Quantum \(\mathcal{D}\)-modules for toric nef complete intersections. (English) Zbl 1373.14056
Let \(Z\) be smooth projective variety over \(\mathbb C\) and let QDM\((Z)\) be the quantum \(\mathcal D\)-module of \(Z\). Givental gives presentation of QDM\((Z)\)in terms of GKZ systems by proving the isomorphism QDM\((Z)\cong \mathcal D/\mathcal G_Z\) in the case that \(Z\) is toric and Fano. Here \(\mathcal G_Z\) is the GKZ ideal associated to \(Z\).
The paper under review studies the case where \(Z\) is a nef complete intersection inside a smooth toric variety \(X\). Let \(\mathcal L_1,\dots \mathcal L_k\) be ample line bundles on \(X\) and \(i:Z\to X\) be the inclusion of the zero locus of a generic section of the vector bundle \(\mathcal E:=\oplus \mathcal L_i\). The subspace \(i^*(H^*(X,\mathbb C))\subseteq H^*(Z,\mathbb C)\), denoted by \(H^*_{\mathrm{amb}}(Z)\), is stable under small quantum cohomology product of \(Z\). Let QDM\(_{\mathrm{amb}}(Z)\) be the corresponding sub-\(\mathcal D\)-module. The main result of the paper under review is the following giving an answer to a question posed by Cox and Katz: Let \(\widehat{c}_{top}\in \mathcal D\) be the operator associated to the top Chern class of the vector bundle \(\mathcal E\), and \((\mathcal G_{X,\mathcal E}:\widehat{c}_{top})\) be the left quotient ideal of the GKZ ideal associated to \(X\) and \(\mathcal E\). If \(\dim_{\mathbb C} X\geq k+3\) then there is an isomorphism of the \(\mathcal D\)-modules QDM\(_{\mathrm{amb}}(Z)\cong \mathcal D/(\mathcal G_{X,\mathcal E}:\widehat{c}_{\mathrm{top}})\).
This result has been used to prove a mirror theorem for non-affine Landau-Ginzburg model.
The paper under review studies the case where \(Z\) is a nef complete intersection inside a smooth toric variety \(X\). Let \(\mathcal L_1,\dots \mathcal L_k\) be ample line bundles on \(X\) and \(i:Z\to X\) be the inclusion of the zero locus of a generic section of the vector bundle \(\mathcal E:=\oplus \mathcal L_i\). The subspace \(i^*(H^*(X,\mathbb C))\subseteq H^*(Z,\mathbb C)\), denoted by \(H^*_{\mathrm{amb}}(Z)\), is stable under small quantum cohomology product of \(Z\). Let QDM\(_{\mathrm{amb}}(Z)\) be the corresponding sub-\(\mathcal D\)-module. The main result of the paper under review is the following giving an answer to a question posed by Cox and Katz: Let \(\widehat{c}_{top}\in \mathcal D\) be the operator associated to the top Chern class of the vector bundle \(\mathcal E\), and \((\mathcal G_{X,\mathcal E}:\widehat{c}_{top})\) be the left quotient ideal of the GKZ ideal associated to \(X\) and \(\mathcal E\). If \(\dim_{\mathbb C} X\geq k+3\) then there is an isomorphism of the \(\mathcal D\)-modules QDM\(_{\mathrm{amb}}(Z)\cong \mathcal D/(\mathcal G_{X,\mathcal E}:\widehat{c}_{\mathrm{top}})\).
This result has been used to prove a mirror theorem for non-affine Landau-Ginzburg model.
Reviewer: Amin Gholampour (College Park)
MSC:
| 14N35 | Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |
Keywords:
quantum differential modules; Gromov-Witten invariants; Batyrev rings; GKZ systems; mirror symmetryReferences:
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