Alpha invariant and K-stability of Q-Fano varieties

By Y. Odaka, Yuji Sano. 2010

86 citations recorded by the source corpus.

Abstract We give a purely algebro–geometric proof that if the α-invariant of a Q -Fano variety X is greater than dim X / ( dim X + 1 ) , then ( X , O X ( − K X ) ) is K-stable. The key of our proof is a relation among the Seshadri constants, the α-invariant and K-stability. It also gives applications concerning the automorphism group.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ba248fc0aa70e378bcdc7b9a2976899c3d64f5d7

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Uniform stability of twisted constant scalar curvature K\"ahler metrics (2014, background): “Again when T = OX , the uniformity that we prove strengthens work of OdakaSano [35] when L = −KX and the author [12] for general L.”
A non-Archimedean approach to K-stability (2018, methodology): “An algebraic version of this was proved by Odaka and Sano [OS12]; see also [Der16, BHJ17].”
K-stability of Fano varieties: an algebro-geometric approach (2020, methodology): “ where it is shown that a del Pezzo manifold Xof degree ﬁve with dimension four or ﬁve are K-unstable. Notes on history For a long time, Tian’s α-invariant criterion, established in [Tia87] (see also [OS12]), and its equivariant version was the main tool that people used to verify the existence of a”
A generalization of the Ross--Thomas slope theory (2009, methodology): “We refer the reader to [22], [23] and [24] as for further appli cations of the formula 3.”
The Calabi conjecture and K-stability (2010, methodology): “our of the Chow weights in [Od09a] with respect to the twists of polarizations (from which we derived the concept of the S-coeﬃcients originally). 2http://www.kurims.kyoto-u.ac.jp/~yodaka 6 YUJIODAKA [OS10] is a joint paper with Yuji Sano whose contents are applicaions of our formula of the Donaldso”
Uniform K-stability, Duistermaat-Heckman measures and singularities of pairs (2015, result): “This result thus gives ‘uniform versions’ of [61, 65].”
K-stability of Fano varieties via admissible flags (2020, unknown): “There are also a few variants [15, 21, 41] of Tian’s criterion, and a notable application is the K-stability of smooth hypersurfaces of Fano index one [21].”