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Mellin–Barnes integrals as Fourier–Mukai transforms. (English) Zbl 1137.14314
Summary: We study the generalized hypergeometric system introduced by I. M. Gel’fand, A. V. Zelevinskij and M. M. Kapranov [Funct. Anal. Appl. 23, No. 2, 94–106 (1989; Zbl 0721.33006)] and its relationship with the toric Deligne–Mumford (DM) stacks recently studied by L. Borisov, L. Chen and G. G. Smith [J. Am. Math. Soc. 18, No. 1, 193–215 (2005; Zbl 1178.14057)]. We construct series solutions with values in a combinatorial version of the Chen–Ruan (orbifold) cohomology and in the \(K\)-theory of the associated DM stacks. In the spirit of the homological mirror symmetry conjecture of Kontsevich, we show that the \(K\)-theory action of the Fourier–Mukai functors associated to basic toric birational maps of DM stacks are mirrored by analytic continuation transformations of Mellin–Barnes type.

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14J32 Calabi-Yau manifolds (algebro-geometric aspects)
14C15 (Equivariant) Chow groups and rings; motives
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E08 \(K\)-theory of schemes
33C70 Other hypergeometric functions and integrals in several variables
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
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