Special Lagrangian submanifolds of log Calabi–Yau manifolds

Abstract

We study the existence of special Lagrangian submanifolds of log Calabi–Yau manifolds equipped with the complete Ricci-flat Kähler metric constructed by Tian and Yau. We prove that if X is a Tian–Yau manifold and if the compact Calabi–Yau manifold at infinity admits a single special Lagrangian, then X admits infinitely many disjoint special Lagrangians. In complex dimension 2, we prove that if Y is a del Pezzo surface or a rational elliptic surface and $D\in |-K_Y|$ is a smooth divisor with $D^2=d$, then $X=Y\setminus D$ admits a special Lagrangian torus fibration, as conjectured by Strominger–Yau–Zaslow and Auroux. In fact, we show that X admits twin special Lagrangian fibrations, confirming a prediction of Leung and Yau. In the special case that Y is a rational elliptic surface or $Y={\mathbb{P}}^2$, we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kähler rotation, X can be compactified to the complement of a Kodaira type $I_d$ fiber appearing as a singular fiber in a rational elliptic surface $\check{\mathrm{\pi }}:\check{Y}\to {\mathbb{P}}^1$.