15 May 2021 Special Lagrangian submanifolds of log Calabi–Yau manifolds
Tristan C. Collins, Adam Jacob, Yu-Shen Lin
Author Affiliations +
Duke Math. J. 170(7): 1291-1375 (15 May 2021). DOI: 10.1215/00127094-2021-0012


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 D2=d, then X=YD 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=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:YˇP1.


Download Citation

Tristan C. Collins. Adam Jacob. Yu-Shen Lin. "Special Lagrangian submanifolds of log Calabi–Yau manifolds." Duke Math. J. 170 (7) 1291 - 1375, 15 May 2021. https://doi.org/10.1215/00127094-2021-0012


Received: 29 October 2019; Revised: 24 September 2020; Published: 15 May 2021
First available in Project Euclid: 14 April 2021

Digital Object Identifier: 10.1215/00127094-2021-0012

Primary: 32Q25
Secondary: 53D37 , 53E10

Keywords: del Pezzo surface , Lagrangian mean curvature flow , mirror symmetry , rational elliptic surface , Special Lagrangian

Rights: Copyright © 2021 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.170 • No. 7 • 15 May 2021
Back to Top