arXiv subjects
Publications and source records attributed to Bo Berndtsson.
We give an elementary proof of the existence of an asymptotic expansion in powers of $k$ of the Bergman kernel associated to $L^k$, where $L$ is a positive line bundle. We also give an algorithm for computing the coefficients in the expansion.
Let $L$ be a (semi)-positive line bundle over a Kahler manifold, $X$, fibered over a complex manifold $Y$. Assuming the fibers are compact and non-singular we prove that the hermitian vector bundle $E$ over $Y$ whose fibers over points $y$ are the spaces of global sections over $X_y$ to $L\gr K_{X/Y}$ endowed with the $L^2$-metric is (semi)-positive in the sense of Nakano. We also discuss various applications, among them a partial result on a conjecture of Griffiths on the positivity of ample bundles. This is a revised and much expanded version of a previous preprint with the title `` Bergman kernels and the curvature of vector bundles''.
Let $D$ be a pseudoconvex domain in $\C^k_t\times\Cn_z$ and let $ϕ$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z; (t,z)\in D\}$, let $ϕ^t$ be the restriction of $ϕ$ to $D_t$ and denote by $K_t(z,ζ)$ the Bergman kernel of $D_t$ with the weight function $ϕ^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $ϕ=0$) we prove that $\log K_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting of $\Rn$.
In a previous paper, \cite{Berndtsson}, we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted $L^2$-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of $L^2$-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sections to line bundles over a locally trivial fibration we also prove that if a holomorphic vector bundle, $V$, over a complex manifold is ample in the sense of Hartshorne, then $V\gr\det V$ has an Hermitian metric with curvature strictly positive in the sense of Nakano.