Special Lagrangian submanifolds of log CalabiYau manifolds
Abstract
We study the existence of special Lagrangian submanifolds of log CalabiYau manifolds equipped with the complete Ricciflat Kähler metric constructed by TianYau. We prove that if $X$ is a TianYau manifold, and if the compact CalabiYau 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 $D\in K_{Y}$ is a smooth divisor with $D^2=d$, then $X= Y\backslash D$ admits a special Lagrangian torus fibration, as conjectured by StromingerYauZaslow and Auroux. In fact, we show that $X$ admits twin special Lagrangian fibrations, confirming a prediction of LeungYau. 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 hyperKähler 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 $\check{\pi}: \check{Y}\rightarrow \mathbb{P}^1$.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 DOI:
 10.48550/arXiv.1904.08363
 arXiv:
 arXiv:1904.08363
 Bibcode:
 2019arXiv190408363C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Symplectic Geometry
 EPrint:
 70 pages. Updates and improvements. To appear in Duke Mathematical Journal