Nadel vanishing theorem
In mathematics, the Nadel vanishing theorem is a global vanishing theorem for multiplier ideals, introduced by A. M. Nadel in 1989. It generalizes the Kodaira [vanishing theorem] using singular metrics with positive curvature, and also it can be seen as an analytical analogue of the Kawamata–Viehweg vanishing theorem.
Statement
The theorem can be stated as follows. Let X be a smooth complex projective variety, D an effective -divisor and L a line bundle on X, and is a multiplier ideal sheaves. Assume that is big and nef. ThenNadel vanishing theorem in the analytic setting: Let be a Kähler manifold such that weakly pseudoconvex, and let F be a holomorphic line bundle over X equipped with a singular hermitian metric of weight. Assume that for some continuous positive function on X. Then
Let arbitrary plurisubharmonic function on, then a multiplier ideal sheaf is a coherent on, and therefore its zero variety is an analytic set.