A vanishing theorem for semipositive line bundles over non-Kähler manifolds

Siu, Yum Tong

doi:10.4310/jdg/1214438686

Cited by 75 publications

(46 citation statements)

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“…Lorsque 0 est un cube, on sait alors expliciter les fonctions propres par une transformation de Fourier partielle qui ramène le problème à celui classique de l'oscillateur harmonique en une variable. L'idée de ce calcul nous a été fortement inspirée par les articles [3] [16] (telle qu'elle est reprise dans [7]), la taille des cubes était choisie égale à k~2; on peut voir aisément que ce choix était critique pour permettre de borner les effets du champ magnétique indépendamment de fc, mais la détermination exacte du spectre devenait alors impossible.…”

Section: Champs Magnétiques Et Inégalités De Morse Pour La (T-cohomolunclassified

“…On peut montrer d'autre part, en utilisant un raisonnement classique de Siegel [15] mis en forme par [16] que dim H^X.E*) < cte. Ce travail a fait l'objet d'une note [8] du même titre, publiée aux Comptes Rendus.…”

Section: Champs Magnétiques Et Inégalités De Morse Pour La (T-cohomolunclassified

“…Fourier, Grenoble 35, 4 (1985), 189 à 229. [16]). On observera que la constante numérique de l'inégalité 0. l(a) est optimale, comme le montre l'exemple du fibre produit tensoriel total E = OWT-^ ^ 0(-iy au-dessus de X = (P^C))".…”

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“…Pour ce fibre, on a en effet X(q) = X et dim H^X^) = (k+ l^-^fe-1)^, k ^ 1, f / i y ,c(E) =(-1)^!. Jx \27t / L'existence d'une majoration du type 0.1 (a) était conjecturée par Y. T. Siu, qui a successivement démontré le cas particulier où fc(E) est > 0 dans le complémentaire d'un ensemble de mesure nulle [16], puis le cas où ic(E) est > 0 sur X [17]. Nous avons d'ailleurs emprunté à Siu une partie des techniques utilisées ici, notamment aux § 3 et § 5.…”

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Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie

Demailly¹

1985

Annales De L’institut Fourier

148

160

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Section: Champs Magnétiques Et Inégalités De Morse Pour La (T-cohomolunclassified

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Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie

Demailly¹

1985

Annales De L’institut Fourier

148

160

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“…The problem in proving this was to obtain Moishezonness of a compact complex space X if it carries an almost positive coherent sheaf. From then a number of characterizations of Moishezonness with additional assumptions to the hypothesis of the conjecture have been obtained [7], [12], [9], [5], [10] . For example, in [8] it was assumed that X is Káhler.…”

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A note on characterization of Moishezon spaces

Rama¹

1990

Publicacions Matemàtiques

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In this note a necessary and sufficient condition for a compact complex space X t.o be Moishezon is obtained ; it can be seen as the existente of a line bundle L on X such that, for some point x E X, the first cohomology groups of X with values respectively in L®m y and L®m', vanish . (Here rn y , denote the ideal sheaf at x) .1. Grauert and Riemenschneider [1], [7] conjectured a characterization of a Moishezon space in tercos of "almost (quasi) positive" coherent sheafs on it. The problem in proving this was to obtain Moishezonness of a compact complex space X if it carries an almost positive coherent sheaf. From then a number of characterizations of Moishezonness with additional assumptions to the hypothesis of the conjecture have been obtained [7], [12], [9], [5], [10] . For example,in [8] it was assumed that X is Káhler. Some of these are in such a way that the proofs can be obtained by using Kodaira's techniques : namely blowups, Kodaira's Vanishing and embedding theorems. Siu [11] has succeeded in proving a stronger version than the conjecture. His proof is by using the powerful theorem of Hirzebruch-Riemann-Roch and giving estimates on the dimensions of the cohomology groups óf X with coefficients in a power of a line bundle which has a non strictly positive curvature form. lt deals, more generally with one of the fundamental questions of obtaining holomorphic sections for non strictly positive line bundles .The present note gives a characterization of Moishezon spaces using Kodaira's techniques.2. Since compact complex analytic space can be desingularized and coherent analytic sheaves can be made free (modulo torsion) by proper modifications, a characterization of Moishezon space X can be stated in tercos of compact complex manifold X and line bundles (locally free sheaves) over X . We prove the following :

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On the Cohomology of Pseudoeffective Line Bundles

Demailly

2015

Complex Geometry and Dynamics

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The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact Kähler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general Kähler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1, 1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact Kähler threefolds without nontrivial subvarieties, following a joint work with F. Campana and M. Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.

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A vanishing theorem for semipositive line bundles over non-Kähler manifolds

Cited by 75 publications

References 15 publications

Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie

Champs magnétiques et inégalités de Morse pour la $d''$-cohomologie

A note on characterization of Moishezon spaces

On the Cohomology of Pseudoeffective Line Bundles

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