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

DOI10.1007/S10476-022-0156-4zbMATH Open1513.32034arXiv2112.14463WikidataQ114227593 ScholiaQ114227593MaRDI QIDQ6386945

Jean-Pierre Demailly

Publication date: 29 December 2021

Abstract: Let $E$ be a holomorphic vector bundle on a projective manifold $X$ such that $detE$ is ample. We introduce three functionals $Ph{i}_P$ 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 $Ph{i}_P(Thet{a}_{E,h})$, where $Thet{a}_{E,h}$ is the Chern curvature tensor of $(E,h)$. These volumes are shown to satisfy optimal Chern class inequalities. We also prove that the functionals $Ph{i}_P$ 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 $Eotimes(detE{)}^t$, where $t>-1/mrankE$, can possibly be investigated by studying the infimum of exponents $t$ for which the Yang-Mills differential system has a solution.

Mathematics Subject Classification ID

Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) (53C07) Differential complexes (58J10) Transcendental methods of algebraic geometry (complex-analytic aspects) (32J25) Complex Monge-Ampère operators (32W20)

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