The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces
Collins, Tristan; Jacob, Adam; Lin, Yu-Shen
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 [25], 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
[40] and settles a folklore conjecture of Yau [43] in this setting. Thirdly,
building on the authors' previous work [9], 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 [9]. We give some applications of these results, including to the study of automorphisms of del Pezzo surfaces fixing an anti-canonical divisor.
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