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Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties. (English) Zbl 0843.14016
Extending known results of P. Candelas, X. C. de la Ossa, P. S. Green and L. Parkes [“A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory” in: Essays on mirror manifolds, 31-95 (1992; Zbl 0826.32016), see also Nuclear Phys., Particle Physics, B 359, No. 10, 21-74 (1991)], D. R. Morrison [“Mirror symmetry and rational curves on quintic threefolds: A guide for mathematicians”, J. Am. Math. Soc. 6, No. 1, 223-247 (1993; Zbl 0843.14005) and “Picard Fuchs equations and mirror maps for hypersurfaces”, in: Essays on mirror manifolds, 241-264 (1992; Zbl 0841.32013)] and many others, the authors formulate interesting conjectures on the mirror symmetry and generalized hypergeometric series for Calabi-Yau complete intersections in toric varieties. To state these conjectures, we need to introduce necessary ingredients as follows: Let \(P_\Sigma\) be a \((d + r)\)-dimensional projective toric variety corresponding to a complete simplicial fan \(\Sigma\) for a free \(\mathbb{Z}\)-module \(N\) of \(\text{rank} d + r\). Denote by \(E = \{v_1, \dots, v_k\}\) the set of primitive generators of one-dimensional cones in the fan \(\Sigma\), and let \(D_j\) be the irreducible torus-invariant Weil divisor on \(P_\Sigma\) corresponding to the one-dimensional cone spanned by \(v_j \in E\). Split \(E\) into a disjoint union \(E = E_1 \cup E_2 \cup \cdots \cup E_r\) and denote also by \(E_i\) the set of indices \(\{j \mid v_j \in E_i\}\). Assume that \(\sum_{j \in E_i} D_j\) for each \(1 \leq i \leq r\) is numerically effective (or equivalently, base-point-free in the present context) and is linearly equivalent to a hypersurface \(V_i \subset P_\Sigma\). Then the complete intersection \(V : = V_1 \cap V_2 \cap \cdots \cap V_r\) is a \(d\)-dimensional Calabi-Yau variety possibly with Gorenstein toroidal singularities, since \(\sum^k_{j = 1} D_j = \sum_i (\sum_{j \in E_i} D_j)\) is an anticanonical divisor of \(P_\Sigma\). When \(P_\Sigma\) is smooth, the kernel of the surjective homomorphism \(Z^k \ni \lambda = (\lambda_1, \dots, \lambda_k) \mapsto \sum^k_{j = 1} \lambda_j v_j \in N\) is known to coincide with the \(\mathbb{Z}\)-module \(R(E)\) of algebraic 1-cycles on \(P_\Sigma\). \(\lambda_j = \langle D_j, \lambda \rangle\) is the intersection number of the algebraic 1-cycle \(\lambda \in R(E)\) with the divisor \(D_j\). Thus \(R^+ (E) : = R(E) \cap (\mathbb{Z}_{\geq 0})^k\) is the submonoid of nef 1-cycles, where \(\mathbb{Z}_{\geq 0}\) is the set of nonnegative integers. We can choose a \(\mathbb{Z}\)-basis \(\{\lambda^{(1)}, \dots, \lambda^{(t)}\}\) of \(R(E)\) so that effective algebraic 1-cycles on \(P_\Sigma\), hence elements in \(R^+ (E)\) in particular, are nonnegative linear combinations of \(\lambda^{(1)}, \dots, \lambda^{(t)}\). Let us introduce a generalized hypergeometric series in complex variables \(u_1, \dots, u_k\) by \[ \Phi_0 (u) : = \sum_{\lambda \in R^+ (E)} \prod^r_{i = 1} \left( \sum_{j \in E_i} \lambda_j \right)! \left( \left.\prod_{j \in E_i} u_j^{ \lambda_j}\right/\lambda_j! \right). \] Let \(T : = \operatorname{Hom}_\mathbb{Z} (N,C^\times)\) be the \((d + r)\)-dimensional algebraic torus with the character group \(N\). Denote by \(X^v\) the Laurent monomial corresponding to \(v \in N\). Then in terms of the Laurent polynomials \(P_{E_i} (X) : = 1 - \sum_{j \in E_i} u_j X^{v_j}\), \(i = 1, 2, \dots, r\), we have an integral representation \[ \Phi_0 (u) = {1 \over (2i \sqrt {-1})^{d + r}} \int_{|X_1 |= 1, \dots, |X_{d + r} |= 1} {1 \over P_{ E_1} (X) \cdots P_{E_r}(X)} {dX_1 \over X_1} \wedge \cdots \wedge {dX_r \over X_r}, \] where \(X_1, \dots, X_{d + r}\) are suitable coordinates for \(T\).
In terms of a new set of complex variables \(z_1, \dots, z_t\) defined by \(z_s : = \prod^r_{i = 1} \prod_{j \in E_i} u^{\lambda_j^{(s)}}_j\), \(s = 1, \dots, t\), \(\Phi_0 (u)\) can be expressed as a power series \[ \Phi_0 (z) = \sum_{\lambda \in R^+ (E)} (\langle V_1, \lambda \rangle! \cdots \langle V_r, \lambda \rangle!/ \langle D_1, \lambda \rangle! \cdots \langle V_k, \lambda \rangle!) z^\lambda, \] where \(z^\lambda = z_1^{c_1} \cdots z_t^{c_t}\) with \(\lambda = c_1 \lambda^{(1)} + \cdots + c_t \lambda^{(t)}\), and \(\langle V_i, \lambda \rangle\) is the intersection number of \(V_i\) with the nef 1-cycle \(\lambda\).
Assume further that \(V\) is smooth and that the restriction map \(\text{Pic} (\mathbb{P}_\Sigma) \leftarrow \text{Pic} (V)\) is injective. There exists a flat “\(A\)-model connection” \(\nabla_{AP}\) on \(H^* (\mathbb{P}_\Sigma, \mathbb{C})\) which defines a quantum variation of Hodge structures on \(H^* (\mathbb{P}_\Sigma, \mathbb{C})\). Likewise, there exists a flat “\(A\)-model connection” \(\nabla_{AV}\) on \(H^* (V,\mathbb{C})\) which defines a quantum variation of Hodge structures on \(H^* (V,\mathbb{C})\). The complex variables \(z_1, \dots, z_t\) can be identified with \(\nabla_{AP}\)-flat coordinates on the image \(\widetilde H^2\) of \(H^2 (\mathbb{P}_\Sigma, \mathbb{C})\) and \(H^2 (V,\mathbb{C})\). – Here are some of the authors’ conjectures in terms of these ingredients:
(1) The generalized hypergeometric series \(\Phi_0 (z)\) in terms of \(z_1, \dots, z_r\) is a solution of the differential system \({\mathcal D}\) defined by the restriction of \(\nabla_{ AV}\) to \(\widetilde H^2\).
(2) The differential system \({\mathcal D}\) has logarithmic solutions of the form \(\Phi_s (z) = (\log z_s) \Phi_0 (z) + \Psi_s (z)\), \(s = 1, \dots, t\), with \(\Psi_s (z)\) holomorphic at \(z = 0\) and \(\Psi_s (0) = 0\). We can then define \(\nabla_{AV}\)-flat coordinates on \(\widetilde H^2\) by \(q_s : = \exp (\Phi_s (z)/ \Phi_0 (z))\), \(s = 1, \dots, t\). The coefficients of \(q_1, \dots, q_s\) with respect to \(z_1, \dots, z_t\) are integers.
(3) The Calabi-Yau variety mirror symmetric to \(V\) is obtained as a Calabi-Yau compactification of the complete intersection \(\{P_{E_1} (X) = 0\} \cap \cdots \cap \{P_{E_r} (X) = 0\}\) of affine hypersurfaces in the \((d + r)\)-dimensional algebraic torus \(T\).
The authors go on to check these conjectures by dealing with many examples of Calabi-Yau threefolds obtained as complete intersections in products of projective spaces.
Reviewer: T.Oda (Sendai)

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
33C20 Generalized hypergeometric series, \({}_pF_q\)
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