A-Hypergeometric Systems and D-Module Functors

Avi Steiner, Purdue University

Abstract

Let A be a d by n integer matrix. Gel’fand et al. proved that most A-hypergeometric systems have an interpretation as a Fourier–Laplace transform of a direct image. The set of parameters for which this happens was later identified by Schulze and Walther as the set of not strongly resonant parameters of A. A similar statement relating A-hypergeometric systems to exceptional direct images was proved by Reichelt. In the first part of this thesis, we consider a hybrid approach involving neighborhoods U of the torus of A and consider compositions of direct and exceptional direct images. Our main results characterize for which parameters the associated A-hypergeometric system is the inverse Fourier–Laplace transform of such a “mixed Gauss-Manin” system. If the semigroup ring of A is normal, we show that every A-hypergeometric system is “mixed Gauss–Manin”. In the second part of this thesis, we use our notion of mixed Gauss–Manin systems to show that the projection and restriction of a normal A-hypergeometric system to the coordinate subspace corresponding to a face are isomorphic up to cohomological shift; moreover, they are essentially hypergeometric. We also show that, if A is in addition homogeneous, the holonomic dual of an A-hypergeometric system is itself Ahypergeometric. This extends a result of Uli Walther, proving a conjecture of Nobuki Takayama in the normal homogeneous case.

Degree

Ph.D.

Advisors

Walther, Purdue University.

Subject Area

Applied Mathematics|Mathematics|Transportation

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