The Branched Deformations of the Special Lagrangian Submanifolds

In this paper, we investigate the branched deformations of immersed compact special Lagrangian submanifolds. If there exists a nondegenerate Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2$$\end{document} harmonic 1-form over a special Lagrangian submanifold L, we construct a family of immersed special Lagrangian submanifolds L~t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{L}_t$$\end{document}, that are diffeomorphic to a branched covering of L and L~t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tilde{L}_t$$\end{document} converge to 2L as current. This answers a question suggested by Donaldson (Deformations of multivalued harmonic functions, 2019. arXiv:1912.08274). As a corollary, we discover examples of special Lagrangian submanifolds that are rigid in the classical sense but exhibit branched deformations. In conjunction with the work of Abouzaid and Imagi in Nearby special lagrangians, 2021. arXiv:2112.10385, we derive constraints on the existence of nondegenerate Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2$$\end{document} harmonic 1-forms.