Linear invariants of complex manifolds and their plurisubharmonic variations

Abstract

For a bounded domain $D$ and a real number $p>0$, we denote by $A^p(D)$ the space of $L^p$ integrable holomorphic functions on $D$, equipped with the $L^p$- pseudonorm. We prove that two bounded hyperconvex domains $D_1\subset \mc^n$ and $D_2\subset \mc^m$ are biholomorphic (in particular $n=m$) if there is a linear isometry between $A^p(D_1)$ and $A^p(D_2)$ for some $0<p<2$. The same result holds for $p>2,p\ne 2,4,\cdots$, provided that the $p$-Bergman kernels on $D_1$ and $D_2$ are exhaustive. With this as a motivation, we show that, for all $p>0$, the $p$-Bergman kernel on a strongly pseudoconvex domain with ${\mathcal{C}}^2$ boundary or a simply connected homogeneous regular domain is exhaustive. These results shows that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important invariants of complex manifolds. The second part of the present work devotes to studying variations of these invariants. We show that the direct image sheaf of the twisted relative $m$-pluricanonical bundle associated to a holomorphic family of Stein manifolds or compact Kähler manifolds is positively curved, with respect to the canonical singular Finsler metric.

Comments and suggestions are welcome. arXiv admin note: text overlap with arXiv:1809.10371

References in corpus (1)

• A new geometric approach to problems in birational geometry

Cited by in corpus (1)

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