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

To every scheme, not necessarily smooth neither proper, we can associate its different mixed realizations (de Rham, Betti, étale, 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 differential graded (dg) categories. This leads to new additive invariants of dg categories, 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.