Rank One Local Systems and Forms of Degree One

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

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On the fundamental groups of normal varieties

Communications in Contemporary Mathematics ◽

10.1142/s0219199715500650 ◽

2016 ◽

Vol 18 (04) ◽

pp. 1550065 ◽

Cited By ~ 3

Author(s):

Donu Arapura ◽

Alexandru Dimca ◽

Richard Hain

Keyword(s):

Fundamental Groups ◽

Algebraic Varieties ◽

Local Systems ◽

Normal Variety ◽

Normal Complex ◽

Normal Varieties ◽

Rank One

We show that the fundamental groups of normal complex algebraic varieties share many properties of the fundamental groups of smooth varieties. The jump loci of rank one local systems on a normal variety are related to the jump loci of a resolution and of a smoothing of this variety.

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GENERIC VANISHING THEORY VIA MIXED HODGE MODULES

Forum of Mathematics Sigma ◽

10.1017/fms.2013.1 ◽

2013 ◽

Vol 1 ◽

Cited By ~ 10

Author(s):

MIHNEA POPA ◽

CHRISTIAN SCHNELL

Keyword(s):

Abelian Varieties ◽

Local Systems ◽

Coherent Sheaves ◽

Natural Categories ◽

Rank One ◽

Generic Vanishing ◽

Mixed Hodge Modules ◽

Irregular Varieties

AbstractWe extend most of the results of generic vanishing theory to bundles of holomorphic forms and rank-one local systems, and more generally to certain coherent sheaves of Hodge-theoretic origin associated with irregular varieties. Our main tools are Saito’s mixed Hodge modules, the Fourier–Mukai transform for $\mathscr{D}$-modules on abelian varieties introduced by Laumon and Rothstein, and Simpson’s harmonic theory for flat bundles. In the process, we also discover two natural categories of perverse coherent sheaves.

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Étale cohomology of rank one $$\ell $$-adic local systems in positive characteristic

Selecta Mathematica ◽

10.1007/s00029-021-00681-y ◽

2021 ◽

Vol 27 (4) ◽

Author(s):

Hélène Esnault ◽

Moritz Kerz

Keyword(s):

Positive Characteristic ◽

Vanishing Theorem ◽

Deformation Spaces ◽

Local Systems ◽

Etale Cohomology ◽

Lefschetz Theorem ◽

Rank One ◽

Étale Cohomology ◽

Generic Vanishing ◽

Hard Lefschetz Theorem

AbstractWe show that in positive characteristic special loci of deformation spaces of rank one $$\ell $$ ℓ -adic local systems are quasi-linear. From this we deduce the Hard Lefschetz theorem for rank one $$\ell $$ ℓ -adic local systems and a generic vanishing theorem.

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Characteristic varieties and logarithmic differential 1-forms

Compositio Mathematica ◽

10.1112/s0010437x09004461 ◽

2010 ◽

Vol 146 (1) ◽

pp. 129-144 ◽

Cited By ~ 6

Author(s):

Alexandru Dimca

Keyword(s):

Elliptic Curve ◽

Recent Work ◽

Configuration Space ◽

Zero Set ◽

Characteristic Variety ◽

Local Systems ◽

Complex Variety ◽

Rank One ◽

Characteristic Varieties ◽

Resonance Varieties

AbstractWe introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety M. A logarithmic resonance variety is also considered and, as an application, we determine the first characteristic variety of the configuration space of n distinct labeled points on an elliptic curve. Finally, for a logarithmic 1-form α on M we investigate the relation between the resonance degree of α and the codimension of the zero set of α on a good compactification of M. This question was inspired by the recent work by Cohen, Denham, Falk and Varchenko.

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