## 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, $${}_pF_q$$

### Citations:

Zbl 0841.32013; Zbl 0826.32016; Zbl 0843.14005
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### References:

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