The rigidity of Dolbeault-type operators and symplectic circle actions

Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce many identities concerning the weights on the fixed points. In particular, it gives a criterion to detemine whether or not a symplectic circle action with isolated fixed points is Hamiltonian. As applications, we simplify the proofs of some known results related to symplectic circle actions, due to Godinho, Tolman-Weitsman and Pelayo-Tolman, and generalize some of them to more general cases.Comment: 8 pages, title changed slightly, final version to be publishe

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