Aspects of functoriality in homological mirror symmetry for toric varieties
This paper is about homological mirror symmetry for toric varieties, which suggests that the bounded derived category of coherent sheaves on a toric variety is equivalent to the Fukaya-Seidel category of its mirror. For a toric variety \(X\), the mirror to it is given by the algebraic torus \(({\mathbb{C}^*})^n\) together with a potential function \(W: ({\mathbb{C}^*})^n \to \mathbb{C}\). The authors prove that the derived category of coherent sheaves on \(X\) is equivalent to the partially wrapped Fukaya category of \(({\mathbb{C}^*})^n\) relative to a closed subset of its contact boundary. When \(X\) is Fano, this closed subset is \(W^{-1}(\infty)\) and the partially wrapped Fukaya category is equivalent to the Fukaya category of \((({\mathbb{C}^*})^n,W)\). This main result also follows from the work of [\textit{B. Fang} et al., Invent. Math. 186, No. 1, 79--114 (2011; Zbl 1250.14011)], who showed that the equivalence between the derived category of coherent sheaves on \(X\) with a category of constructible sheaves, and then applied results of [\textit{S. Ganatra} et al., ``Microlocal Morse theory of wrapped Fukaya categories'', Preprint, \url{arXiv:1809.08807}] relating the category of constructible sheaves to the Fukaya category. The authors of this paper follow a direct approach, which does not involve constructible sheaves.
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