Local Topological Obstruction For Divisors

Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in $H^2(\mathcal{O}_X)$. In this article, we replace $H^2(\mathcal{O}_X)$ by $H^2_D(\mathc...

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Detalhes bibliográficos
Autores: Biswas, I., Dan, A.
Formato: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2020
País:España
Recursos:Basque Center for Applied Mathematics (BCAM)
Repositorio:BIRD. BCAM's Institutional Repository Data
OAI Identifier:oai:bird.bcamath.org:20.500.11824/1201
Acesso em linha:http://hdl.handle.net/20.500.11824/1201
https://doi.org/10.1007/s13163-020-00376-6
Access Level:acceso abierto
Palavra-chave:Obstruction theories
Hodge locus
semi-regularity map
deformation of linear systems
Noether-Lefschetz locus
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spelling Local Topological Obstruction For DivisorsBiswas, I.Dan, A.Obstruction theoriesHodge locussemi-regularity mapdeformation of linear systemsNoether-Lefschetz locusGiven a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in $H^2(\mathcal{O}_X)$. In this article, we replace $H^2(\mathcal{O}_X)$ by $H^2_D(\mathcal{O}_X)$ and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of $D$ as an effective Cartier divisor of a first order infinitesimal deformations of $X$). We apply this to study the jumping locus of families of linear systems and the Noether-Lefschetz locus. Finally, we give examples of first order deformations $X_t$ of $X$ for which the cohomology class $[D]$ deforms as a Hodge class but $D$ \emph{does not} lift as an effective Cartier divisor of $X_t$.202020202020info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttp://hdl.handle.net/20.500.11824/1201https://doi.org/10.1007/s13163-020-00376-6reponame:BIRD. BCAM's Institutional Repository Datainstname:Basque Center for Applied Mathematics (BCAM)Ingléshttps://link.springer.com/article/10.1007/s13163-020-00376-6info:eu-repo/grantAgreement/EC/FP7/615655Reconocimiento-NoComercial-CompartirIgual 3.0 Españahttp://creativecommons.org/licenses/by-nc-sa/3.0/es/info:eu-repo/semantics/openAccessoai:bird.bcamath.org:20.500.11824/12012026-06-19T12:47:47Z
dc.title.none.fl_str_mv Local Topological Obstruction For Divisors
title Local Topological Obstruction For Divisors
spellingShingle Local Topological Obstruction For Divisors
Biswas, I.
Obstruction theories
Hodge locus
semi-regularity map
deformation of linear systems
Noether-Lefschetz locus
title_short Local Topological Obstruction For Divisors
title_full Local Topological Obstruction For Divisors
title_fullStr Local Topological Obstruction For Divisors
title_full_unstemmed Local Topological Obstruction For Divisors
title_sort Local Topological Obstruction For Divisors
dc.creator.none.fl_str_mv Biswas, I.
Dan, A.
author Biswas, I.
author_facet Biswas, I.
Dan, A.
author_role author
author2 Dan, A.
author2_role author
dc.subject.none.fl_str_mv Obstruction theories
Hodge locus
semi-regularity map
deformation of linear systems
Noether-Lefschetz locus
topic Obstruction theories
Hodge locus
semi-regularity map
deformation of linear systems
Noether-Lefschetz locus
description Given a smooth, projective variety $X$ and an effective divisor $D\,\subseteq\, X$, it is well-known that the (topological) obstruction to the deformation of the fundamental class of $D$ as a Hodge class, lies in $H^2(\mathcal{O}_X)$. In this article, we replace $H^2(\mathcal{O}_X)$ by $H^2_D(\mathcal{O}_X)$ and give an analogous topological obstruction theory. We compare the resulting local topological obstruction theory with the geometric obstruction theory (i.e., the obstruction to the deformation of $D$ as an effective Cartier divisor of a first order infinitesimal deformations of $X$). We apply this to study the jumping locus of families of linear systems and the Noether-Lefschetz locus. Finally, we give examples of first order deformations $X_t$ of $X$ for which the cohomology class $[D]$ deforms as a Hodge class but $D$ \emph{does not} lift as an effective Cartier divisor of $X_t$.
publishDate 2020
dc.date.none.fl_str_mv 2020
2020
2020
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.11824/1201
https://doi.org/10.1007/s13163-020-00376-6
url http://hdl.handle.net/20.500.11824/1201
https://doi.org/10.1007/s13163-020-00376-6
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv https://link.springer.com/article/10.1007/s13163-020-00376-6
info:eu-repo/grantAgreement/EC/FP7/615655
dc.rights.none.fl_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Reconocimiento-NoComercial-CompartirIgual 3.0 España
http://creativecommons.org/licenses/by-nc-sa/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:BIRD. BCAM's Institutional Repository Data
instname:Basque Center for Applied Mathematics (BCAM)
instname_str Basque Center for Applied Mathematics (BCAM)
reponame_str BIRD. BCAM's Institutional Repository Data
collection BIRD. BCAM's Institutional Repository Data
repository.name.fl_str_mv
repository.mail.fl_str_mv
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