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-Yau. We prove that if X is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty 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∈|−KY| is a smooth divisor with D2=d, then X=Y∖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-Yau. In the special case that Y is a rational elliptic surface, or Y=ℙ2 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 Id fiber appearing as a singular fiber in a rational elliptic surface πˇ:Yˇ→ℙ1.