Abstract: In Kähler geometry, especially in the questions of existence of canonical Kähler metrics, the volume form \(\omega^n\) associated with a Kähler form \(\omega\) plays a central role. In weighted Kähler geometry, we consider a Kähler 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 Kähler metrics: weighted solitons and weighted cscK metrics (introduced by Lahdili), which include most classical canonical Kähler 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 Kähler geometry is that, under assumptions on the principal bundle, the Kähler geometry of the total space reduces to the weighted Kähler geometry of the fiber.

Biography: Thibaut Delcroix is Maître 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 Kähler metrics on manifolds with many symmetries.

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