On the approximation of positive closed currents on compact Kaehler manifolds

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 $L^p\otimes K_X$.