Positivity of holomorphic vector bundles in terms of $L^p$-conditions of $\bar{\partial }$

Abstract

We study the positivity properties of Hermitian (or even Finsler) holomorphic vector bundles in terms of $L^p$-estimates of $\bar{\partial }$ and $L^p$-extensions of holomorphic objects. To this end, we introduce four conditions, called the optimal $L^p$-estimate condition, the multiple coarse $L^p$-estimate condition, the optimal $L^p$-extension condition, and the multiple coarse $L^p$-extension condition, for a Hermitian (or Finsler) vector bundle $(E,h)$. The main result of the present paper is to give a characterization of the Nakano positivity of $(E,h)$ via the optimal $L^2$-estimate condition. We also show that $(E,h)$ is Griffiths positive if it satisfies the multiple coarse $L^p$-estimate condition for some $p>1$, the optimal $L^p$-extension condition, or the multiple coarse $L^p$-extension condition for some $p>0$. These results can be roughly viewed as converses of Hörmander's $L^2$-estimate of $\bar{\partial }$ and Ohsawa-Takegoshi type extension theorems. As an application of the main result, we get a totally different method to Nakano positivity of direct image sheaves of twisted relative canonical bundles associated to holomorphic families of complex manifolds.

30 pages, comments and suggestions are welcome

References in corpus (2)

• A converse of Hörmander's $L^2$-estimate and new positivity notions for vector bundles
• Linear invariants of complex manifolds and their plurisubharmonic variations

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• Nakano positivity of singular Hermitian metrics and vanishing theorems of Demailly-Nadel-Nakano type
• Curvature positivity of invariant direct images of Hermitian vector bundles
• Characterization of Curvature positivity of Riemannian metrics on flat vector bundles
• Nakano Positivity of direct image sheaves of adjoint line bundles with mild singularities
• Optimal $L^2$-extensions on tube domains and a simple proof of Prékopa's theorem