Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Non-Abelian Lefschetz hyperplane theorems

Author: Daniel Litt
Journal: J. Algebraic Geom. 27 (2018), 593-646
Published electronically: May 17, 2018
MathSciNet review: 3846549
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Abstract | References | Additional Information

Abstract: Let $ X$ be a smooth projective variety over the complex numbers, and let $ D\subset X$ be an ample divisor. For which spaces $ Y$ is the restriction map

$\displaystyle r: \textup {Hom}(X, Y)\to \textup {Hom}(D, Y)$    

an isomorphism?

Using positive characteristic methods, we give a fairly exhaustive answer to this question. An example application of our techniques is: if $ \dim (X)\geq 3$, $ Y$ is smooth, $ \Omega ^1_Y$ is nef, and $ \dim (Y)< \dim (D),$ the restriction map $ r$ is an isomorphism. Taking $ Y$ to be the classifying space of a finite group $ BG$, the moduli space of pointed curves $ \mathscr {M}_{g,n}$, the moduli space of principally polarized Abelian varieties $ \mathscr {A}_g$, certain period domains, and various other moduli spaces, one obtains many new and classical Lefschetz hyperplane theorems.

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Additional Information

Daniel Litt
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027

Received by editor(s): April 6, 2016
Received by editor(s) in revised form: November 28, 2016
Published electronically: May 17, 2018
Article copyright: © Copyright 2018 University Press, Inc.