【摘要】
In K?hler geometry, especially in the questions of existence of canonical K?hler metrics, the volume form \(\omega^n\) associated with a K?hler form \(\omega\) plays a central role. In weighted K?hler geometry, we consider a K?hler manifold \(X\) equipped with a Hamiltonian action of a torus \(T\), a moment map \(\mu\) and associated moment polytope \(\Delta=\mu(X)\). We fix a weight function \(v:\Delta \to (0,+\infty)\), then replace the volume form \(\omega^n\) by \(v\circ \mu \omega^n\). One can then define new canonical K?hler metrics: weighted solitons and weighted cscK metrics (introduced by Lahdili), which include most classical canonical K?hler metrics.
I will first introduce this weighted setting, then the (analytic) weighted delta invariant, a number that encodes the existence of weighted solitons. I will then present a sufficient condition of existence of weighted cscK metrics, in line with the J-flow approach of Song-Weinkove. Then I will focus on the semisimple principal fibration cosntruction, a construction of varieties from a principal torus bundle and a fiber. The link with weighted K?hler geometry is that, under assumptions on the principal bundle, the K?hler geometry of the total space reduces to the weighted K?hler geometry of the fiber.
【报告人介绍】
Thibaut Delcroix is Ma?tre de Conférences at Univ. Montpellier, France, since 2019. He obtained his PhD thesis in 2015 under the supervision of Philippe Eyssidieux in Grenoble. His research focuses on complex geometry, especially the existence of canonical K?hler metrics on manifolds with many symmetries.
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