The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces
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2020
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| author | Collins, Tristan C. Jacob, Adam Lin, Yu-Shen |
| author_facet | Collins, Tristan C. Jacob, Adam Lin, Yu-Shen |
| contents | 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 Kähler class. Secondly, we prove a uniqueness theorem to the effect that, modulo automorphisms, every Kähler class on $Y\setminus D$ admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional Kähler moduli space of Calabi-Yau metrics on $Y\setminus D$. 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 $ω_{CY}$ admits a special Lagrangian fibration whenever the de Rham cohomology class of $ω_{CY}$ 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 Kähler 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2012_05416 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces Collins, Tristan C. Jacob, Adam Lin, Yu-Shen Differential Geometry Algebraic Geometry 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 Kähler class. Secondly, we prove a uniqueness theorem to the effect that, modulo automorphisms, every Kähler class on $Y\setminus D$ admits a unique asymptotically semi-flat Calabi-Yau metric. This result yields a finite dimensional Kähler moduli space of Calabi-Yau metrics on $Y\setminus D$. 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 $ω_{CY}$ admits a special Lagrangian fibration whenever the de Rham cohomology class of $ω_{CY}$ 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 Kähler 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. |
| title | The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces |
| topic | Differential Geometry Algebraic Geometry |
| url | https://arxiv.org/abs/2012.05416 |