论文标题
del pezzo表面和理性椭圆表面的SYZ镜子对称性猜想
The SYZ mirror symmetry conjecture for del Pezzo surfaces and rational elliptic surfaces
论文作者
论文摘要
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,合理椭圆表面$ y $的$（y，d）$，以及$ d $ a单数纤维的kodaira type $ i_k $。关于后者对$（y，d）$建立了三个主要结果。首先，调整Hein \ cite {hein}的作品，我们证明了在$ y \ y \ setminus d $渐变上存在一个完整的calabi-yau指标。其次，我们证明了唯一的定理，即Modulo自动形态，$ y \ setminus d $ of y y \ setminus d $的每个kählerclass都承认了一个独特的渐近半flat calabi-yau指标。该结果产生了$ y \ setminus d $的Calabi-yau指标的有限尺寸Kähler模量。此外，在这种情况下，此结果回答了Tian-Yau和Yau的问题。第三，在作者之前的作品的基础上，我们证明了配备了渐近的半流量的calabi-yau公制$ω__{cy} $的$ y \ setminus d $，每当$ω_{cy} $的de rham同胞级时，每当$ω_{cy} $都不会受到替代。 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 $(\check{Y}, \ check {d}）$先前由作者构建。我们给出了这些结果的一些应用，包括研究固定抗典型分裂的Del Pezzo表面的自动形态。
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.
