On the approximation of positive closed currents on compact Kaehler manifolds

TL;DR

This paper develops methods to approximate positive closed currents on compact Kähler manifolds using zeros of holomorphic sections, and studies their convergence and distribution properties, with applications to complex geometry.

Contribution

It introduces new approximation techniques for positive closed currents via zeros of high tensor powers of line bundles with singular metrics on Kähler manifolds.

Findings

Approximation of wedge powers of curvature currents by zero sets of sections

Convergence results for Fubini-Study currents

Equidistribution of zeros of sections of adjoint bundles

Abstract

Let $L$ be a holomorphic line bundle over a compact K\"ahler manifold $X$ endowed with a singular Hermitian metric $h$ with curvature current $c_1(L,h)\geq0$. In certain cases when the wedge product $c_1(L,h)^k$ is a well defined current for some positive integer $k\leq\dim X$, we prove that $c_1(L,h)^k$ can be approximated by averages of currents of integration over the common zero sets of $k$-tuples of holomorphic sections over $X$ of the high powers $L^p:=L^{\otimes p}$. In the second part of the paper we study the convergence of the Fubini-Study currents and the equidistribution of zeros of $L^2$-holomorphic sections of the adjoint bundles $L^p\otimes K_X$, where $L$ is a holomorphic line bundle over a complex manifold $X$ endowed with a singular Hermitian metric $h$ with positive curvature current. As an application, we obtain an approximation theorem for the current $c_1(L,h)^k$ using currents of integration over the common zero sets of $k$-tuples of sections of $L^p\otimes K_X$.