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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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