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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.

##### MSC:
 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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