Modified mixed realizations, new additive invariants, and periods of dg categories

TL;DR

This paper extends classical mixed realizations and period rings from schemes to dg categories, introducing new additive invariants and a theory of periods for dg categories, with applications to invariance under duality.

Contribution

It generalizes mixed realizations and periods to dg categories, providing explicit descriptions and new invariants, expanding the scope of these concepts beyond schemes.

Findings

Extended mixed realizations to dg categories

Computed invariants for differential operators

Proved invariance of period rings under projective homological duality

Abstract

To every scheme, not necessarily smooth neither proper, we can associate its different mixed realizations (de Rham, Betti, etale, Hodge, etc) as well as its ring of periods. In this note, following an insight of Kontsevich, we prove that, after suitable modifications, these classical constructions can be extended from schemes to the broad setting of dg categories. This leads to new additive invariants, which we compute in the case of differential operators, as well as to a theory of periods of dg categories. Among other applications, we prove that the ring of periods of a scheme is invariant under projective homological duality. Along the way, we explicitly describe the modified mixed realizations using the Tannakian formalism.