## Kähler Metric with Constant Weighted Scalar Curvature and Weighted K-stability

Time: 2018-10-01
Published By: He Liu

**Speaker(s): ** Abdellah Lahdili (Université du Québec à Montréal)

**Time: ** 15:00-17:00 October 10, 2018

**Venue: ** Room 29, Quan Zhai, BICMR

We introduce a notion of a Kähler metric with constant weighted scalar curvature on a compact Kähler manifold X, depending on a fixed real torus T in the reduced group of automorphisms of X, and two smooth (weight) functions, defined on the momentum image (with respect to a given Kähler class on X) of X in the dual Lie algebra of T. A number of natural problems in Kähler geometry, such as the existence of extremal Kähler metrics and conformally Kähler, Einstein{Maxwell metrics, or Kähler-Ricci solitons reduce to the search of Kähler metrics with constant weighted scalar curvature in a given Kähler class, for special choices of the weight functions.

We define a notion of weighted Mabuchi energy adapted to our setting, and of a weighted Futaki invariant of a T-compatible smooth Kähler test configuration associated to (X; T), and show that the boundedness from below of the weighted Mabuchi energy implies a suitable notion of a weighted K-semistability.

We define a notion of weighted Mabuchi energy adapted to our setting, and of a weighted Futaki invariant of a T-compatible smooth Kähler test configuration associated to (X; T), and show that the boundedness from below of the weighted Mabuchi energy implies a suitable notion of a weighted K-semistability.