$L^2$-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics

Yuta Watanabe

complex geometry

$L^{2}$ -Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics

arXiv:2602.03332

summary

The paper develops an $L^{2}$ -theory for the $ar\partial$-operator on complex spaces with singular Hermitian metrics, providing $L^{2}$ -Dolbeault resolutions and estimates, and uses these to extend Nadel vanishing results.

Abstract

In this paper, in order to develop a more general $L^{2}$ -theory for the $\partial$ -operator on complex spaces, we provide $L^{2}$ -Dolbeault fine resolutions and isomorphisms, and $L^{2}$ -estimates, for holomorphic line bundles on complex spaces equipped with singular Hermitian metrics. As applications, we obtain several generalizations of the Nadel vanishing theorem.

v2: 22 pages. Mainly revised the introduction and Section 4 in light of prior work, and removed the former Section 4. v1: 24 pages

Topics & keywords

#l2 theory #dolbeault cohomology #singular hermitian metrics #weakly pseudoconvex spaces #nadel vanishing$L^2$-Dolbeault resolutionsingular Hermitian metricweakly pseudoconvex$\overline{\partial}$-operatorNadel vanishing theorem

L2-Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics

Abstract

Topics & keywords

$L^{2}$ -Dolbeault resolutions and Nadel vanishing on weakly pseudoconvex complex spaces with singular Hermitian metrics