Geometry, Topology and Dynamics Seminar
Weighted cscK metrics and weighted K-stability
Abstract: We will introduce a notion of a K\"ahler metric with constant weighted scalar curvature on a compact K\"ahler 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 of $X$. We will also define a notion of weighted Mabuchi energy adapted to our setting, and of a weighted Futaki invariant of a $\T$-compatible smooth K\"ahler test configuration associated to $(X, \T)$. After that, using the geometric quantization scheme of Donaldson, we will show that if a projective manifold admits in the corresponding Hodge K\"ahler class a K\"ahler metric with constant weighted scalar curvature, then this metric minimizes the weighted Mabuchi energy, which implies a suitable notion of weighted K-semistability. As an application, we describe the K\"ahler classes on a geometrically ruled complex surface of genus greater than 2, which admits conformally K\"ahler Einstein-Maxwell metrics.
Monday September 9, 2019 at 3:00 PM in 636 SEO