English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a bounded domain $D$ and a real number $p>0$, we denote by $A^p(D)$ thespace 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 thereis a linear isometry between $A^p(D_1)$ and $A^p(D_2)$ for some $0<p<2$. Thesame result holds for $p>2, p\neq 2,4,\cdots$, provided that the $p$-Bergmankernels on $D_1$ and $D_2$ are exhaustive. With this as a motivation, we showthat, for all $p>0$, the $p$-Bergman kernel on a strongly pseudoconvex domainwith $\mathcal C^2$ boundary or a simply connected homogeneous regular domainis exhaustive. These results shows that spaces of pluricanonical sections ofcomplex manifolds equipped with canonical pseudonorms are important invariantsof complex manifolds. The second part of the present work devotes to studyingvariations of these invariants. We show that the direct image sheaf of thetwisted relative $m$-pluricanonical bundle associated to a holomorphic familyof Stein manifolds or compact K\"ahler manifolds is positively curved, withrespect to the canonical singular Finsler metric.

Linear invariants of complex manifolds and their plurisubharmonic variations