The Geometry of Three-Forms on Symplectic Six-Manifolds

TL;DR

This paper explores the rich geometric structures associated with 3-forms on symplectic six-manifolds, revealing connections to Calabi-Yau degenerations and the SYZ conjecture, with implications for understanding canonical structures via the Type IIA flow.

Contribution

It introduces a new perspective on the geometry of 3-forms on symplectic 6-manifolds and links their degenerations to Calabi-Yau structures and the SYZ conjecture.

Findings

Rich geometric structures linked to unstable 3-forms

Degeneration of Calabi-Yau structures influences geometry

Type IIA flow detects canonical structures

Abstract

In this paper, we investigate the geometries associated with 3-forms of various orbital types on a symplectic 6-manifold. We show that there are extremely rich geometric structures attached to certain unstable 3-forms arising naturally from degeneration of Calabi-Yau structures, which in turn provides us a new perspective towards the SYZ conjecture. We give concrete examples and demonstrate that the limiting behavior of the Type IIA flow can be used to detect canonical geometric structures on symplectic manifolds.