Hodge-Riemann balanced structures on non-Kähler manifolds, ICMS seminar talk by Asia Mainenti,

A Hodge-Riemann balanced structure on a complex manifold is the datum of a balanced metric whose (n-1)-th power can be decomposed into the wedge product of two differential forms, satisfying the classical Hodge-Riemann bilinear relations. Such structures were introduced by X. Chen and R. Wentworth, to generalize the nonabelian Hodge correspondence to non-Kähler Hermitian metrics. However, there are no known examples of Hodge-Riemann balanced structures on non-Kähler manifolds. The aim of this talk is to address this lack of examples, highlighting the relation with p-Kähler structures and discuss some obstruction results in the class of solvmanifolds. Lastly, we will present the first example of such a structure on a non-Kähler, non-compact complex manifold obtained as the product of the Iwasawa manifold by C.

This is joint work with A. Fino.