A pullback operation on a class of currents

Abstract

For any holomorphic map $f\colon X\to Y$ between a complex manifold $X$ and a complex Hermitian manifold $Y$ we extend the pullback $f^*$ from smooth forms to a class of currents in a cohomologically sound way. We provide a basic calculus for this pullback. The class of currents we consider contains in particular the Lelong current of any analytic cycle. Our pullback depends in general on the Hermitian structure of $Y$ but coincides with the usual pullback of currents in case $f$ is a submersion. The construction is based on the Gysin mapping in algebraic geometry.Comment: Theorem 1.2 is improve