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Quantum cohomology rings of toric manifolds. (English) Zbl 0806.14041
Journées de géométrie algébrique d’Orsay, France, juillet 20-26, 1992. Paris: Société Mathématique de France, Astérisque. 218, 9-34 (1993).
Let $$\Sigma$$ be a finite complete nonsingular fan for $$N = \mathbb{Z}^ d$$. Then the toric variety $$\mathbb{P}_ \Sigma$$ over $$\mathbb{C}$$ corresponding to $$\Sigma$$ is a $$d$$-dimensional compact nonsingular algebraic variety. Let $$n$$ be the cardinality of the set of one-dimensional cones in $$\Sigma$$ and denote by $$G(\Sigma) = \{v_ 1, \dots, v_ n\}$$ the set of primitive lattice vectors $$v_ j \in N$$ such that $$\mathbb{R}_{\geq 0} v_ 1, \dots, \mathbb{R}_{\geq 0} v_ n$$ are the one-dimensional cones in $$\Sigma$$, where $$\mathbb{R}_{\geq 0}$$ is the set of nonnegative real numbers.
J. Jurkiewich and V. I. Danilov determined the ordinary cohomology ring $$H^ \bullet (\mathbb{P}_ \Sigma, \mathbb{Z})$$ as the quotient $$H^ \bullet (\mathbb{P}_ \Sigma, \mathbb{Z}) = \mathbb{Z} [z_ 1, \dots, z_ n]/ \bigl( P (\Sigma)_ \mathbb{Z} + \text{SR} (\Sigma)_ \mathbb{Z} \bigr)$$ of the polynomial ring $$\mathbb{Z} [z_ 1, \dots, z_ n]$$, where $$P(\Sigma)_ \mathbb{Z}$$ is the ideal generated by the linear forms $$\sum_{j = 1}^ n \langle m,v_ j \rangle z_ j$$ for $$m \in M : = \operatorname{Hom}_ \mathbb{Z} (N, \mathbb{Z})$$ with $$\langle m,m \rangle : M \times N \to \mathbb{Z}$$ the canonical bilinear pairing, while $$\text{SR} (\Sigma)_ \mathbb{Z}$$ is the Stanley- Reisner ideal generated by $$z_{j_ 1} z_{j_ 2} \dots z_{j_ s}$$ with $$\{j_ 1,j_ 2, \dots,j_ s\}$$ running through the subsets of $$\{1,2,\dots,n\}$$ such that the cone $$\sum_{l = 1}^ s \mathbb{R}_{\geq 0} v_{j_ i}$$ does not belong to the fan $$\Sigma$$.
Let $$\text{PL} (\Sigma)_ \mathbb{R}$$ be the $$\mathbb{R}$$-vector space consisting of $$\mathbb{R}$$-valued functions $$\varphi$$ on $$N_ \mathbb{R} : N \otimes_ \mathbb{Z} \mathbb{R} = \bigcup_{\sigma \in \Sigma} \sigma$$ which are linear on each cone $$\sigma \in \Sigma$$, and denote by $$\text{PL} (\Sigma)_ \mathbb{Z}$$ the $$\mathbb{Z}$$-submodule consisting of those $$\varphi$$’s satisfying $$\varphi (N) \subset \mathbb{Z}$$. Obviously, $$M$$ is a $$\mathbb{Z}$$-submodule of $$\text{PL} (\Sigma)_ \mathbb{Z}$$. The map which sends $$\varphi \in \text{PL} (\Sigma)_ \mathbb{Z}$$ to $$\sum_{j = 1}^ n \varphi (v_ j) z_ j$$ is known to give rise to an exact sequence $0 \to M \to \text{PL} (\Sigma)_ \mathbb{Z} \to H^ 2(\mathbb{P}_ \Sigma, \mathbb{Z}) \to 0.$ The cone in $$\text{PL} (\Sigma)_ \mathbb{R}$$ consisting of the $$\varphi$$’s which are strictly convex with respect to the fan $$\Sigma$$ (resp. which are convex) is known to be mapped onto the cone $$K^ 0(\mathbb{P}_ \Sigma) \subset H^ 2 (\mathbb{P}_ \Sigma, \mathbb{R})$$ of Kähler classes (resp. the closed Kähler cone $$K(\mathbb{P}_ \Sigma) \subset H^ 2 (\mathbb{P}_ \Sigma, \mathbb{R}))$$. (Beware of the confusion in the paper as to the distinction between convexity and upper convexity.)
For each $$\varphi \in \text{PL} (\Sigma)_ \mathbb{C} : = \text{PL} (\Sigma)_ \mathbb{R} \otimes_ \mathbb{R} \mathbb{C}$$, the author defines the quantum cohomology ring of the toric variety $$\mathbb{P}_ \Sigma$$ with respect to $$\varphi$$ to be $QH^ \bullet_ \varphi (\mathbb{P}_ \Sigma, \mathbb{C}) : = \mathbb{C} [z_ 1, \dots, z_ n]/ \bigl( P (\Sigma)_ \mathbb{C} + Q_ \varphi (\Sigma) \bigr),$ where $$P(\Sigma)_ \mathbb{C} : = P (\Sigma)_ \mathbb{Z} \otimes_ \mathbb{Z} \mathbb{C}$$ and $$Q_ \varphi (\Sigma)$$ is the ideal in $$\mathbb{C} [z_ 1, \dots, z_ n]$$ generated by the binomials $\exp \left( \sum_{j = 1}^ n a_ j \varphi (v_ j) \right) \prod_{j = 1}^ n z_ j^{a_ j} - \exp \left( \sum_{l = 1}^ n b_ l \varphi (v_ l) \right) \prod_{l = 1}^ n z_ l^{b_ l}$ with $$\sum_{j = 1}^ n a_ j v_ j = \sum_{l = 1}^ n b_ lv_ l$$ running through all the linear relations among $$v_ 1, \dots, v_ n$$ such that $$a_ j$$’s and $$b_ l$$’s are nonnegative integers.
By the very definition, the quantum cohomology ring depends not on the fan $$\Sigma$$ but only on the set $$G(\Sigma)$$ of the generators of one- dimensional cones in $$\Sigma$$. In particular, compact nonsingular toric varieties which are isomorphic in codimension one have the same quantum cohomology ring.
Among other things, the author shows the following when $$\mathbb{P}_ \Sigma$$ is a Kähler manifold, that is, $$K^ 0(\mathbb{P}_ \Sigma) \neq \emptyset$$:
(1) As the positive real number $$t$$ tends to $$\infty$$, the limit of $$QH^ \bullet_{t \varphi} (\mathbb{P}_ \Sigma, \mathbb{C})$$ exists in an appropriate sense if and only if $$\varphi$$ belongs to the complexified Kähler cone $$K(\mathbb{P}_ \Sigma)_ \mathbb{C} : = K (\mathbb{P}_ \Sigma) + iH^ 2(\mathbb{P}_ \Sigma, \mathbb{R})$$. In that case, the limit coincides with the ordinary cohomology ring $$H^ \bullet (\mathbb{P}_ \Sigma, \mathbb{C})$$.
(2) If $$\varphi \in \text{PL} (\Sigma)_ \mathbb{R}$$ is strictly convex with respect to $$\Sigma$$ so that it gives rise to a Kähler class, then $$QH^ \bullet_ \varphi (\mathbb{P}_ \Sigma, \mathbb{C})$$ is shown to coincide with the quantum cohomology ring of $$\mathbb{P}_ \Sigma$$ considered by physicists in terms of the topological sigma model of $$\mathbb{P}_ \Sigma$$, namely, the moduli space of holomorphic maps from the complex projective line $$\mathbb{C} \mathbb{P}^ 1$$ to $$\mathbb{P}_ \Sigma$$ [E. Witten in Proc. Conf., Cambridge 1990, Surv. Differ. Geom., J. Differ. Geom., Suppl. 1, 243-310 (1991; Zbl 0757.53049)].
(3) When the first Chern class of $$\mathbb{P}_ \Sigma$$ belongs to the Kähler cone so that $$\mathbb{P}_ \Sigma$$ is a toric Fano manifold, the quantum cohomology ring $$QH^ \bullet_ \varphi (\mathbb{P}_ \Sigma, \mathbb{C})$$ is shown to be canonically isomorphic to the Jacobian ring of the Laurent polynomial $f_ \varphi (X_ 1, \dots, X_ d) = - 1 + \sum_{j = 1}^ n \exp \bigl( - \varphi (v_ j)\bigr) \prod_{l = 1}^ d X_ l^{ v_{jl}},$ with $$v_ j = (v_{j1}, \dots, v_{jd})$$ in the coordinate system $$N = \mathbb{Z}^ d$$. In the sense of the author’s earlier work [“Dual polyhedra and the mirror symmetry for Calabi-Yau hypersurfaces in toric varieties,” J. Algebr. Geom. 3, No. 3, 493-535 (1994)], the equation $$f_ \varphi = 0$$ defines in the Fano variety dual to $$\mathbb{P}_ \Sigma$$ a Calabi-Yau hypersurface which is mirror symmetric to the Calabi-Yau hypersurface in $$\mathbb{P}_ \Sigma$$ defined as a general member of the anticanonical linear system.
For the entire collection see [Zbl 0790.00001].
Reviewer: T.Oda (Sendai)

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C80 Applications of global differential geometry to the sciences
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