TRANSCENDENTAL PROOF OF A GENERALIZED KAWAMATA VIEHWEG VANISHING THEOREM
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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 .
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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
TRANSCENDENTAL PROOF OF A GENERALIZED KAWAMATAVIEHWEG VANISHING THEOREM
JeanPierre DEMAILLY Universite´deGrenobleI, Institut Fourier, BP 74, Laboratoireassoci´eauC.N.R.S.n˚188, F38402SaintMartind’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 KodairaNakano and theLestimates for∂yield a very elementary proof of the KawamataViehweg theorem [7,13,15]: ifLis numerically effective, then q−1s H(X,L) = 0forq < s, wheresis the largest integer such thatc1(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) ifc1(L)↾Γ≥0 for every curve Γ inX. Then, it is d know≥0 for any subvarietyY⊂Xof dimensiond. On the n [10] thatc1(L)↾Y other hand, the wellknown NakaiMoishezon criterion says that a line bundleH d >variety0 f Yof dimensiond. IfHis is ample if and only ifc1(H)↾Yor 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(Vm) the set of common zeros of all 0m sections inVm=H(X,F) . TheKodaira dimensionκ(F) is the maximum whenmruns over all positive integers of the generic rank of the analytic morphism ⋆ Φm:X\Z(Vm)−→P(V) which maps a pointx∈X\Z(Vm) to the hyperplane of m sectionsσ∈Vmsuch thatσ(x) = 0 . As usual, we setκ(F) =−∞ifVm={0}for allm. The following definition (related to the concept of logterminal singularities) gives a way of measuring how singular is a divisor. P Definition.—We say that a divisorD=αjDjwith rational coefficients Q −2αj αj∈Qis integrable at a pointx0∈Xif the function|gj|equal to the product of local generatorsgjof the ideal ofDjatx0is integrable on a neighborhood ofx0.
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