9 nov 2022 -- 11:30
Aula Tricerri, DIMaI, Firenze
Seminario di Geometria del Dini
Abstract.
A systematic treatment of a Hilbert criterion for stability theory for an action of a real reductive group $G$ on a real submanifold $X$ of a K\"ahler manifold $Z$ is presented. More precisely, suppose that the action of a compact Lie group with Lie algebra $\mathfrak u$ extends holomorphically to an action of the complexified group $U^{\mathbb{C}}$ and that the $U$-action on $Z$ is Hamiltonian. If $G \subset U^{\mathbb{C}}$ is compatible, there is a corresponding gradient map $\mu_{\mathfrak p}: X \to \mathfrak p$, where $\mathfrak g = \mathfrak k \oplus \mathfrak p$ is a Cartan decomposition of the Lie algebra of $G$. The concept of energy complete action of $G$ on $X$ is introduced. For such actions, one can characterize stability, semistability and polystability of a point by a numerical criteria using a $G$-equivariant function called maximal weight. We prove the classical Hilbert-Mumford criteria for semistability and polystability conditions. This is joint work with Leonardo Biliotti.