关于环带退化内蕴同调镜像对称性
On intrinsic homological mirror symmetry for toric degenerations
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中文总结 AI 辅助
本文针对Calabi--Yau流形的极大单能退化，利用固定点Floer上同调构造候选镜像族，并在热带拉格朗日截面存在时建立导出范畴到Fukaya范畴的完全忠实嵌入，验证了若干环带退化情形。
中文摘要 AI 辅助
本文研究了由Perutz和Siebert的提议以及Gross--Siebert内蕴镜像对称性纲领所启发的同调镜像对称性的Floer理论方面。给定一个在穿孔圆盘上的光滑射影Calabi--Yau流形的极大单能退化，我们利用该退化的单值迭代的固定点Floer上同调群，配备裤子积，构造了一个环。在此环可交换的假设下，我们可以考虑由相对Proj构造定义的候选镜像族。进一步假设一个光滑纤维$X_t$包含一个所谓的热带拉格朗日截面，我们构造了一个从候选镜像族上的完美复形的导出范畴到$X_t$的Fukaya范畴的完全忠实嵌入。我们验证了这两个假设对于某些Batyrev--Borisov环带退化以及一些来自Gross--Siebert重构算法的Calabi--Yau三维流形的环带退化成立。这两个几何假设都是为了支持对Calabi--Yau流形的极大单能退化的镜像对称性的一般研究，将证明同调镜像对称性的辛输入在很大程度上归结为构造热带拉格朗日截面的问题。
英文摘要
This paper studies the Floer-theoretic aspects of homological mirror symmetry inspired by proposals of Perutz and Siebert and the Gross--Siebert intrinsic mirror symmetry program. Given a maximally unipotent degeneration of smooth projective Calabi--Yau manifolds over the punctured disk, we construct a ring using the fixed point Floer cohomology groups of the iterates of the monodromy of the degeneration equipped with the pair of pants product. Under the assumption that this ring is commutative, we can consider a candidate mirror family defined by the relative Proj construction. Further assuming that a smooth fiber $X_t$ contains a so-called tropical Lagrangian section, we construct a fully faithful embedding from the derived category of perfect complexes on our candidate mirror family into the Fukaya category of $X_t$. We verify both of these assumptions for certain Batyrev--Borisov toric degenerations, as well as some toric degenerations of Calabi--Yau threefolds coming from the Gross--Siebert reconstruction algorithm. These two geometric hypotheses are both phrased to support the general study of mirror symmetry for maximally unipotent degenerations of Calabi--Yau manifolds, largely reducing the symplectic inputs for proving homological mirror symmetry to the problem of constructing tropical Lagrangian sections.
