The nondegenerate generalized Kähler Calabi–Yau problem

Abstract

Abstract We formulate a Calabi–Yau-type conjecture in generalized Kähler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized Kähler structures generalizing the notion of Kähler class, we conjecture unique solvability of Gualtieri’s Calabi–Yau equation within this class. We establish the uniqueness, and moreover show that all such solutions are actually hyper-Kähler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeroes of a moment map associated to a Hamiltonian action and finding a Kempf–Ness functional. Lastly we indicate the naturality of generalized Kähler–Ricci flow in this setting, showing that it evolves within the given Hamiltonian deformation class, and that the Kempf–Ness functional is monotone, so that the only possible fixed points for the flow are hyper-Kähler metrics. On a hyper-Kähler background, we establish global existence and weak convergence of the flow.