We consider mirror symmetry for (essentially arbitrary) hypersurfaces in (possibly noncompact) toric varieties from the perspective of the Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface \(H\) in a toric variety \(V\) we construct a Landau-Ginzburg model which is SYZ mirror to the blowup of \(V\times \mathbf {C}\) along \(H\times0\), under a positivity assumption. This construction also yields SYZ mirrors to affine conic bundles, as well as a Landau-Ginzburg model which can be naturally viewed as a mirror to \(H\). The main applications concern affine hypersurfaces of general type, for which our results provide a geometric basis for various mirror symmetry statements that appear in the recent literature. We also obtain analogous results for complete intersections.

中文翻译：

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复曲面变种爆炸的拉格朗日纤维化和超表面的镜面对称

我们从Strominger-Yau-Zaslow（SYZ）猜想的角度考虑（可能是非紧凑的）复曲面品种中的（基本上任意）超曲面的镜像对称性。给定复曲面变种\（V \）中的超曲面\（H \），我们构造了Landau-Ginzburg模型，该模型是SYZ镜像，沿着\（H \ times0分解\\ V \ times \ mathbf {C} \） \），在积极假设下。这种构造还产生了仿射圆锥束的SYZ反射镜，以及可以自然地视为\（H \）的反射镜的Landau-Ginzburg模型。主要应用涉及一般类型的仿射超曲面，为此我们的研究结果为近期文献中出现的各种镜像对称性陈述提供了几何基础。我们还获得了完整交集的相似结果。