A Künneth theorem for the cyclic homology of A-infinity algebras

The cyclic homology of a $\mathbb{Z}\mathrm{/}2\mathbb{Z}$-graded, smooth and proper $A_{\mathrm{\infty }}$ category satisfying the Hodge-to-de-Rham degeneration property carries the structure of a polarized semi-infinite Hodge structure or the so-called EP-structure. Given two $A_{\mathrm{\infty }}$ algebras $A$ and $B$ with the above conditions, we construct a Künneth map from the tensor product of their cyclic homologies to the cyclic homology of the $A_{\mathrm{\infty }}$ tensor product $A{\otimes }_{\mathrm{\infty }}B$ and show that it respects the EP-structures. As an application, we show that if $A$ and $B$ are equipped with weak Calabi-Yau structures, then $A{\otimes }_{\mathrm{\infty }}B$ also inherits a weak Calabi-Yau structure. Also, we show that the Künneth quasi-isomorphism respects good splittings of the Hodge filtration on $A$ and $B$ compatible with the weak Calabi-Yau structure. Our explicit calculations rely on the combinatorial (tree) description of the tensor product of $A_{\mathrm{\infty }}$ algebras developed in \cite{amorim2016tensor}.