Torsion points on cohomology support loci: From D-modules to Simpson's theorem

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Abstract We study cohomology support loci of regular holonomic D -modules on complex abelian varieties, and obtain conditions under which each irreducible component of such a locus contains a torsion point. One case is that both the D -module and the corresponding perverse sheaf are defined over a number field; another case is that the D -module underlies a graded-polarizable mixed Hodge module with a ℤ-structure. As a consequence, we obtain a new proof for Simpson's result that Green–Lazarsfeld sets are translates of subtori bytorsion points. 1 Overview 1.1 Introduction Let X be a projective complex manifold. In their two influential papers about the generic vanishing theorem [6, 7], Green and Lazarsfeld showed that the so-called cohomology support loci are finite unions of translates of subtori of Pic 0 ( X ). Beauville and Catanese [2]conjectured that the translates are always by torsion points , and this was proved by Simpson [19] with the help of the Gelfond–Schneider theorem from transcendental number theory. There is also a proof using positive characteristic methods by Pink and Roessler [13]. Over the past 10 years, the results of Green and Lazarsfeld have been reinterpreted and generalized several times [8, 14, 17], and we now understand that they are consequences of a general theory of holonomic D -modules on abelian varieties. The purpose of this paper is to investigate under what conditions the result about torsion points on cohomology support loci remains true in that setting. One application is a new proof for the conjecture by Beauville and Catanese that does not use transcendental number theory or reduction to positive characteristic. Note: In a recent preprint [20], Wang extends Theorem 1.4 to polarizable Hodge modules on compact complex tori; as a corollary, he proves the conjecture of Beauville and Catanese for arbitrary compact Kahler manifolds. 1.2 Cohomology support loci for D -modules Let A be a complex abelian variety, and let M be a regular holonomic D A -module; recall that a D -module is called holonomic if its characteristic varietyis a union of Lagrangian subvarieties of the cotangent bundle.