The nondegenerate generalized K\"ahler Calabi-Yau problem

Abstract

We formulate a Calabi-Yau type conjecture in generalized K\"ahler geometry, focusing on the case of nondegenerate Poisson structure. After defining natural Hamiltonian deformation spaces for generalized K\"ahler structures generalizing the notion of K\"ahler 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-K\"ahler metrics. We furthermore establish a GIT framework for this problem, interpreting solutions of this equation as zeros of a moment map associated to a Hamiltonian action and finding a Kempf-Ness functional. Lastly we indicate the naturality of generalized K\"ahler-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-K\"ahler metrics. On a hyper-K\"ahler background, we establish global existence and weak convergence of the flow.