Multiplier ideal

Summary

The multiplier ideal of a plurisubharmonic function $\varphi$ is the sheaf of holomorphic germs $f$ for which $\abs{f}^2 e^{-2\varphi}$ is locally integrable. It measures the singularities of $\varphi$: the more singular $\varphi$ is, the smaller the ideal. Nadel proved that it is a coherent analytic sheaf. The Nadel vanishing theorem, a consequence of Hörmander's L2 estimates, makes it a basic tool in complex and algebraic geometry. In algebraic geometry, the multiplier ideal of an ideal sheaf with a rational exponent is defined through a log resolution. It agrees with the analytic definition.