A pullback operation on a class of currents

For any holomorphic mapping $f\text{\hspace{0.17em}⁣}:X\to Y$ between a complex manifold $X$ and a complex Hermitian manifold $Y$ we extend the pullback $f^{\ast }$ from smooth forms to a class of currents. We provide a basic calculus for this pullback and show under quite mild assumptions that it is cohomologically sound. 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.