The degeneration of Calabi-Yau 3-folds via 3-forms and the SYZ conjecture
Teng Fei
TL;DR
The paper develops a framework to study degenerations of Calabi–Yau 3-folds via 3-forms on a 6-dimensional symplectic manifold, emphasizing unstable orbits and their integrability through invariants $K(\varphi)$, $F(\varphi)$ and $Q(\varphi)$. It introduces the notion of a wonderful degeneration, where the real part of a holomorphic volume form converges to an $F$-harmonic orbit $\mathcal{O}_0^+$ on an open dense set, yielding a canonical Lagrangian foliation and, in the torus-fibration case, a semi-flat SYZ picture with Hessian base metrics. Leaves of the foliation carry dual Hessian structures and parallel volume forms, and the base $B$ inherits an $L^2$-type Hessian metric; the framework provides a geometric path from nonlinear Hodge theory to SYZ mirror symmetry, with explicit constructions and nontrivial examples. Through a detailed treatment of the orbit structure, integrability conditions, and degeneration mechanics, the work connects the algebraic classification of 3-forms to concrete geometric structures that realize SYZ-type fibrations in degenerate Calabi–Yau settings.
Abstract
In this paper, we investigate the geometries associated with 3-forms of various orbital types on a symplectic 6-manifold. We demonstrate that certain unstable 3-forms, which naturally emerge from specific degenerations of Calabi-Yau structures, exhibit remarkably rich geometric properties. This, in turn, offers a novel perspective on the SYZ conjecture.