A Kawamata-Viehweg Vanishing Theorem on Compact Kähler Manifolds

By J. Demailly, T. Peternell. 2002

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We prove a Kawamata-Viehweg vanishing theorem on a normal compact Kahler space X : if L is a nef line bundle with L 6= 0, then H(X,KX + L) = 0 for q ≥ dimX − 1. As an application we complete a part of the abundance theorem for minimal Kahler threefolds: if X is a minimal Kahler threefold, then the Kodaira dimension κ(X) is nonnegative. §0. Introduction In this paper we establish the following Kawamata-Viehweg type vanishing theorem on a compact Kahler manifold or, more generally, a normal compact Kahler space. 0.1 Theorem. Let X be a normal compact Kahler space of dimension n and L a nef line bundle on X. Assume that L 6= 0. Then H(X,KX + L) = 0 for q ≥ n− 1. In general, one expects a vanishing H(X,KX + L) = 0 for q ≥ n + 1 − ν(L), where ν(L) is the numerical Kodaira dimension of the nef line bundle L, i.e. ν(L) is the largest integer ν such that L 6= 0. Of course, when X is projective, Theorem 0.1 is contained in the usual KawamataViehweg vanishing theorem, but the methods of proof in the algebraic case clearly fail in the general Kahler setting. Instead we proceed in the following way. Clearly we may assume that X is smooth and by Serre duality, only the cohomology group H is of interest. Take a singular metric h on L with positive curvature current T with local weight function h. By [Si74, De93a] there exists a decomposition T = ∑ λjDj +G, (D) where λj ≥ 1 are irreducible divisors, and G is a pseudo-effective current such that G|Di is pseudo-effective for all i. Consider the multiplier ideal sheaf I(h). We associate to h another, “upper regularized” multiplier ideal sheaf I+(h) by setting I+(h) := lim e→0+ I(h) = lim e→0+ I (

Subfields: Applied Mathematics, Geometry and Topology

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Citations in the corpus

A proof of Demailly's strong openness conjecture (2015, background): “In the present article, we discuss a more general conjecture — the strong openness conjecture about multiplier ideal sheaves for plurisubharmonic functions, which was posed by Demailly in [7] and [8] (see also [11]):”
A sharp effectiveness result of Demailly's strong openness conjecture (2017, background): “ and Kollar. 1. Introduction The multiplier ideal sheaf related to a plurisubharmonic function plays an important role in complex geometry and algebraic geometry, which was widely discussed (see e.g. [34, 24, 29, 6, 7, 3, 8, 22, 31, 32, 4]). We recall the deﬁnition as follows. Let ϕbe a plurisubharm”
Abelian fibred holomorphic symplectic manifolds (2003, background): “We really only need h(D) = 0, and in fact this holds for X non-projective too, provided it is compact and Kähler (see Demailly and Peternell [9]).”
A TRANSCENDENTAL APPROACH TO KOLL AR'S INJECTIVITY THEOREM II (2007, background): “More precisely, it is a generalization of Enoki’s injectivity theorem, which is an analytic version of Kollár’s theorem (see [6, Theorem 0.2] and [13, Corollary 1.4]).”
Ohsawa-Takegoshi Extension Theorem for Compact Kähler Manifolds and Applications (2014, background): “ending only on CU. The theorem is thus proved. 4. Appendix For the reader’s convenience, we give the proof of (41) and (42), which is a rather standard estimate (cf. [Dem12, Prop. 12.4, Remark 12.5], [DP03] or [Li12]). Set g := gm,k, η := ηǫ, B := Bǫ,k and δ := δk for simplicity. Let Yk be a subvari”