  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 $p>2, p\neq 2,4,\cdots$, 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.
