The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces
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Citation (published version)Tristan Collins, Adam Jacob, Yu-Shen Lin, Yu-Shen Lin. "The SYZ Mirror Symmetry Conjecture for Del Pezzo Surfaces and Rational Elliptic Surfaces." https://arxiv.org/abs/2012.05416
We prove the Strominger-Yau-Zaslow mirror symmetry conjecture for non-compact Calabi-Yau surfaces arising from, on the one hand, pairs (Ŷ, Ď) of a del Pezzo surface Ŷ and Ď a smooth anticanonical divisor and, on the other hand, pairs (Y, D) of a rational elliptic surface Y, and D a singular fiber of Kodaira type I_k. Three main results are established concerning the latter pairs (Y, D). First, adapting work of Hein , 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 an optimal uniqueness theorem to the effect that, modulo automorphisms, every Kähler class on Y n 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 a question of Tian-Yau  and settles a folklore conjecture of Yau  in this setting. Thirdly, building on the authors' previous work , we prove that Y ⧵D equipped with an asymptotically semi-flat Calabi-Yau metric !CY admits a special Lagrangian fibration whenever the de Rham cohomology class of WCY is not topologically obstructed. Combining these results we define a mirror map from the moduli space of del Pezzo pairs (Ŷ, Ď) to the complexified Kähler moduli of (Y, D) and prove that the special Lagrangian fibration on (Ŷ, Ď) is T-dual to the special Lagrangian fibration on (Ŷ, Ď) previously constructed by the authors in . We give some applications of these results, including to the study of automorphisms of del Pezzo surfaces fixing an anti-canonical divisor.