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A mirror theorem for toric complete intersections. (English) Zbl 0936.14031
Kashiwara, Masaki (ed.) et al., Topological field theory, primitive forms and related topics. Proceedings of the 38th Taniguchi symposium, Kyoto, Japan, December 9-13, 1996 and the RIMS symposium with the same title, Kyoto, Japan, December 16-19, 1996. Boston, MA: Birkhäuser. Prog. Math. 160, 141-175 (1998).
Let \(X\) be a non-singular compact symplectic toric variety. Then \(X\) can be obtained by the symplectic reduction of \({\mathbb C}^N\) with its standard symplectic structure \(\text{Im}\sum d\bar{z}_i\wedge dz_i/2\) by the action of a subtorus \(T^k\) in the maximal torus \(T^N\) of diagonal unitary matrices on a generic level of the momentum map. In this situation \(k\) is called the Picard number of \(X\). Let \(u_1,u_2,\ldots,u_N\in H^2(X)\) denote the Poincaré duals of the fundamental cycles determined by the \(N\) compact toric hypersurfaces in \(X\) which are the \(T^k\)-reductions of the \(T^N\)-invariant coordinate hyperplanes in \({\mathbb C}^N\). Let \(Y\subset X\) be a non-singular complete intersection defined by global sections of \(\ell\) (\(\ell\geq 0\)) non-negative line bundles \(\mathcal L_a\), \(a=1,2,\ldots,\ell\), with first Chern classes \(v_1,v_2,\ldots,v_{\ell}\in H^2(X)\). Write \(\mathcal V=\bigoplus_a\mathcal L_a\). Then the bundle \(\mathcal V\) is convex. It can be endowed with the action of \(T^{\ell}\) by scalar multiplication on the fibers of each summand \(\mathcal L_a\). Then one may study the equivariant Gromov-Witten theory of the pair \((X,\mathcal V)\) with respect to the action of the torus \(T=T^N\times T^{\ell}\). Write \(H^*(\mathcal V)\subset H^*(Y)\) for the subring multiplicatively generated by the \(u_i\)’s. One has \(c_1(\mathcal T_X)=u_1+\cdots +u_N\), and \(c_1(Y)=u_1+\cdots +u_N-v_1-\cdots -v_{\ell}\), where \(v_a=c_1(\mathcal L_a)\).
Gromov-Witten theory associates to \(Y\) its quantum cohomology \(\mathcal D\)-module and its quantum cohomology algebra. Let \((t_1,\ldots,t_k)\) denote coordinates on \(H^2(Y)\) with respect to a basis \((p_1,\ldots,p_k)\) of integral symplectic classes. \(t_0\) will denote the coordinate on \(H^0(Y)\). The \(p_i\) define quantum operators \(p_i^*\) acting on vector functions \(s=s(t_0,t)\) with values in \(H^*(Y,{\mathbb C})\). They give rise to a system of linear PDE’s with associated \(\mathcal D\)-module generated by a single formal vector function \(J_Y=J_Y(t,\hbar^{-1})\) with coefficients in \(H^*(Y,{\mathbb Q})\). One can give a construction of \(J_Y\) in terms of intersection theory on moduli spaces of stable maps. Write \(J\) for the orthogonal projection of \(J_Y\) to the subalgebra \(H^*(\mathcal V,{\mathbb Q})\subset H^*(Y,{\mathbb Q})\). Let \(\Lambda\) be the semigroup of classes of compact holomorphic curves in \(Y\subset X\). For \(d\in\Lambda\) write \(D_j(d)=\int_{[d]}u_j\), \(L_a(d)=\int_{[d]}v_a\). Consider the formal vector function \(I=I(t,\hbar^{-1})\) with values in \(H^*(\mathcal V,{\mathbb Q})\) (‘hypergeometric series’): \[ \begin{split} I=\exp\left((t_0+\sum_{i=1}^kp_it_i)/\hbar\right)\sum_{d\in\Lambda} \exp\left(\sum_{j=1}^kt_jd_j\right)\times\\ \times{{\prod_{a=1}^{\ell} \prod_{m=-\infty}^{L_a(d)}(v_a+m\hbar)\prod_{j=1}^N\prod_{m=-\infty} ^0(u_j+m\hbar)}\over{\prod_{a=1}^{\ell}\prod_{m=-\infty}^0(v_a+m\hbar) \prod_{j=1}^N\prod_{m=-\infty}^{D_j(d)}(u_j+m\hbar)}} \end{split} \] One can now state the main result
\((*)\): Let \(Y\) be a non-singular toric complete intersection with non-negative first Chern class. Then \(I\) and \(J\) with coefficients in \(H^*(\mathcal V,{\mathbb Q})\) coincide up to a triangular weighted homogeneous change of variables \[ t_0\mapsto t_0+f_0(q)\hbar+h(q),\quad\log q_i\mapsto\log q_i+f_i(q), \] where \(q=e^t\), and \(h,f_0,\ldots,f_k\) are weighted homogeneous power series supported on \(\Lambda-\{0\}\), and \(\deg f_0=\deg f_i=0\).
As a matter of fact one may prove a more general result depending on additional variables \(\lambda\). The above result can be interpreted as the specialization for \(\lambda\rightarrow 0\). By a result of V. Batyrev one deduces that \((*)\) confirms the mirror conjecture for Calabi-Yau toric complete intersections.
For the entire collection see [Zbl 0905.00081].

MSC:
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14M10 Complete intersections
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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