Symmetric tensors and geometry of ℙN subvarieties

Fedor Bogomolov, Bruno De Oliveira

Research output: Contribution to journalArticle

Abstract

This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials H 0(X, SmΩX 1 ⊗ script OX(k)) on subvarieties X ⊂ ℙN, with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N - 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, Qα,m(Xt) = dim Η0(X, SmXt 1 ⊗ αKXt)) along smooth families of projective varieties Xt are not invariant even for α arbitrarily large.

Original languageEnglish (US)
Pages (from-to)637-656
Number of pages20
JournalGeometric and Functional Analysis
Volume18
Issue number3
DOIs
StatePublished - Sep 2008

Fingerprint

Tensor
Geometric Approach
Projective Variety
Quadric
Tangent line
Invariant
Range of data
Family

Keywords

  • Quadrics
  • Symmetric differentials
  • Trisecant variety
  • Vanishing and nonvanishing theorems

ASJC Scopus subject areas

  • Geometry and Topology
  • Analysis

Cite this

Symmetric tensors and geometry of ℙN subvarieties. / Bogomolov, Fedor; De Oliveira, Bruno.

In: Geometric and Functional Analysis, Vol. 18, No. 3, 09.2008, p. 637-656.

Research output: Contribution to journalArticle

Bogomolov, Fedor ; De Oliveira, Bruno. / Symmetric tensors and geometry of ℙN subvarieties. In: Geometric and Functional Analysis. 2008 ; Vol. 18, No. 3. pp. 637-656.
@article{c1dd6801894f47559777723dbdc8ddb3,
title = "Symmetric tensors and geometry of ℙN subvarieties",
abstract = "This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials H 0(X, SmΩX 1 ⊗ script OX(k)) on subvarieties X ⊂ ℙN, with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N - 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, Qα,m(Xt) = dim Η0(X, Sm(ΩXt 1 ⊗ αKXt)) along smooth families of projective varieties Xt are not invariant even for α arbitrarily large.",
keywords = "Quadrics, Symmetric differentials, Trisecant variety, Vanishing and nonvanishing theorems",
author = "Fedor Bogomolov and {De Oliveira}, Bruno",
year = "2008",
month = "9",
doi = "10.1007/s00039-008-0666-7",
language = "English (US)",
volume = "18",
pages = "637--656",
journal = "Geometric and Functional Analysis",
issn = "1016-443X",
publisher = "Birkhauser Verlag Basel",
number = "3",

}

TY - JOUR

T1 - Symmetric tensors and geometry of ℙN subvarieties

AU - Bogomolov, Fedor

AU - De Oliveira, Bruno

PY - 2008/9

Y1 - 2008/9

N2 - This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials H 0(X, SmΩX 1 ⊗ script OX(k)) on subvarieties X ⊂ ℙN, with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N - 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, Qα,m(Xt) = dim Η0(X, Sm(ΩXt 1 ⊗ αKXt)) along smooth families of projective varieties Xt are not invariant even for α arbitrarily large.

AB - This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials H 0(X, SmΩX 1 ⊗ script OX(k)) on subvarieties X ⊂ ℙN, with k ≤ m. Emphasis is given to the case of k = m which is special and whose nonvanishing results on the dimensional range dim X > 2/3(N - 1) are related to the space of quadrics containing X and the variety of all tangent trisecant lines of X. The paper ends with an application showing that the twisted symmetric plurigenera, Qα,m(Xt) = dim Η0(X, Sm(ΩXt 1 ⊗ αKXt)) along smooth families of projective varieties Xt are not invariant even for α arbitrarily large.

KW - Quadrics

KW - Symmetric differentials

KW - Trisecant variety

KW - Vanishing and nonvanishing theorems

UR - http://www.scopus.com/inward/record.url?scp=52449130806&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=52449130806&partnerID=8YFLogxK

U2 - 10.1007/s00039-008-0666-7

DO - 10.1007/s00039-008-0666-7

M3 - Article

VL - 18

SP - 637

EP - 656

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 3

ER -