Injectivity theorem for pseudo-effective line bundles and its applications

Abstract

We formulate and establish a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár's torsion-freeness, Kollár's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use $L^2$-harmonic forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.

34 pages, v3: minor revisions, completely revised version

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Cited by in corpus (8)

• On semipositivity, injectivity and vanishing theorems
• Logarithmic vanishing theorems on compact Kähler manifolds I
• Nadel-Nakano vanishing theorems of vector bundles with singular Hermitian metrics
• On the modified ideal sheaf
• Some Kollar-Enoki type injectivity and Nadel type vanishing theorems on compact Kahler manifolds
• The harmonic theory on a vector bundle with singular Hermitian metrics and positivity
• Nefness of the direct images of pluricanonical bundles
• Global generation and very ampleness for adjoint linear series