Special Lagrangian submanifolds of log Calabi–Yau manifolds
Special Lagrangian submanifolds of log Calabi–Yau manifolds
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DOI：
10.1215/00127094-2021-0012
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发表时间：
2019-04
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影响因子：
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通讯作者：
Tristan C. Collins;Adam Jacob;Yu-Shen Lin
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文献类型：
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作者：
Tristan C. Collins;Adam Jacob;Yu-Shen Lin
我们研究了配备了由Tian-Yau建造的完整ricci-flat Kahler指标的日志calabi-yau歧管的特殊拉格朗日submanifolds的存在。我们证明，如果$ x $是田（Tian-Yau）的歧管，如果Infinty的紧凑型卡拉比（Calabi-Yau）歧管承认一个特殊的拉格朗日人，那么$ x $承认了许多与众不同的特殊Lagrangians。在复杂的尺寸$ 2 $中，我们证明，如果$ y $是del pezzo表面或理性椭圆表面，而$ d \ in | -k_ {y} | $是平滑的除数，则$ d^2 = d $ ，然后$ x = y \ backslash d $承认了特殊的拉格朗日圆环纤维化，这是由Strominger-yau-Zaslow和Auroux猜想的。实际上，我们表明$ x $接纳了双胞胎特殊的拉格朗日纤维，证实了Leung-Yau的预测。在特殊情况下，$ y $是一个有理椭圆形的表面，或$ y = \ mathbb {p}^2 $，我们确定了通用数据的单数纤维，从而确认了两个auroux的猜想。最后，我们证明，在超级旋转后，$ x $可以被压实到kodaira型$ i_ {d} $纤维的补充中，在有理椭圆表面$ \ check {\ pi}中以奇异纤维的形式出现。 \ check {y} \ rightarrow \ mathbb {p}^1 $。
We study the existence of special Lagrangian submanifolds of log Calabi-Yau manifolds equipped with the complete Ricci-flat Kahler metric constructed by Tian-Yau. We prove that if $X$ is a Tian-Yau manifold, and if the compact Calabi-Yau manifold at infinty admits a single special Lagrangian, then $X$ admits infinitely many disjoint special Lagrangians. In complex dimension $2$, we prove that if $Y$ is a del Pezzo surface, or a rational elliptic surface, and $D\in |-K_{Y}|$ is a smooth divisor with $D^2=d$, then $X= Y\backslash D$ admits a special Lagrangian torus fibration, as conjectured by Strominger-Yau-Zaslow and Auroux. In fact, we show that $X$ admits twin special Lagrangian fibrations, confirming a prediction of Leung-Yau. In the special case that $Y$ is a rational elliptic surface, or $Y= \mathbb{P}^2$ we identify the singular fibers for generic data, thereby confirming two conjectures of Auroux. Finally, we prove that after a hyper-Kahler rotation, $X$ can be compactified to the complement of a Kodaira type $I_{d}$ fiber appearing as a singular fiber in a rational elliptic surface $\check{\pi}: \check{Y}\rightarrow \mathbb{P}^1$.