Quantum 𝒟-modules for toric nef complete intersections

Abstract

Let X be a smooth projective toric variety with k ample line bundles. Let Z be the zero locus of k generic sections. It is well-known that the ambient quantum D-module of Z is cyclic i.e., is defined by an ideal of differential operators. In this paper, we give an explicit construction of this ideal as a quotient ideal of a GKZ system associated to the toric data of X and the line bundles. This description can be seen as a "left cancellation procedure". We consider some examples where this description enables us to compute generators of this ideal, and thus to give a presentation of the ambient quantum D-module.

Quantum Serre Theorem as a Duality Between QuantumD-Modules: Table 1.

“…All the properties of Proposition 2.5 are true for the limit (see [MM11,§2] for a more precise statement).…”

“…This paper arose out of our previous works [Iri11,MM11] on quantum D-modules of (toric) complete intersections. The embedding of QDM amb (Z) into QDM(E ∨ ) appeared in [Iri11,Remark 6.14] in the case where X is a weak Fano toric orbifold and E ∨ = K X ; in [MM11, Theorem 1.1], QDM amb (Z) was presented as the quotient QDM (e,E) (X) by Ker(e(E)∪) when E is a direct sum of ample line bundles.…”

Hypergeometric Hodge modules

“…Notice that this corollary depends in an essential way on our main Theorem 5.35 since the strictness of the duality morphism with respect to the order filtrations holds only because the latter are (up to a shift) the Hodge filtrations of mixed Hodge modules. The identification with the reduced quantum D-module relies on the explicit description of the latter from [MM17] (already used extensively in [RS17]).…”

“…We refer to [MM17] for a detailed discussion of the definition of QDM(X Σ , E); a short version can be found in [RS17, Section 4.1]. Notice that in loc.…”

“…From here on, we will need to consider a Zariski-open subset of KM • ⊂ (C * ) b 2 (X Σ ) which contains B * ε . We recall the main result from [MM17], which gives a GKZ-type description of QDM(X Σ , E). We present it in a slightly different form, taking into account [RS17, Proposition 6.9].…”