K-stability of smooth Fano SL2-threefolds

Abstract: There has been much interest in K-stability since it was shown to be equivalent to the existence of Kähler-Einstein metrics by Chen-Donaldson-Sun. The theory of K-stability is now well developed, but practical methods to check whether a given variety is K-stable are hard to come by. Equivariant K-stability, introduced by Datar-Székelyhidi, makes finding such criteria easier for varieties with large automorphism groups.

If an algebraic group G acts on a variety X, the complexity of the action is the minimal codimension in X of the orbits of a Borel subgroup B of G (e.g. if T is a torus, the complexity zero T-varieties are the toric varieties). Conditions for K-stability have been found for toric varieties by Wang-Zhu, for complexity one T-varieties by Ilten-Süss and for all complexity zero varieties by Delcroix.

We will discuss the combinatorial description due to Timashev of complexity one G-varieties, and describe a practical method to check K-stability in the particular case of smooth Fano SL2-threefolds. In particular, this method proves the K-stability of several varieties not previously known to be K-stable, e.g. projective 3-space blown up along three disjoint lines. This is joint work with Hendrik Süss.

algebraic geometry

Audience: researchers in the topic

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ZAG (Zoom Algebraic Geometry) seminar

Series comments: Description: ZAG seminar

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