# zbMATH — the first resource for mathematics

Non-affine Landau-Ginzburg models and intersection cohomology. (Modèles de Landau-Ginzburg non affines et cohomologie d’intersection.) (English. French summary) Zbl 1395.14033
The paper aims at constructing mirror models for complete intersections in smooth toric varieties in the case when these subvarieties have a numerically effective anticanonical bundle. Particular cases of such are toric Fano manifolds, whose mirror is usually described by oscillating integrals defined by Laurent polynomials, and Calabi-Yau hypersurfaces in toric Fano manifolds. Both occur as special cases of non-affine Landau-Ginzburg models (LGMs). The authors construct LGMs for numerically effective intersections of toric manifolds as partial compactifications of families of Laurent polynomials. They show that the quantum $$\mathcal{D}$$-module of the ambient part of the cohomology of the submanifold is isomorphic to an intersection cohomology $$\mathcal{D}$$-module defined from the partial compactification. They deduce Hodge properties of these differential systems.

##### MSC:
 14J33 Mirror symmetry (algebro-geometric aspects) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects) 14D07 Variation of Hodge structures (algebro-geometric aspects) 34Mxx Ordinary differential equations in the complex domain 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
Full Text: