Linear invariants of complex manifolds and their plurisubharmonic variations

Abstract

For a bounded domain D and a real number p>0, denote by Ap(D) the linear space of Lp integrable holomorphic functions on D, equipped with the Lp-pseudonorm. We prove that two bounded hyperconvex domains D1⊂Cn and D2⊂Cm are biholomorphic (in particular n=m) if there is a linear isometry between Ap(D1) and Ap(D2) for some 0<p<2. The same result holds for p>2,p≠2,4,⋯, provided that the p-Bergman kernels on D1 and D2 are exhaustive. With this as a motivation, we show that, for all p>0, the p-Bergman kernel on a strongly pseudoconvex domain with C2 boundary or a simply connected homogeneous regular domain is exhaustive. These results show that spaces of pluricanonical sections of complex manifolds equipped with canonical pseudonorms are important linear 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 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.