Preprints, Working Papers, ... Year : 2021

Monge-Ampère functionals for the curvature tensor of a holomorphic vector bundle

Jean-Pierre Demailly

Abstract

Let E be a holomorphic vector bundle on a projective manifold X such that det E is ample. We introduce three functionals Φ_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-Ampère integrals of Φ_P (Θ_{E,h}), where Θ_{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 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.
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Dates and versions

hal-03500059 , version 1 (23-12-2021)
hal-03500059 , version 2 (02-02-2022)

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Jean-Pierre Demailly. Monge-Ampère functionals for the curvature tensor of a holomorphic vector bundle. 2021. ⟨hal-03500059v2⟩
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