The branched deformations of the special Lagrangian submanifolds

Abstract

The branched deformations of immersed compact special Lagrangian submanifolds are studied in this paper. If there exists a nondegenerate ${\mathbb{Z}}_2$ harmonic 1-form over a special Lagrangian submanifold $L$, we construct a family of immersed special Lagrangian submanifolds ${\tilde{L}}_t$, that are diffeomorphic to a branched covering of $L$ and ${\tilde{L}}_t$ convergence to $2L$ as current. This answers a question suggested by Donaldson. Combining with the work of Abouzaid and Imagi, we obtain constraints for the existence of nondegenerate ${\mathbb{Z}}_2$ harmonic 1-forms.