Lefschetz theorems on quasi-projective varieties

Hamm, Helmut A.; Tráng, Lê Dũng

doi:10.24033/bsmf.2023

Cited by 40 publications

(37 citation statements)

References 7 publications

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“…This is guaranteed by the results of Flenner and Langer (see the discussion in the beginning of the current setup). (3.2.3) We have the isomorphism π 1 (S reg ) ∼ = π 1 (X reg ), which follows from Lefschetz-hyperplane theorem [HL85] for quasi-projective varieties. 3.B.…”

Section: A General Setupmentioning

confidence: 87%

A Characterization of Finite Quotients of Abelian Varieties

Taji

2016

Int Math Res Notices

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ABSTRACT. We provide a characterization of quotients of Abelian varieties by finite groups actions that are free in codimension-one via vanishing conditions on the orbifold Chern classes. The characterization is given among a class of varieties with singularities that are more general than quotient singularities, namely among the class of klt varieties. Furthermore, for a semistable (respectively stable) reflexive O X -module E with zero first and second orbifold Chern classes over such a variety X, we show that E | X an reg is locallyfree and flat, given by a linear (irreducible unitary) representation of π 1 (X reg ), and that it extends over a finite Galois cover X of X étale over X reg to a locally-free and flat sheaf given by an equivariant linear (irreducible unitary) representation of π 1 ( X). These are generalizations to the singular setting that is more general than any orbifold strengthenings of the classical correspondences of Donaldson-Uhlenbeck-Yau-Simpson.

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Section: A General Setupmentioning

confidence: 87%

A Characterization of Finite Quotients of Abelian Varieties

Taji

2016

Int Math Res Notices

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“…Nous e Âtudions l'homotopie de varie Âte Âs quasi-projectives (c'est-a Á-dire de sousensembles de l'espace projectif complexe P n C de la forme X n A, ou Á X et A sont des sous-ensembles alge Âbriques ferme Âs avec A Ì X ) selon la me Âthode de Lefschetz [23], c'est-a Á-dire en conside Ârant leurs sections par les hyperplans d'un pinceau (tomographie). En particulier, nous aboutissons a Á un the Âore Áme du type de Lefschetz (the Âore Áme 2.2) qui ge Âne Âralise dans une certaine direction les meilleurs re Âsultats connus dus a Á Hamm, Le Ã, Goresky et MacPherson [16,11,12]. Ce the Âore Áme est de Âmontre Â par re Âcurrence sur la dimension de l'espace projectif ambiant qui contient la varie Âte Â a Á partir d'un the Âore Áme sur les «pinceaux de Lefschetz» (the Âore Áme 2.5) qui constitue le re Âsultat principal de l'article.…”

Section: Introductionunclassified

Tomographie Des Variétés Singulières Et Théorèmes De Lefschetz

Eyral¹

2001

Proceedings of the London Mathematical Society

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Nous étudions l'homotopie d'une variété quasi‐projective dans un espace projectif complexe selon la méthode de Lefschetz, c'est‐à‐dire en considérant ses sections par les hyperplans d'un pinceau (tomographie). En particulier, nous aboutissons à un théorème du type de Lefschetz qui généralise dans une certaine direction les meilleurs résultats connus dus à Hamm, Lê, Goresky et MacPherson. Ce théorème est démontré par récurrence sur la dimension de l'espace projectif ambiant à partir d'un théorème sur les pinceaux d'axe générique qui constitue le résultat principal de l'article. Ce dernier compare la topologie de la variété à celle de sa section par un hyperplan générique du pinceau sur la base des comparaisons (section hyperplane générique – section par l'axe du pinceau) et (sections hyperplanes exceptionnelles – section par l'axe); l'incidence des singularités est mesurée par un invariant appelé ‘profondeur homotopique rectifiée globale’ (analogue global de la notion de profondeur homotopique rectifiée de Grothendieck). We study the homotopy of a quasi‐projective variety in a complex projective space following Lefschetz's method, that is, by considering its sections by the hyperplanes of a pencil (tomography). Specifically, we obtain a theorem of Lefschetz type which generalizes in a certain direction the best‐known results due to Hamm, Lê, Goresky and MacPherson. This theorem is proved by induction on the dimension of the ambient projective space with the help of a theorem on pencils with generic axis which is the main result of the paper. The latter compares the topology of the variety with that of its section by a generic hyperplane of the pencil, on the basis of the following comparisons: section by a generic hyperplane with section by the axis of the pencil; and sections by the exceptional hyperplanes with section by the axis. The effect of the singularities is measured by an invariant called ‘global rectified homotopical depth’ (a global analogue of the notion of rectified homotopical depth of Grothendieck). eyral@cmi.univ‐mrs.fr 2000 Mathematics Subject Classification: 32S50, 14F35, 14F17.

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“…Indeed, when d ¿ 3, Corollary 5.3 does not give extra information compared with the non-singular version of the Lefschetz hyperplane section theorem (cf. [12,13,15]) which asserts that if X is non-singular then the pair (X;…”

Section: Statement Of the Main Resultsmentioning

confidence: 99%

“…The singular version of the Lefschetz hyperplane section theorem (cf. [11][12][13]15]) asserts that the pair (X; L ∩ X ) is m-connected for some integer m depending on the singularities of X . In this paper, we are interested in the class of varieties X for which the best known integer m is 1.…”

Section: Introductionmentioning

confidence: 99%

Fundamental groups of singular quasi-projective varieties

Eyral

2004

Topology

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Lefschetz theorems on quasi-projective varieties

Cited by 40 publications

References 7 publications

A Characterization of Finite Quotients of Abelian Varieties

A Characterization of Finite Quotients of Abelian Varieties

Tomographie Des Variétés Singulières Et Théorèmes De Lefschetz

Fundamental groups of singular quasi-projective varieties

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