Poisson K-stability and the semiclassical Yau--Tian--Donaldson correspondence

Abstract

We introduce a notion of K-polystability for compact Kähler holomorphic Poisson manifolds. On the one hand, this notion of stability is well-adapted to constructions of moduli spaces. For instance, when the underlying manifold is K-polystable with reductive reduced automorphism group, Poisson K-stability is equivalent to geometric invariant theoretic stability in the space of Poisson bivectors, but there also exist K-unstable varieties that become stable after incorporating a Poisson structure. On the other hand, the Poisson K-stability condition interacts well with generalized Käher metrics -- the background geometry of (2,2) supersymmetric string theory. In particular, we conjecture that Poisson K-polystability characterizes the existence of constant scalar curvature symplectic generalized Kähler structures with a sufficiently small Poisson tensor -- a natural extension of the Yau--Tian--Donaldson (YTD) conjecture. Our main result is a proof of the existence part of this ``semiclassical YTD conjecture'' for Poisson structures on Kähler--Einstein Fano manifolds, using infinite-dimensional momentum map techniques. In this way, we obtain the existence of many new examples of symplectic generalized Kähler structure of constant scalar curvature, and prove the conjecture completely in the case of the projective plane.