A converse of Hörmander's $L^2$-estimate and new positivity notions for vector bundles

Abstract

We study conditions of Hörmander's $L^2$-estimate and the Ohsawa-Takegoshi extension theorem. Introducing a twisted version of Hörmander-type condition, we show a converse of Hörmander $L^2$-estimate under some regularity assumptions on an $n$-dimensional domain. This result is a partial generalization of the 1-dimensional result obtained by Berndtsson. We also define new positivity notions for vector bundles with singular Hermitian metrics by using these conditions. We investigate these positivity notions and compare them with classical positivity notions.

15 pages

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