Effective nonvanishing, effective global generation

Cataldo, Mark Andrea A. de

doi:10.5802/aif.1658

Cited by 6 publications

(6 citation statements)

References 13 publications

(52 reference statements)

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“…Statements involving higher jets can be proved as well. Details will appear in [5]. Question 5.2.6 Let X, E and L be as above.…”

Section: Effective Results On Vector Bundlesmentioning

confidence: 99%

“…The tensor powers T n (E) are t-nef by virtue of (6). S n (E) and ∧ n (E) are both direct summands of T n (E) so that they are t-nef by (5). Recall that Γ a E is a direct summand of the vector bundle ⊗ r i=1 S ai (∧ i E) which is t-nef by what above, (5) and (6).…”

Section: (7)mentioning

confidence: 99%

“…A natural algebraic approach to the case of higher rank is to consider an analogous question for the tautological line bundle ξ of the projectivized bundle π : P(E) → X. The results I obtain with the algebraic approach are for P ⊗ det E; compare Remark 5.2.5 with the sample effective global generation result presented below; see [5]. On the analytic side, the problem is that the nefness of a vector bundle E does not seem to be linked to a curvature condition on E itself.…”

Section: Introductionmentioning

confidence: 99%

“…I am indebted to J. Kollár for posing a question similar to Question 0.0.1. I thank L. Ein and R. Lazarsfeld for convincing me to think about an algebraic proof of the results of Theorem 5.2.2; this has lead me to the statements of Remark 5.2.5; see [5]. It is a pleasure to thank the participants of the lively algebraic geometry seminar at Washington University in St. Louis for their encouragment and useful criticisms: V. Masek, T. Nguyen, P. Rao and D. Wright.…”

mentioning

confidence: 99%

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Singular hermitian metrics on vector bundles

Cataldo¹

1998

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We introduce a notion of singular hermitian metrics (s.h.m.) for holomorphic vector bundles and define positivity in view of L 2 -estimates. Associated with a suitably positive s.h.m. there is a (coherent) sheaf 0-th kernel of a certain d ′′ -complex. We prove a vanishing theorem for the cohomology of this sheaf. All this generalizes to the case of higher rank known results of Nadel for the case of line bundles. We introduce a new semi-positivity notion, t-nefness, for vector bundles, establish some of its basic properties and prove that on curves it coincides with ordinary nefness. We particularize the results on s.h.m. to the case of vector bundles of the form E = F ⊗ L, where F is a t-nef vector bundle and L is a positive (in the sense of currents) line bundle. As applications we generalize to the higher rank case 1) Kawamata-Viehweg Vanishing Theorem, 2) the effective results concerning the global generation of jets for the adjoint to powers of ample line bundles, and 3) Matsusaka Big Theorem made effective.

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“…Statements involving higher jets can be proved as well. Details will appear in [5]. Question 5.2.6 Let X, E and L be as above.…”

Section: Effective Results On Vector Bundlesmentioning

confidence: 99%

Section: (7)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Singular hermitian metrics on vector bundles

Cataldo¹

1998

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Observe that, by (3.10) and by using some results in [10], one can immediately determine some conditions for the regularity of [s] ∈ V δ (F ). Indeed, if F is a globally generated rank-two vector bundle on X which generates the 0-jets at Σ = {p 1 , .…”

Section: Description Of the Tangent Space T [S] (V δ (F )) And Regula...mentioning

confidence: 99%

Families of nodal curves on projective threefolds and their regularity via postulation of nodes

Flamini

2003

Trans. Amer. Math. Soc.

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Abstract. The main purpose of this paper is to introduce a new approach to study families of nodal curves on projective threefolds. Precisely, given a smooth projective threefold X, a rank-two vector bundle E on X, and integers k ≥ 0, δ > 0, denote by V δ (E(k)) the subscheme of P(H 0 (E(k))) parametrizing global sections of E(k) whose zero-loci are irreducible δ-nodal curves on X. We present a new cohomological description of the tangent space. This description enables us to determine effective and uniform upper bounds for δ, which are linear polynomials in k, such that the family V δ (E(k)) is smooth and of the expected dimension (regular, for short). The almost sharpness of our bounds is shown by some interesting examples. Furthermore, when X is assumed to be a Fano or a Calabi-Yau threefold, we study in detail the regularity property of a point [s] ∈ V δ (E(k)) related to the postulation of the nodes of its zero-locus C = V (s) ⊂ X. Roughly speaking, when the nodes of C are assumed to be in general position either on X, or on an irreducible divisor of X having at worst log-terminal singularities or to lie on a l.c.i. and subcanonical curve in X, we find upper bounds on δ which are, respectively, cubic, quadratic and linear polynomials in k ensuring the regularity of V δ (E(k)) at [s]. Finally, when X = P 3 , we also discuss some interesting geometric properties of the curves given by sections parametrized by V δ (E ⊗ O X (k)).

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Positivity of the schur-weyl invariants of projective varieties

Nakashima

2002

Abh.Math.Semin.Univ.Hambg.

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Effective nonvanishing, effective global generation

Abstract: We prove a multiple-points higher-jets nonvanishing theorem by the use of local Seshadri constants. Applications are given to effectivity problems such as constructing rational and birational maps into Grassmannians, and the global generation of vector bundles.

Cited by 6 publications

References 13 publications

Singular hermitian metrics on vector bundles

Singular hermitian metrics on vector bundles

Families of nodal curves on projective threefolds and their regularity via postulation of nodes

Positivity of the schur-weyl invariants of projective varieties

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