Generic special Lagrangian moduli spaces of a non-K\"ahler Calabi--Yau threefold

Abstract

Given a (possibly non-K\"ahler) Calabi--Yau threefold $(X,\Omega)$, we introduce the notion of a (perturbed) special Lagrangian (SL) submanifold of $(X,\omega,\Omega)$, where $\omega$ is a Hermitian metric on $X$. The equations defining this class of submanifolds reduce to the usual SL equations when $\omega$ is a K\"ahler metric. Using the Sard--Smale technique, we prove the existence of a comeagre set of Hermitian metrics $\omega$ on $X$ such that the moduli space of perturbed SL submanifolds in $(X,\omega,\Omega)$ consists of isolated points.