Kähler–Einstein metrics along the smooth continuity method

Abstract

Abstract. We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kähler-Einstein metric. This is a strengthening of the solution of the Yau-Tian-Donaldson conjecture for Fano manifolds by Chen-DonaldsonSun [17], and can be used to obtain new examples of Kähler-Einstein manifolds. We also give analogous results for twisted Kähler-Einstein metrics and Kahler-Ricci solitons.

On K-stability of finite covers

“…The equivariant version of Theorem 2.12 also follows from the work of Datar-Székelyhidi [11], while the non-equivariant version is also a consequence of the proof of the Yau-Tian-Donaldson conjecture [8,36].…”

“…The first, due to DatarSzékelyhidi [11], states that it is enough to consider equivariant test configurations.…”

“…We will need some simplifications of the K-stability condition. The first, due to Datar-Székelyhidi [11], states that it is enough to consider equivariant test configurations. Definition 2.8.…”

Greatest lower bounds on Ricci curvature for Fano T-manifolds of complexity one

“…In [7] Li determined a simple formula for R(X Δ ), where X Δ is the polarized toric Fano manifold determined by a reflexive lattice polytope Δ. This result was later recovered in [4], by Datar and Székelyhidi, using notions of G-equivariant K-stability. Using this same method we obtain an effective formula for manifolds with a torus action of complexity one, in terms of the combinatorial data of its divisorial polytope.…”

Extremal metrics on fibrations

“…The analogue of the Yau–Tian–Donaldson conjecture described in predicts that the existence of a twisted Kähler–Einstein metric (hence twisted cscK) on is equivalent to the map being equivariantly K‐stable. There are now several ways of producing explicit examples of (equivariantly) K‐stable Fano varieties , for example, the threefolds with a two‐dimensional torus action of Ilten–Süss , which admit Kähler–Einstein metrics by the work of Chen–Donaldson–Sun and Datar–Székelyhidi . Thus by analogy, assuming an analogue of the Yau–Tian–Donaldson conjecture for Fano maps can be proved, we hope to produce cscK metrics on fibrations, where the base is a Fano map.…”