FROBENIUS AMPLITUDE AND STRONG VANISHING THEOREMS FOR VECTOR BUNDLES

The primary goal of this paper is to systematically exploit the method of Deligne-Illusie to obtain Kodaira type vanishing theorems for vector bundles and more generally coherent sheaves on algebraic varieties. The key idea is to introduce a number which provides a cohomological measure of the positivity of a coherent sheaf called the Frobenius or F-amplitude. The F- amplitude enters into the statement of the basic vanishing theorem, and this leads to the problem of calculating, or at least estimating, this number. Most of the work in this paper is devoted to doing this various situations.

[1]  D. Keeler.  Ample filters of invertible sheaves , 2001, math/0108068.

[2]  S. Kovács.  A characterization of rational singularities , 2000 .

[3]  D. Arapura.  A class of sheaves satisfying Kodaira's vanishing theorem , 1999, math/9903137.

[4]  M. A. Cataldo.  Vanishing via Lifting to Second Witt Vectors and a Proof of an Isotriviality Result , 1998, math/9811151.

[5]  Nobuo Hara.  A characterization of rational singularities in terms of injectivity of Frobenius maps , 1998 .

[6]  L. Migliorini.  Some observations on cohomologicallyp-ample bundles , 1993 .

[7]  L. Illusie.  Réduction semi-stable et décomposition de complexes de de Rham à coefficients , 1990 .

[8]  I. Bauer,et al.  On the Hodge spectral sequence for some classes of non-complete algebraic manifolds , 1989 .

[9]  L. Illusie,et al.  Relèvements modulop2 et décomposition du complexe de de Rham , 1987 .

[10]  H. Esnault,et al.  Logarithmic de Rham complexes and vanishing theorems , 1986 .

[11]  V. B. Mehta,et al.  Frobenius splitting and cohomology vanishing for Schubert varieties , 1985 .

[12]  Y. Kawamata.  A generalization of Kodaira-Ramanujam's vanishing theorem , 1982 .

[13]  J. Potier.  Annulation de la cohomologie à valeurs dans un fibré vectoriel holomorphe positif de rang quelconque , 1975 .

[14]  R. Carter,et al.  On the modular representations of the general linear and symmetric groups , 1974 .

[15]  T. Willmore.  Algebraic Geometry , 1973, Nature.

[16]  D. Gieseker.  P-Ample Bundles and Their Chern Classes , 1971, Nagoya Mathematical Journal.

[17]  C. Barton.  Tensor Products of Ample Vector Bundles in Characteristic p , 1971 .

[18]  S. Kleiman.  Toward a numerical theory of ampleness , 1966 .

[19]  G. Horrocks.  Vector Bundles on the Punctured Spectrum of a Local Ring , 1964 .

[20]  Robert Steinberg,et al.  Representations of Algebraic Groups , 1963, Nagoya Mathematical Journal.

[21]  S. Mardešić.  The vanishing theorems , 2000 .

[22]  I. Bauer,et al.  Some aspects of hodge theory on non-complete algebraic manifolds , 1991 .

[23]  D. Arapura.  Local cohomology of sheaves of differential forms and Hodge theory. , 1990 .

[24]  A. Sommese,et al.  Vanishing Theorems on Complex Manifolds , 1985 .

[25]  Robin Hartshorne,et al.  Ample subvarieties of algebraic varieties , 1970 .

[26]  R. Hartshorne.  Ample vector bundles , 1966 .

[27]  David Mumford,et al.  Lectures on curves on an algebraic surface , 1966 .

[28]  Alexander Grothendieck,et al.  Elements de geometrie algebrique III: Etude cohomologique des faisceaux coherents , 1961 .

[29]  A. Grothendieck,et al.  Éléments de géométrie algébrique , 1960 .