Generic vanishing for semi-abelian varieties and integral Alexander modules

Advances in Mathematics

Wang

2018

Self Cite

For arbitrary field coefficients K, we show that K-perverse sheaves on a complex affine torus satisfy the so-called propagation package, i.e., the generic vanishing property and the signed Euler characteristic property hold, and the corresponding cohomology jump loci satisfy the propagation property and codimension lower bound. The main ingredient used in the proof is Gabber-Loeser's Mellin transformation functor for K-constructible complexes on a complex affine torus, and the fact that it behaves well with respect to perverse sheaves.As a concrete topological application of our sheaf-theoretic results, we study homological duality properties of complex algebraic varieties, via abelian duality spaces. We provide new obstructions on abelian duality spaces by showing that their cohomology jump loci satisfy a propagation package. This is then used to prove that complex abelian varieties are the only complex projective manifolds which are abelian duality spaces. We also construct new examples of abelian duality spaces. For example, we show that if a smooth quasi-projective variety X satisfies a certain Hodge-theoretic condition and it admits a proper semi-small map (e.g., a closed embedding or a finite map) to a complex affine torus, then X is an abelian duality space.

mentioning

confidence: 60%

“…Assume that n > r(f ). Then it was shown in [LMW17] that the induced homomorphism f * : π 1 (X) → π 1 (T ) is non-trivial, and let us denote its image by Z m (this image is a free abelian group).…”

Section: 2mentioning

confidence: 99%

Mellin transformation, propagation, and abelian duality spaces

Liu

Advances in Mathematics

Wang

2018

Self Cite

“…The proof of (1) in [8] is derived from a Riemann-Roch-type formula for constructible sheaf complexes on a semi-abelian variety. The result also follows from the generic vanishing theorems for perverse sheaves [19,22], see also [1,29,32] for the case of abelian varieties, while the corresponding statement for the Euler characteristic of perverse sheaves on a complex affine torus was proved in [10,23].…”

mentioning

confidence: 71%

On the signed Euler characteristic property for subvarieties of abelian varieties

Elduque

Geske

2018

jsing

Self Cite

We give an elementary proof of the fact that a pure-dimensional closed subvariety of a complex abelian variety has a signed intersection homology Euler characteristic. We also show that such subvarieties which, moreover, are local complete intersections, have a signed Euler-Poincaré characteristic. Our arguments rely on the construction of circle-valued Morse functions on such spaces, and use in an essential way the stratified Morse theory of Goresky-MacPherson. Our approach also applies (with only minor modifications) for proving similar statements in the analytic context, i.e., for subvarieties of compact complex tori. Alternative proofs of our results can be given by using the general theory of perverse sheaves.Classically, much of the manifold theory, e.g., Morse theory, Lefschetz theorems, Hodge decompositions, and especially Poincaré Duality, is recovered in the singular stratified context if, instead of the usual (co)homology, one uses Goresky-MacPherson's intersection homology groups [12,13]. We recall here the definition of intersection homology of complex analytic (or algebraic) varieties, and discuss some preliminary results concerning the corresponding intersection homology Euler characteristic. For more details on intersection homology, the reader may consult, e.g., [9] and the references therein.Let X be a purely n-dimensional complex analytic (or algebraic) variety with a fixed Whitney stratification. All strata of X are of even real (co)dimension. By [11], X admits a triangulation which is compatible with the stratification, so X can also be viewed as a PL stratified pseudomanifold. Let (C * (X), ∂) denote the complex of finite PL chains on X, with Z-coefficients. The intersection homology groups of X, denoted IH i (X), are the homology groups of a complex of "allowable chains", defined by imposing restrictions on how chains meet the singular strata. Specifically, the chain complex (IC * (X), ∂) of allowable finite PL chains is defined as follows: if ξ ∈ C i (X) has support |ξ|, then ξ ∈ IC i (X) if, and only if, dim(|ξ| ∩ S) < i − s and dim(|∂ξ| ∩ S) < i − s − 1, for each stratum S of complex codimension s > 0. The boundary operator on allowable chains is induced from the usual boundary operator on chains of X. (The second condition above ensures that ∂ restricts to the complex of allowable chains.)

“…One particular interesting case is when alb is proper and semi-small, in which case r (alb) = 0. It was shown in [23,Remark 1.3] that X admits a proper semi-small map f : X → G to some complex semi-abelian variety G if and only if the Albanese map E n with complement X := E n \ A. Then X is a complex n-dimensional affine variety.…”

Section: Proof (A)mentioning

confidence: 99%

“…This partly motivates our study of cohomology jump loci of constructible complexes, with a view towards a complete characterization of perverse sheaves on complex semi-abelian varieties. Besides providing new obstructions on the cohomology jump loci (hence also on the homotopy type) of smooth complex quasi-projective varieties, such a characterization has other important topological applications, such as finiteness properties of Alexander-type invariants (see, e.g., [23]), or for the study of homological duality properties of complex algebraic varieties.…”

Section: Introductionmentioning

confidence: 99%

Perverse sheaves on semi-abelian varieties

Yong-qiang

Wang

2021

Sel. Math. New Ser.

Self Cite

We give a complete (global) characterization of C-perverse sheaves on semi-abelian varieties in terms of their cohomology jump loci. Our results generalize Schnell's work on perverse sheaves on complex abelian varieties, as well as Gabber-Loeser's results on perverse sheaves on complex affine tori. We apply our results to the study of cohomology jump loci of smooth quasi-projective varieties, to the topology of the Albanese map, and in the context of homological duality properties of complex algebraic varieties.