A converse of Hörmander’s L2-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.

The harmonic theory on a vector bundle with singular Hermitian metrics and positivity

“…H,ω is integrable in some neighbourhood of y} for a given point y ∈ Y . Moreover, it is proved in [4,24] that E(H) is coherent [22] when (E, H) possesses certain positivity. We also prove the coherence in a general setting here.…”

“…is integrable in some neighbourhood of y} for a given point y ∈ Y . Although it has already been proved in [4,24] that E(H) is coherent [22] in several situations, we would like to present a general version that describes the coherence in Proposition 2.3. In particular, E(H) will always be coherent in our context in the view of Proposition 2.3.…”

Curvature operator of holomorphic vector bundles and $L^2$-estimate condition for $(n,q)$ and $(p,n)$-forms

“…The aim of the present paper is to study the relation between the positivity properties of curvature operators for holomorphic Hermitian vector bundles and L 2 -estimates by using the (p, q)-L 2 -estimate condition (see Definition 1.3). This type of condition was firstly introduced in [HI20], which was named as the twisted Hörmander condition. After that, in a paper [DNWZ20], Deng et al generalized this concept as the optimal L 2 -estimate condition, which is a special case of the twisted Hörmander condition when p = 2, and gave a new characterizations of Nakano semi-positivity by L 2 -estimate.…”

From Hörmander’s L 2 -estimates to partial positivity

“…Taking an arbitrary weight, we can estimate the curvature −1Θ (L,h) . This type of condition was firstly introduced in [11], which was called the twisted Hörmander condition. Next, in [7] and [8], Deng et al generalized this notion and introduced the optimal L p -estimate condition, which corresponded to the particular case of the twisted Hörmander condition when p = 2.…”