Global Division of Cohomology Classes via Injectivity Lawrence Ein and Mihnea Popa

The aim of this note is to remark that the injectivity theorems of Kollár and EsnaultViehweg can be used to give a quick algebraic proof of a strengthening (by dropping the positivity hypothesis) of the Skoda-type division theorem for global sections of adjoint line bundles vanishing along suitable multiplier ideal sheaves proved in [EL], and to extend this result to higher cohomology classes as well (cf. Theorem 4.1). For global sections, this is a slightly more general statement of the algebraic version of an analytic result of Siu [Siu] based on the original Skoda theorem. In §4 we list a few consequences of this type of result, like the surjectivity of various multiplication or cup-product maps and the corresponding version of the geometric effective Nullstellensatz.