The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces

Abstract: 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 $(\check{Y},\check{D})$ of a del Pezzo surface $\check{Y}$ and $\check{D}$ 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 $I_k$. Three main results are established concerning the latter pairs $(Y,D)$. First, adapting work of Hein \cite{Hein}, we prove the existence of a complete Calabi-Yau metric on $Y\setminus D$ asymptotic to a (generically non-standard) semi-flat metric in every K\"ahler class. Secondly, we prove a uniqueness theorem to the effect that, modulo automorphisms, every K\"ahler class on $Y\setminus D$ admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional K\"ahler moduli space of Calabi-Yau metrics on $Y\setminus D$. Further, this result answers, in this setting, questions of Tian-Yau and Yau. Thirdly, building on the authors' previous work, we prove that $Y\setminus D$ equipped with an asymptotically semi-flat Calabi-Yau metric $\omega_{CY}$ admits a special Lagrangian fibration whenever the de Rham cohomology class of $\omega_{CY}$ is not topologically obstructed. Combining these results we define a mirror map from the moduli space of del Pezzo pairs $(\check{Y}, \check{D})$ to the complexified K\"ahler moduli of $(Y,D)$ and prove that the special Lagrangian fibration on $(Y,D)$ is $T$-dual to the special Lagrangian fibration on $(\check{Y}, \check{D})$ 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.