The generalized K\"ahler Calabi-Yau problem

TL;DR

This paper extends the Calabi conjecture to generalized K"ahler geometry, proving existence, uniqueness, and convergence of solutions and flows, with applications to hyperK"ahler manifolds and new structural lemmas.

Contribution

It introduces a generalized Calabi-Yau problem, establishes a transgression formula, and proves global existence and convergence of the generalized K"ahler-Ricci flow.

Findings

Solutions are classical K"ahler Calabi-Yau structures.

The flow converges to a unique fixed point in the generalized K"ahler class.

Established a new $dd^c$-lemma for commuting-type structures.

Abstract

We formulate an extension of the Calabi conjecture to the setting of generalized K\"ahler geometry. We show a transgression formula for the Bismut Ricci curvature in this setting, which requires a new local Goto/Kodaira-Spencer deformation result, and use it to show that solutions of the generalized Calabi-Yau equation on compact manifolds are classically K\"ahler, Calabi-Yau, and furthermore unique in their generalized K\"ahler class. We show that the generalized K\"ahler-Ricci flow is naturally adapted to this conjecture, and exhibit a number of a priori estimates and monotonicity formulas which suggest global existence and convergence. For initial data in the generalized K\"ahler class of a K\"ahler Calabi-Yau structure we prove the flow exists globally and converges to this unique fixed point. This has applications to understanding the space of generalized K\"ahler structures, and as a special case yields the topological structure of natural classes of Hamiltonian symplectomorphisms on hyperK\"ahler manifolds. In the case of commuting-type generalized K\"ahler structures we establish global existence and convergence with arbitrary initial data to a K\"ahler, Calabi-Yau metric, which yields a new $d d^c$-lemma for these structures.