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• W2911704172 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 $D_1subset mc^n$ and $D_2subset mc^m$ 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 $p>2, pneq 2,4,cdots$, 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 Kahler manifolds is positively curved, with respect to the canonical singular Finsler metric." @default.
• W2911704172 created "2019-02-21" @default.
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• W2911704172 date "2019-01-24" @default.
• W2911704172 modified "2023-09-23" @default.
• W2911704172 title "Linear invariants of complex manifolds and their plurisubharmonic variations" @default.
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• W2911704172 doi "https://doi.org/10.48550/arxiv.1901.08920" @default.
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