10.25394/PGS.8008955.v1
Avram W Steiner
<i>A</i>-Hypergeometric Systems and <i>D</i>-Module Functors
2019
Purdue University Graduate School
Algebraic geometry
Commutative algebra
D-modules
Local cohomology
Affine semigroup rings
Toric varieties
GKZ systems
2019-05-15 19:27:40
article
https://hammer.figshare.com/articles/_i_A_i_-Hypergeometric_Systems_and_i_D_i_-Module_Functors/8008955
<div>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". </div><div><br></div><div>If the semigroup ring of A is normal, we show that every A-hypergeometric system is "mixed Gauss–Manin". </div><div><br></div><div>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.</div>