Thibaut Delcroix, Weighted Kähler geometry and semisimple principal fibrations (Lecture I)

In Kähler geometry, especially in the questions of existence of canonical Kähler metrics, the volume form ωⁿ associated with a Kähler form ω 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 μ and associated moment polytope Δ=μ(X). We fix a weight function v:Δ → (0,+∞), then replace the volume form ωⁿ by v ◦ μ ωⁿ. 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.