  We prove that the cyclic homology of a saturated $A_\infty$ category admits
the structure of a `polarized variation of Hodge structures', building heavily
on the work of many authors: the main point of the paper is to present complete
proofs, and also explicit formulae for all of the relevant structures. This
forms part of a project of Ganatra, Perutz and the author, to prove that
homological mirror symmetry implies enumerative mirror symmetry.
