Non-Abelian Lefschetz hyperplane theorems

Litt, Daniel

doi:10.1090/jag/704

Author: Daniel Litt
Journal: J. Algebraic Geom. 27 (2018), 593-646
DOI: https://doi.org/10.1090/jag/704
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 \begin{equation*}r: \mathrm {Hom}(X, Y)\to \mathrm {Hom}(D, Y) \end{equation*} 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
MR Author ID: 916147
ORCID: 0000-0003-2273-4630
Email: dlitt@math.columbia.edu

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.