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 $0 2, p\neq 2,4,\cdots$, 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 K\"ahler manifolds is positively curved, with respect to the canonical singular Finsler metric.

中文翻译：

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复流形的线性不变量及其多次谐波变化

对于有界域 $D$ 和实数 $p>0$，我们用 $A^p(D)$ 表示 $D$ 上 $L^p$ 可积全纯函数的空间，配备 $L^ p$- 伪范数。我们证明了两个有界超凸域 $D_1\subset\mc^n$ 和 $D_2\subset\mc^m$ 是双全纯的（特别是 $n=m$），如果 $A^p(D_1 )$ 和 $A^p(D_2)$ 为一些 $0 2, p\neq 2,4,\cdots$，前提是 $D_1$ 和 $D_2$ 上的 $p$-Bergman 内核是详尽的。以此为动机，我们证明，对于所有 $p>0$，具有 $\mathcal C^2$ 边界的强伪凸域或简单连接的齐次正则域上的 $p$-Bergman 核是详尽的。这些结果表明，配备规范伪范的复杂流形的多经典截面空间是复杂流形的重要不变量。本工作的第二部分致力于研究这些不变量的变化。我们证明了与 Stein 流形的全纯族或紧致的 K\"ahler 流形相关的扭曲相对 $m$-pluricanonical 丛的直接像层相对于规范奇异 Finsler 度量是正弯曲的。