Hermitian-symplectic and Kahler structures on degenerate twistor deformations

TL;DR

This paper demonstrates that degenerate twistor deformations of compact holomorphic symplectic Kähler manifolds remain Kähler, establishing their Hermitian symplectic nature and linking them to bimeromorphic equivalence via Teichmüller space analysis.

Contribution

It proves that degenerate twistor deformations are Hermitian symplectic and Kähler, extending known structures and applying Teichmüller space techniques to these deformations.

Findings

Degenerate twistor deformations are Hermitian symplectic.

Such deformations of compact holomorphic symplectic Kähler manifolds are also Kähler.

Application of Huybrechts's theorem relates non-separated points to bimeromorphic manifolds.

Abstract

Let $(M, \Omega)$ be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $\pi: M \to B$, and $\eta$ a closed $(1,1)$-form on $B$. Then $\Omega+ \pi^* \eta$ 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 K\"ahler manifolds are also K\"ahler. 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 Teichm\"uller space of holomorphically symplectic manifolds correspond to bimeromorphic manifolds if they are Hermitian symplectic.