Monge-Amp\`ere functionals for the curvature tensor of a holomorphic vector bundle

TL;DR

This paper introduces new Monge-Ampère functionals for holomorphic vector bundles on projective manifolds, leading to novel volume concepts, inequalities, and differential systems related to bundle positivity and curvature.

Contribution

It defines three new functionals associated with different positivity notions, establishing volume measures, optimal inequalities, and elliptic systems for curvature analysis.

Findings

Established volume bounds satisfying Chern class inequalities.

Linked positivity thresholds to solutions of Hermitian-Yang-Mills type systems.

Proposed new tools for studying bundle positivity and curvature properties.

Abstract

Let $E$ be a holomorphic vector bundle on a projective manifold $X$ such that $\det E$ is ample. We introduce three functionals $\Phi_P$ 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-Amp\`ere integrals of $\Phi_P(\Theta_{E,h})$, where $\Theta_{E,h}$ is the Chern curvature tensor of $(E,h)$. These volumes are shown to satisfy optimal Chern class inequalities. We also prove that the functionals $\Phi_P$ 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 $E\otimes(\det E)^t$, where $t>-1/{\rm rank} E$, can possibly be investigated by studying the infimum of exponents $t$ for which the Yang-Mills differential system has a solution.