2020/12/10 by Collins, Tristan C., Jacob, Adam, Lin, Yu-Shen · 1 citation
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.05416
We prove a version of the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs (\checkY,\checkD) of a del Pezzo surface \checkY and \checkD a smooth anti-canonical divisor and, on the other hand, pairs (Y,D) of a rational elliptic surface Y, and D a singular fiber of Kodaira type Ik. Three main results are established concerning the latter pairs (Y,D). First, adapting work of Hein \citeHein, we prove the existence of a complete Calabi-Yau metric on Y∖ D asymptotic to a (generically non-standard) semi-flat metric in every Kähler class. Secondly, we prove a uniqueness theorem to the effect that, modulo automorphisms, every Kähler class on Y∖ D admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional Kähler moduli space of Calabi-Yau metrics on Y∖ D. Further, this result answers, in this setting, questions of Tian-Yau and Yau. Thirdly, building on the authors' previous work, we prove that Y∖ D equipped with an asymptotically semi-flat Calabi-Yau metric ωCY admits a special Lagrangian fibration whenever the de Rham cohomology class of ωCY is not topologically obstructed. Combining these results we define a mirror map from the moduli space of del Pezzo pairs (\checkY, \checkD) to the complexified Kähler moduli of (Y,D) and prove that the special Lagrangian fibration on (Y,D) is T-dual to the special Lagrangian fibration on (\checkY, \checkD) previously constructed by the authors. We give some applications of these results, including to the study of automorphisms of del Pezzo surfaces fixing an anti-canonical divisor.