On weak Landau-Ginzburg models for complete intersections in Grassmannians. (English. Russian original) Zbl 1319.14047

Russ. Math. Surv. 69, No. 6, 1129-1131 (2014); translation from Usp. Mat. Nauk 69, No. 6, 181-182 (2014).
The well-known Givental’s construction describes a Landau-Ginzburg (LG) model for a complete intersection \(Y\) in a toric variety as a complete intersection in a torus equipped with a function on the torus called the superpotential. Oscillating integrals associated to the LG model are mirror to Gromov-Witten invariants of \(Y\). In the spirit of Givental’s construction, LG models for complete intersections in Grassmannians (and partial flag varieties) are described by V. V. Batyrev et al. [Nucl. Phys., B 514, No. 3, 640–666 (1998; Zbl 0896.14025); Acta Math. 184, No. 1, 1–39 (2000; Zbl 1022.14014)] via toric degenerations, where the LG models are given as certain complete intersections in complex tori equipped with superpotentials. This paper announces that for a smooth Fano complete intersection \(Y\) in a Grassmannian of planes, the LG model constructed in [loc. cit.] is birational to a torus and the superpotential can be given as a Laurent polynomial. The authors expect that this result can be generalized to complete intersections in an arbitrary Grassmannian.


14J33 Mirror symmetry (algebro-geometric aspects)
14J45 Fano varieties
14M10 Complete intersections
14M15 Grassmannians, Schubert varieties, flag manifolds
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