TRANSCENDENTAL PROOF OF A GENERALIZED KAWAMATA VIEHWEG VANISHING THEOREM

Jean-Pierre Demailly

Description

Niveau: Supérieur, Licence, Bac+2
TRANSCENDENTAL PROOF OF A GENERALIZED KAWAMATA-VIEHWEG VANISHING THEOREM Jean-Pierre DEMAILLY Universite de Grenoble I, Institut Fourier, BP 74, Laboratoire associe au C.N.R.S. n˚ 188, F-38402 Saint-Martin d'Heres Dedicated to Professor Leon Ehrenpreis on his sixtieth birthday Abstract.— Let L be a holomorphic line bundle over a projective algebraic manifold X . It is shown that the differential geometric technique of Bochner- Kodaira-Nakano and the L2 estimates for ∂ yield a very elementary proof of the Kawamata-Viehweg theorem [7,13,15]: if L is numerically effective, then Hq(X,L?1) = 0 for q < s , where s is the largest integer such that c1(L)s 6= 0 . More generally, our method implies a vanishing result when L is tensorized with an effective Q–divisor which may have non normal crossings, under a natural integrability hypothesis for the divisor. 1. Statement of results. Recall that a line bundle L over a projective algebraic manifold X is said to be numerically effective (nef) if c1(L)?? ≥ 0 for every curve ? in X . Then, it is known [10] that c1(L)d?Y ≥ 0 for any subvariety Y ? X of dimension d . On the other hand, the well-known Nakai-Moishezon criterion says that a line bundle H is ample if and only if c1(H)d?Y > 0 for any subvariety Y of dimension d .

• has normal


• crossings norm


• kawamata-viehweg vanishing


• normal crossings


• f?1 ??


• x0 ?


• known bochner-kodaira- nakano


• f?1

Sujets

• Ehrenpreis
• Université Grenoble-I
• Demailly
• Normal crossings

Informations

• Publié par: profil-vieg-2012
• Nombre de lectures: 11
• Langue: English

Extrait

TRANSCENDENTAL PROOF OF A GENERALIZED KAWAMATAVIEHWEG VANISHING THEOREM

JeanPierre DEMAILLY Universite´deGrenobleI, Institut Fourier, BP 74, Laboratoireassoci´eauC.N.R.S.n˚188, F38402SaintMartind’H`eres

Dedicated to Professor Leon Ehrenpreis on his sixtieth birthday

Abstract.—LetLbe a holomorphic line bundle over a projective algebraic manifoldX. It is shown that the differential geometric technique of Bochner 2 KodairaNakano and theLestimates for∂yield a very elementary proof of the KawamataViehweg theorem [7,13,15]: ifLis numerically effective, then q−1s H(X,L) = 0forq < s, wheresis the largest integer such thatc 1 (L)6= 0. More generally, our method implies a vanishing result whenLis tensorized with an effectiveQ–divisor which may have non normal crossings, under a natural integrability hypothesis for the divisor.

1. Statement of results.

Recall that a line bundleLover a projective algebraic manifoldXis said to benumerically effective(nef) ifc 1 (L) ↾Γ ≥0 for every curve Γ inX. Then, it is d know≥0 for any subvarietyY⊂Xof dimensiond. On the n [10] thatc 1 (L) ↾Y other hand, the wellknown NakaiMoishezon criterion says that a line bundleH d >variety0 f Yof dimensiond. IfHis is ample if and only ifc 1 (H) ↾Y or any sub a given ample line bundle, it follows easily that a line bundleLis nef if and only m ifHL ⊗ is ample for every integerm≥0 . LetFbe a line bundle overXandZ(V m ) the set of common zeros of all 0m sections inV m =H(X,F) . TheKodaira dimensionκ(F) is the maximum whenmruns over all positive integers of the generic rank of the analytic morphism ⋆ Φ m :X\Z(V m )−→P(V) which maps a pointx∈X\Z(V m ) to the hyperplane of m sectionsσ∈V m such thatσ(x) = 0 . As usual, we setκ(F) =−∞ifV m ={0}for allm. The following definition (related to the concept of logterminal singularities) gives a way of measuring how singular is a divisor. P Definition.—We say that a divisorD=α j D j with rational coefficients Q −2α j α j ∈Qis integrable at a pointx 0 ∈Xif the function|g j |equal to the product of local generatorsg j of the ideal ofD j atx 0 is integrable on a neighborhood ofx 0 .

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