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The Frobenius structure theorem for affine log Calabi-Yau varieties containing a torus. (English) Zbl 1527.14085

Let \(U\) be a smooth affine log Calabi-Yau variety containing an open algebraic torus. Let \(U \subset Y\) be any simple normal crossing compactification. In this paper, the authors show that the naive counts of rational curves in \(U\) uniquely determine a commutative associative algebra \(A\), which is called the mirror algebra, equipped with a compatible multilinear form. The algebra structure of \(A\) is defined by giving non-negative integer structure constants as naive counts of non-archimedean analytic disks. Moreover, the spectrum of \(A\) is shown to be the total space of a family whose generic fiber is an affine Calabi-Yau variety of the same dimension of \(U\) with at worst log canonical singularities.
Let \(k\) be any field of characteristic zero. Let \(U\) be a connected affine smooth log Calabi-Yau variety over \(k\) with volume form \(\omega\). Let \(U \subset Y\) be a compactification with \(Y \setminus U\) a simple normal crossing divisor. Let \(k(U)\) be the function field of \(U\). Let \[ \mathrm{Sk}(U,\mathbb{Z}) :=\{0\} \sqcup \{m \nu \mid \textrm{\(m \in \mathbb{N}_{>0}\), \(\nu\) is a divisorial valuation on \(k(U)\) where \(\omega\) has a pole}\}. \] Let \begin{align*} R&:=\bigoplus_{\beta \in \mathrm{NE}(Y)} \mathbb{Z} \cdot z^{\beta} \colon \textrm{the monoid ring of \(\mathrm{NE}(Y)\)}, \\ A&:= \bigoplus_{P \in \mathrm{Sk}(U,\mathbb{Z})} R \cdot \theta_P \colon \textrm{the free \(R\)-module with basis \(\mathrm{Sk}(U,\mathbb{Z})\).} \end{align*} For a tree \((P_1,\ldots,P_n)\) with \(P_i \in \mathrm{Sk}(U,\mathbb{Z})\) and a curve \(\beta \in \mathrm{NE}(Y) \subset N_1(Y,\mathbb{Z})\), a number \(\eta(P_1,\ldots,P_n,\beta)\), which counts rational curves in \(U\), is defined. In the Introduction, the authors say that the simple definition of counts is the heart of this work. For \(n \geq 2\), a multilinear map \(\langle-,\ldots,-\rangle_n \colon A^n \to R\) is defined by \[ \langle \theta_{P_1},\ldots,\theta_{P_n}\rangle_n =\sum_{\beta \in \mathrm{NE}(Y)} \eta(P_1,\ldots,P_n,\beta)z^{\beta}. \] Then the following is proved in this paper.
(1)
For \(n \geq 2\), the \(R\)-multilinear map \(\langle-,\ldots,-\rangle_n \colon A^n \to R\) is non-degenerate.
(2)
There exists a unique finitely generated commutative associative \(R\)-algebra structure on \(A\) such that \(\theta_0=1\) and \[ \langle a_1,\ldots,a_n\rangle_n =\mathrm{Trace}(a_1 \cdots a_n) \] for every \(n \geq 2\), where \(\mathrm{Trace} \colon A \to R\) takes the coefficient of \(\theta_0\).
(3)
The restriction of the family \(\mathrm{Spec}\ A \to \mathrm{Spec}\ R\) over \(\mathbb{Q}\) is a flat family of affine varieties of same dimension as \(U\), and each fiber is Gorenstein, semi-log-canonical and \(K\)-trivial. The generic fiber is log canonical and log Calabi-Yau.

MSC:

14J33 Mirror symmetry (algebro-geometric aspects)
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