Special Lagrangian submanifolds of log Calabi–Yau manifolds

Tristan C. Collins Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Adam Jacob Department of Mathematics, University of California, Davis, Davis, California, USA Yu-Shen Lin Department of Mathematics, Boston University, Boston, Massachusetts, USA

Differential Geometry mathscidoc:2203.10003

Duke Math. J., 170, (7), 1291-1375, 2021.5
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∈|-KY| is a smooth divisor with D^2=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 and Yau. In the special case that Y is a rational elliptic surface or Y=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 π: Y →P^1.
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  title={Special Lagrangian submanifolds of log Calabi–Yau manifolds},
  author={Tristan C. Collins, Adam Jacob, and Yu-Shen Lin},
  booktitle={Duke Math. J.},
Tristan C. Collins, Adam Jacob, and Yu-Shen Lin. Special Lagrangian submanifolds of log Calabi–Yau manifolds. 2021. Vol. 170. In Duke Math. J.. pp.1291-1375. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220316105944543472976.
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