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On stringy cohomology spaces. (English) Zbl 1292.14029
The first obstacle in generalizing mirror symmetry as equality of the Hodge numbers in higher dimensions is that singular Calabi-Yau varieties often do not admit crepant resolutions. A solution to this was obtained by V. V. Batyrev and D. I. Dais [Topology 35, No. 4, 901–929 (1996; Zbl 0864.14022)] through introducing the \(E\)-function for the varieties with log-terminal singularities. For toroidal varieties the \(E\)-function is a polynomial and one can define the stringy Hodge numbers as its coefficients which are the same as the usual Hodge numbers of a crepant resolution (if exists). The equality of the stringy Hodge numbers have been proven for a large class of examples of mirror pairs given as complete intersections in Gorenstein toric Fano varieties. In the setting of toric mirror symmetry, L. A. Borisov and A. R. Mavlyutov [Adv. Math. 180, No. 1, 355–390 (2003; Zbl 1055.14044)] proposed a combinatorial definition for the stringy cohomology vector spaces. They are proven to have natural double grading such that the dimensions of the double graded components recover the stringy Hodge numbers. In some cases, Borisov and Mavlyutov were able to identify the stringy cohomology spaces with the chiral rings of certain \(N=2\) vertex algebras. The proof of this contained a minor gap, however if proven, it allows one to define two different graded supercommutative associative products on the stringy cohomology spaces as pursued further by L. A. Borisov [Math. Z. 248, No. 3, 567–591 (2004; Zbl 1067.14034)].
The paper under review fills in the gap in the proof of what mentioned above by proving a new result in intersection cohomology of polyhedral fans. The main result of the paper under review relates the stringy cohomology spaces to Gelgand-Kapranov-Zelevisky (GKZ) hypergeometric system of partial differential equations. This is obtained by modifying the definition of the families of \(A\) and \(B\) stringy cohomology spaces associated to a pair of dual reflexive Gorenstein cones given by Borisov and Mavlyutov and showing they are isomorphic. The proof of the main result is competed by proving the existence of a natural flat connection on the \(B\) stringy cohomology bundle which is related to GKZ hypergeometric system.

14J33 Mirror symmetry (algebro-geometric aspects)
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)
17B69 Vertex operators; vertex operator algebras and related structures
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[1] G. Barthel, J.-P. Brasselet, K.-H. Fieseler, and L. Kaup, Combinatorial intersection cohomology for fans , Tohoku Math. J. (2) 54 (2002), 1-41. · Zbl 1055.14024
[2] V. V. Batyrev, Variations of the mixed Hodge structure of affine hypersurfaces in algebraic tori , Duke Math. J. 69 (1993), 349-409. · Zbl 0812.14035
[3] V. V. Batyrev, “Stringy Hodge numbers of varieties with Gorenstein canonical singularities” in Integrable Systems and Algebraic Geometry (Kobe/Kyoto, 1997) , World Scientific, River Edge, N.J., 1998, 1-32. · Zbl 0963.14015
[4] V. V. Batyrev and L. A. Borisov, Mirror duality and string-theoretic Hodge numbers , Invent. Math. 126 (1996), 183-203. · Zbl 0872.14035
[5] V. V. Batyrev and D. I. Dais, Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry , Topology 35 (1996), 901-929. · Zbl 0864.14022
[6] V. V. Batyrev and B. Nill, “Combinatorial aspects of mirror symmetry” in Integer Points in Polyhedra - Geometry, Number Theory, Representation Theory, Algebra, Optimization, Statistics (Snowbird, Utah, 2006) , Contemp. Math. 452 , Amer. Math. Soc., Providence, 2008, 35-66. · Zbl 1161.14037
[7] L. A. Borisov, Vertex algebras and mirror symmetry , Comm. Math. Phys. 215 (2001), 517-557. · Zbl 0990.17023
[8] L. A. Borisov, Chiral rings of vertex algebras of mirror symmetry , Math. Z. 248 (2004), 567-591. · Zbl 1067.14034
[9] L. A. Borisov, Berglund-Hübsch mirror symmetry via vertex algebras , Comm. Math. Phys. 320 (2013), 73-99. · Zbl 1317.17032
[10] L. A. Borisov and R. P. Horja, On the better behaved version of the GKZ hypergeometric system , Math. Ann. 357 (2013), 585-603. · Zbl 1284.33012
[11] L. A. Borisov and R. M. Kaufmann, On CY-LG correspondence for \((0,2)\) toric models , Adv. Math. 230 (2012), 531-551. · Zbl 1271.14074
[12] L. A. Borisov and A. R. Mavlyutov, String cohomology of Calabi-Yau hypersurfaces via mirror symmetry , Adv. Math. 180 (2003), 355-390. · Zbl 1055.14044
[13] P. Bressler and V. A. Lunts, Intersection cohomology on nonrational polytopes , Compos. Math. 135 (2003), 245-278. · Zbl 1024.52005
[14] K. Karu, Hard Lefschetz theorem for nonrational polytopes , Invent. Math. 157 (2004), 419-447. · Zbl 1077.14071
[15] F. Malikov, V. Schechtman, and A. Vaintrob, Chiral de Rham complex , Comm. Math. Phys. 204 (1999), 439-473. · Zbl 0952.14013
[16] R. Stanley, “Generalized \(H\)-vectors, intersection cohomology of toric varieties, and related results” in Commutative Algebra and Combinatorics (Kyoto, 1985) , Adv. Stud. Pure Math. 11 , North-Holland, Amsterdam, 1987, 187-213. · Zbl 0652.52007
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