A generalization of Kodaira-Ramanujam's vanishing theorem

Kawamata, Yūjirō

doi:10.1007/bf01456407

Cited by 302 publications

(169 citation statements)

References 2 publications

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“…In this context, we get the following important vanishing theorem, which can be seen as a generalization of the Kawamata-Viehweg vanishing theorem (see [Kaw82], [Vie82], [EV86]). …”

Section: Hörmander's L 2 Estimates and Existence Theoremsmentioning

confidence: 99%

Homological Algebra of Mirror Symmetry

Kontsevich

1995

Proceedings of the International Congress of Mathematicians

880

1,045

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Abstract. One important problem arising in algebraic geometry is the computation of effective bounds for the degree of embeddings in a projective space of given algebraic varieties. This problem is intimately related to the following question: Given a positive (or ample) line bundle L on a projective manifold X, can one compute explicitly an integer m 0 such that mL is very ample for m m 0 ? It turns out that the answer is much easier to obtain in the case of adjoint line bundles 2(K X + mL), for which universal values of m 0 exist. We indicate here how such bounds can be derived by a combination of powerful analytic methods: theory of positive currents and plurisubharmonic functions (Lelong), L 2 estimates for ∂ (Andreotti-Vesentini, Hörmander, Bombieri, Skoda), Nadel vanishing theorem, Aubin-Calabi-Yau theorem, and holomorphic Morse inequalities.

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“…In this context, we get the following important vanishing theorem, which can be seen as a generalization of the Kawamata-Viehweg vanishing theorem (see [Kaw82], [Vie82], [EV86]). …”

Section: Hörmander's L 2 Estimates and Existence Theoremsmentioning

confidence: 99%

Homological Algebra of Mirror Symmetry

Kontsevich

1995

Proceedings of the International Congress of Mathematicians

880

1,045

View full text Add to dashboard Cite

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“…Ramanujam [51] (in whose paper the method of coverings already appears), Y. Miyaoka [45] (the first who works with integral parts of Q l divisors, in the surface case), by Y. Kawamata [36] and the second author [63]. All results mentioned replace the condition "ample" in Kodaira's result by weaker conditions.…”

Section: Introductionmentioning

confidence: 99%

Lectures on Vanishing Theorems

Esnault¹,

Viehweg²

1992

419

337

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“…For example, if a defines a smooth subvariety of codimension d, then J (X, a m ) = a m−d . The proof Ein and Lazarsfeld's result heavily uses the Kawamata-Viehweg vanishing theorem [Kaw82,Vie82]. Another way to interpret the subspace…”

Section: Theorem C (Effective Computation Of Test Ideals Theorem 54)mentioning

confidence: 99%

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

Schwede

Tucker

2014

Journal de Mathématiques Pures et Appliquées

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Abstract. Given an ideal a ⊆ R in a (log) Q-Gorenstein F -finite ring of characteristic p > 0, we study and provide a new perspective on the test ideal τ (R, a t ) for a real number t > 0. Generalizing a number of known results from the principal case, we show how to effectively compute the test ideal and also describe τ (R, a t ) using (regular) alterations with a formula analogous to that of multiplier ideals in characteristic zero. We further prove that the F -jumping numbers of τ (R, a t ) as t varies are rational and have no limit points, including the important case where R is a formal power series ring. Additionally, we obtain a global division theorem for test ideals related to results of Ein and Lazarsfeld from characteristic zero, and also recover a new proof of Skoda's theorem for test ideals which directly mimics the proof for multiplier ideals.

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A generalization of Kodaira-Ramanujam's vanishing theorem

Cited by 302 publications

References 2 publications

Homological Algebra of Mirror Symmetry

Homological Algebra of Mirror Symmetry

Lectures on Vanishing Theorems

Test ideals of non-principal ideals: Computations, jumping numbers, alterations and division theorems

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