Mellin transformation, propagation, and abelian duality spaces

TL;DR

This paper demonstrates that perverse sheaves on complex affine tori satisfy propagation properties, and applies these results to study homological duality and abelian duality spaces, including new examples and obstructions.

Contribution

It establishes propagation properties for perverse sheaves on affine tori using Mellin transformation and applies these to characterize abelian duality spaces and their obstructions.

Findings

Perverse sheaves on affine tori satisfy propagation and vanishing properties.

Complex abelian varieties are uniquely characterized as abelian duality spaces among projective manifolds.

New examples of abelian duality spaces are constructed under certain conditions.

Abstract

For arbitrary field coefficients $\mathbb{K}$, we show that $\mathbb{K}$-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 $\mathbb{K}$-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.