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Tautological systems, homogeneous spaces and the holonomic rank problem

Paul Görlach, Thomas Reichelt, Christian Sevenheck, Avi Steiner, Uli Walther

Abstract

Many hypergeometric differential systems that arise from a geometric setting can be endowed with the structure of mixed Hodge modules. We generalize this fundamental result to the tautological systems associated to homogeneous spaces by giving a functorial construction for them. As an application, we solve the holonomic rank problem for such tautological systems in full generality.

Tautological systems, homogeneous spaces and the holonomic rank problem

Abstract

Many hypergeometric differential systems that arise from a geometric setting can be endowed with the structure of mixed Hodge modules. We generalize this fundamental result to the tautological systems associated to homogeneous spaces by giving a functorial construction for them. As an application, we solve the holonomic rank problem for such tautological systems in full generality.
Paper Structure (19 sections, 7 theorems, 236 equations)

This paper contains 19 sections, 7 theorems, 236 equations.

Table of Contents

  1. Introduction
  2. Mixed Hodge modules on line bundles
  3. Fourier--Laplace transformation
  4. Fourier--Laplace transformation on vector bundles
  5. Fourier--Laplace transform of extensions of O_{L^*}^\\ beta
  6. Twisted cohomology of hyperplane sections
  7. Non-vanishing criteria for tautological systems
  8. D-modules from group actions
  9. D-modules from equivariant line bundles
  10. Equivariant line bundles as A_Y-modules:
  11. Twist by characters and non-vanishing of tautological systems
  12. Application to projective homogeneous spaces
  13. Representation theoretic criterion
  14. Formula for beta(e)
  15. Geometric interpretation
  16. ...and 4 more sections

Key Result

Theorem 1.1

In the above situation, the following statements hold true.

Theorems & Definitions (54)

  • Theorem 1.1: \ref{['thm:restrictedTauHatDescription']}, \ref{['theo:tau_is_MHM']} and \ref{['cor:holRankProb']}
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 44 more