Symmetric tensors and geometry of ℙN subvarieties

Fedor Bogomolov; Bruno De Oliveira

doi:10.1007/s00039-008-0666-7

Symmetric tensors and geometry of ℙ^N subvarieties

Fedor Bogomolov, Bruno De Oliveira

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

This paper using a geometric approach produces vanishing and nonvanishing results concerning the spaces of twisted symmetric differentials H ⁰(X, S^mΩ_X¹ ⊗ script O^X(k)) on subvarieties X ⊂ ℙ^N, 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, Q_α,m(X_t) = dim Η⁰(X, S^m(Ω_Xt¹ ⊗ αK_Xt)) along smooth families of projective varieties X_t are not invariant even for α arbitrarily large.

Original language	English (US)
Pages (from-to)	637-656
Number of pages	20
Journal	Geometric and Functional Analysis
Volume	18
Issue number	3
DOIs	https://doi.org/10.1007/s00039-008-0666-7
State	Published - Sep 2008

Keywords

Quadrics
Symmetric differentials
Trisecant variety
Vanishing and nonvanishing theorems

ASJC Scopus subject areas

Analysis
Geometry and Topology

Access to Document

10.1007/s00039-008-0666-7

Cite this

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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, SmΩX1 ⊗ 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(ΩXt1 ⊗ 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",

note = "Funding Information: Keywords and phrases: Symmetric differentials, vanishing and nonvanishing theorems, quadrics, trisecant variety AMS Mathematics Subject Classification: Primary: 14F10; Secondary: 14N15, 14F17, 14M07 F.B. partially supported by the NSF grant DMS-0404715. B.O. Partially supported by the NSF grant DMS-0707097.",

year = "2008",

month = sep,

doi = "10.1007/s00039-008-0666-7",

language = "English (US)",

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pages = "637--656",

journal = "Geometric and Functional Analysis",

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AU - Bogomolov, Fedor

AU - De Oliveira, Bruno

N1 - Funding Information: Keywords and phrases: Symmetric differentials, vanishing and nonvanishing theorems, quadrics, trisecant variety AMS Mathematics Subject Classification: Primary: 14F10; Secondary: 14N15, 14F17, 14M07 F.B. partially supported by the NSF grant DMS-0404715. B.O. Partially supported by the NSF grant DMS-0707097.

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, SmΩX1 ⊗ 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(ΩXt1 ⊗ 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, SmΩX1 ⊗ 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(ΩXt1 ⊗ 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

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