Hermitian-symplectic and Kahler structures on degenerate twistor deformations
Andrey Soldatenkov, Misha Verbitsky
TL;DR
The paper proves that degenerate twistor deformations of compact holomorphically symplectic manifolds with a holomorphic Lagrangian fibration remain Kähler. It first shows each fiber is Hermitian symplectic via positive-current methods and Sullivan’s criterion, then uses a Teichmüller-space non-separation argument (à la Huybrechts) together with Barlet-space and $d$-commendability techniques to propagate Kählerness along the entire degenerate twistor family. A Tate–Shafarevich twist viewpoint is discussed, providing an independent perspective beyond non-M-speciality assumptions. The results extend known Kählerness results for Tate–Shafarevich twists to arbitrary compact hyperkähler manifolds and contribute to the deformation theory of holomorphic symplectic manifolds, with implications for the Streets–Tian conjecture and Fujiki class C stability under degenerations.
Abstract
Let $(M, Ω)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $π: M \to B$, and $η$ a closed $(1,1)$-form on $B$. Then $Ω+ π^* η$ is a holomorphically symplectic form on a complex manifold which is called the degenerate twistor deformation of $M$. We prove that degenerate twistor deformations of compact holomorphically symplectic Kähler manifolds are also Kähler. First, we prove that degenerate twistor deformations are Hermitian symplectic, that is, tamed by a symplectic form; this is shown using positive currents and an argument based on the Hahn--Banach theorem, originally due to Sullivan. Then we apply a version of Huybrechts's theorem showing that two non-separated points in the Teichmüller space of holomorphically symplectic manifolds correspond to bimeromorphic manifolds if they are Hermitian symplectic.