Cyclic A-infinity Algebras and Calabi--Yau Structures in the Analytic Setting

This paper considers $A_\infty$-algebras whose higher products satisfy an
analytic bound with respect to a fixed norm. We define a notion of right
Calabi--Yau structures on such $A_\infty$-algebras and show that these give
rise to cyclic minimal models satisfying the same analytic bound. This
strengthens a theorem of Kontsevich--Soibelman, and yields a flexible method
for obtaining analytic potentials of Hua-Keller.
We apply these results to the endomorphism DGAs of polystable sheaves
considered by Toda, for which we construct a family of such right CY structures
obtained from analytic germs of holomorphic volume forms on a projective
variety. As a result, we can define a canonical cyclic analytic
$A_\infty$-structure on the Ext-algebra of a polystable sheaf, which depends
only on the analytic-local geometry of its support. This shows that the results
of Toda can be extended to the quasi-projective setting, and yields a new
method for comparing cyclic $A_\infty$-structures of sheaves on different
Calabi--Yau varieties.