K-semistability for irregular Sasakian manifolds

Abstract

We introduce a notion of K-semistability for Sasakian manifolds. This extends to the irregular case the orbifold K-semistability of Ross-Thomas. Our main result is that a Sasakian manifold with constant scalar curvature is necessarily K-semistable. As an application, we show how one can recover the volume minimization results of Martelli-Sparks-Yau, and the Lichnerowicz obstruction of Gauntlett-Martelli-Sparks-Yau from this point of view.

Contact Structures of Sasaki Type and Their Associated Moduli

“…However, to treat the general Sasaki metrics one needs to work on the affine cone as described in Section 2.2. This was done by Collins and Székelyhidi [CS18] and it is this approach that we follow here. We begin by defining two important functionals, the total volume and the total transverse scalar curvature, viz.…”

An application of the Duistermaat–Heckman theorem and its extensions in Sasaki geometry

“…Thus, the Sasaki cone is associated with an isotopy class of contact structures of Sasaki type that is invariant under T . This fact has given rise to several equivalent definitions of t + , see [26,14,13], and is also called the Reeb cone.…”

“…The general problem motivating our work is: given a polarized Sasaki type manifold (N 2n+1 , ξ), does there exist a compatible constant scalar curvature Sasaki (cscS for short) metric? This is a hard problem and the answer is conjecturally related to some notion of K-stability see [14] and is closely related to the analogous problem in (compact) Kähler geometry, see for eg. [16,17,35,36].…”

Volume minimization and obstructions to solving some problems in Kähler geometry

“…In Kähler geometry, the Yau-Tian-Donaldson conjecture relates the existence problem of constant scalar curvature Kähler (cscK for short) metrics to K-stablity. Simlilarly in Sasakian geometry, the existence problem of constant scalar curvature Sasaki (cscS for short) metrics is related to K-stablity, see [14], [15], [49], [10] for example.…”