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

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