G-uniform stability and Kähler–Einstein metrics on Fano varieties

By Chi Li. Inventiones Mathematicae, 2019

78 citations recorded by the source corpus.

Let X be any $${{\mathbb {Q}}}$$ Q -Fano variety and $$\mathrm{Aut}(X)_0$$ Aut ( X ) 0 be the identity component of the automorphism group of X . Let $${\mathbb {G}}$$ G be a connected reductive subgroup of $$\mathrm{Aut}(X)_0$$ Aut ( X ) 0 that contains a maximal torus of $$\mathrm{Aut}(X)_0$$ Aut ( X ) 0 . We prove that X admits a Kähler–Einstein metric if and only if X is $${\mathbb {G}}$$ G -uniformly K-stable. This proves a version of Yau–Tian–Donaldson conjecture for arbitrary singular Fano varieties. A key new ingredient is a valuative criterion for $${\mathbb {G}}$$ G -uniform K-stability.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b96b334355c9b4523a76b0c2fc7926007dc2fcef

How researchers see this paper

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In Applied Mathematics

1. Ziquan Zhuang
2. Chenyang Xu
3. Chi Li
4. Feng Wang
5. G. Tian
6. Robert J. Berman
7. Yuchen Liu
8. Mattias Jonsson

In Geometry and Topology

1. Chenyang Xu
2. Ziquan Zhuang
3. Chi Li
4. Feng Wang
5. G. Tian
6. Yuchen Liu
7. Robert J. Berman
8. H. Blum

Citations in the corpus

• K-stability of Fano varieties: an algebro-geometric approach (2020, background): “The equivalence between (1) and (2) are given in [Li19] for test configurations.”
• The Uniform Version of Yau–Tian–Donaldson Conjecture for Singular Fano Varieties (2019, background): “The results of this paper have been generalized to the general case when Aut(X)0 is reductive but not necessarily discrete ([45], see also [40]), and to more general equations of twisted Kähler–Einstein metrics and generalized solitons (see [39]).”
• On positivity of the CM line bundle on K-moduli spaces (2019, methodology): “By [Li19, Lemma 3.15], for any T -equivariant finitely generated filtration F on R0 there exists some ξ ∈ NR such that DNA(F) ≥ η · JNA(Fξ); since (X0,∆0) is K-semistable, we also have S(Fξ) = S(F).”