Twistor property of GKZ-hypergeometric systems

Abstract

We study the mixed twistor D-modules associated to meromorphic functions. In particular, we describe their push-forward and specialization under some situations. We apply the results to study the twistor property of a type of better behaved GKZ-hypergeometric systems, and to study their specializations. As a result, we obtain some isomorphisms of mixed TEP-structures in the local mirror symmetry.

The mirror conjecture for minuscule flag varieties

“…3.11] concerning the matching of nc-Hodge structures on both sides of (12.11.1), in particular on the right-hand side we have identified the irregular Hodge filtration constructed by Deligne and J.-D. Yu [42]. For the case of certain toric mirror pairs this matching and much more is established in [113] and [106].…”

The Künneth Formula for the Twisted de Rham and Higgs Cohomologies

“…We follow the approach of [11] by putting a filtration on the de Rham complex or the Higgs complex via a certain compactification of the pair (U, f ). Here we introduce the notion of a non-degenerate compactification, which is weaker than that of a good compactification used in [11], but appears naturally in the later section (see also [7,Section 4], [9, Section 7.3], [5]). In order to compare the cohomologies with the filtrations of the summands (U i , f i ) and of their product (U, f ), we construct a particular compactification of (U, f ) from the fixed ones of (U i , f i ) in Section 3.…”

A mirror construction for the big equivariant quantum cohomology of toric manifolds

“…As noted in the Introduction, Givental [24,22] and Hori-Vafa [33] proposed Landau-Ginzburg mirrors for toric varieties. There are studies on non-compact case (local mirror symmetry) [41,21,8,39,45], Frobenius manifold [3,17,36,51], semi-simplicity [35,48], toric orbifolds [13,37,31,16,27,12,7,60], an approach using Lagrangian Floer theory [18,19,20,6], tropical geometry [29], quantum Kirwan maps [58,27] and quasimap spaces [10,7], etc. In non-semipositive case, we need to take a certain "Q-adic" completion of the Gauss-Manin system; this has been pointed out by the author [36], [35,Theorem 1.2].…”

“…In the analytic setting (in semipositive case), results analogous to Theorem 4.2 are given, e.g. in [37,Proposition 4.8], [51,Theorem 4.11], [16,Theorem 5.1.1], [45,Theorem 7.43]. 4.2.…”