Absolute sets and the Decomposition Theorem
arXiv:1702.06267
Abstract
We give a framework to produce constructible functions from natural functors between categories, without need of a morphism of moduli spaces to model the functor. We show using the Riemann-Hilbert correspondence that any natural (derived) functor on constructible sheaves on smooth complex algebraic varieties can be used to construct a special kind of constructible sets, called absolute sets, generalizing a notion introduced by Simpson in presence of moduli. We conjecture that the absolute sets of local systems satisfy a "special varieties package", among which is an analog of the Manin-Mumford, Mordell-Lang, and André-Oort conjectures. The conjecture gives a simple proof of the Decomposition Theorem for all semi-simple perverse sheaves, assuming the Decomposition Theorem for the geometric ones. We prove the conjecture in the rank one case by showing that the closed absolute sets in this case are finite unions of torsion-translated affine tori. This extends a structure result of the authors for cohomology jump loci to any other natural jump loci. For example, to jump loci of intersection cohomology and Leray filtrations. We also show that the Leray spectral sequence for the open embedding in a good compactification degenerates for all rank one local systems at the usual page, not just for unitary local systems.
Last page with corrections to the published version is added. This does not affect the main results and conjectures
References in corpus (7)
- Asymptotic behaviour of tame harmonic bundles and an application to pure twistor -modules
- Rank 3 rigid representations of projective fundamental groups
- A non-archimedean Ax-Lindemann theorem
- Recent results on cohomology jump loci
- Cohomology jump loci of quasi-compact Kähler manifolds
- On the length of perverse sheaves and D-modules
- A note on o-minimal flows and the Ax--Lindemann--Weierstrass theorem for abelian varieties over