Extending the categorical Hodge theory beyond geometric origin

Establish a Hodge-theoretic framework for abstract smooth proper S-linear categories (not assumed to be of geometric origin), extending the construction of the variation of Hodge structures Ktop[0](C/S) and its properties currently known for categories that admit embeddings as semiorthogonal components of derived categories of smooth proper S-schemes.

Background

The paper relies on the Hodge theory for smooth proper S-linear categories of geometric origin developed in prior work, which provides a variation of Hodge structures Ktop0 compatible with fibers and the classical theory for varieties.

The authors note that extending this theory to abstract smooth proper S-linear categories would broaden the scope of applications, but this extension is currently conjectural.