The Alexandrov-Fenchel type inequalities, revisited

Various Alexandrov-Fenchel type inequalities have appeared and played important roles in convex geometry, matrix theory and complex algebraic geometry. It has been noticed for some time that they share some striking analogies and have intimate relationships. The purpose of this article is to shed new light on this by comparatively investigating them in several aspects. \emph{The principal result} in this article is a complete solution to the equality characterization problem of various Alexandrov-Fenchel type inequalities for intersection numbers of nef and big classes on compact K\"{a}hler manifolds, extending earlier results of Boucksom-Favre-Jonsson, Fu-Xiao and Xiao-Lehmann. Our proof combines a result of Dinh-Nguy\^{e}n on K\"{a}hler geometry and an idea in convex geometry tracing back to Shephard. In addition to this central result, we also give a geometric proof of the complex version of the Alexandrov-Fenchel type inequality for mixed discriminants and a determinantal type generalization of various Alexandrov-Fenchel type inequalities.Comment: 18 pages, slightly revised version stressing our principal result, comments welcom

Special Hermitian metrics on compact solvmanifolds

We review some constructions and properties of complex manifolds admitting pluriclosed and balanced metrics. We prove that for a 6-dimensional solvmanifold endowed with an invariant complex structure J having holomorphically trivial canonical bundle the pluriclosed flow has a long time solution for every invariant initial datum. Moreover, we state a new conjecture about the existence of balanced and SKT metrics on compact complex manifolds. We show that the conjecture is true for nilmanifolds of dimension 6 and 8 and for 6-dimensional solvmanifolds with holomorphically trivial canonical bundle.Comment: 16 pages. To appear in a special issue of the Journal of Geometry and Physic