Purdue University Graduate School
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A-Hypergeometric Systems and D-Module Functors

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posted on 2019-05-15, 19:27 authored by Avram W SteinerAvram W Steiner
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 A-hypergeometric. This extends a result of Uli Walther, proving a conjecture of Nobuki Takayama in the normal homogeneous case.


National Science Foundation DMS-1401392


Degree Type

  • Doctor of Philosophy


  • Mathematics

Campus location

  • West Lafayette

Advisor/Supervisor/Committee Chair

Uli Walther

Additional Committee Member 2

Linquan Ma

Additional Committee Member 3

Donu Arapura

Additional Committee Member 4

Jaroslaw Wlodarczyk