differential system has a solution.