D-modules in geometry and physics

• Applicant: Privatdozent Dr. Thomas Reichelt
• Subject Area: Mathematics
• Term: from 2014 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 268670937

• Final Report Year: 2023

Final Report Abstract

The notion of a D-module which is a generalization of a linear partial differential equation was introduced by Kashiwara at the beginning of the 70’s. Through the Riemann-Hilbert correspondence they are closely linked with perverse sheaves whose basic building blocks are the intersection complexes of Goresky and MacPherson. D-modules turned out to be an indispensable tool to study singularities of maps between algebraic varieties. These D-modules of ”geometric origin” carry an extra structures, namely so-called Hodge and weight filtrations which are notoriously difficult to compute. One of the major achievements of this project so far was the explicit calculation of these two filtrations in the case of the Gelfand-Kapranov-Zelevinsky hypergeometric systems which are closely related to the geometry of Laurent polynomials in several variables. These maps play a prominent role in mirror symmetry since they serve as Landau- Ginzburg mirror partners for complete intersections in toric varieties. Here mirror symmetry provides a mysterious duality between enumerative geometry of algebraic varieties, captured by the quantum cohomology, on one side and highly transcendental period integrals, captured by D-modules, on the other side. The explicit knowledge of the Hodge filtration on the GKZ-system enabled us to prove a conjecture of Katzarkov-Kontsevich-Pantev on the existence of a non-commutative Hodge structure on the quantum cohomology of a projective variety.

Publications

• On theb-Functions of Hypergeometric Systems. International Mathematics Research Notices, 2018(21), 6535-6555.
  Reichelt, Thomas; Sevenheck, Christian & Walther, Uli
  
• Examples of hypergeometric twistor D-modules. Algebra & Number Theory, 13(6), 1415-1442.
  Castaño, Domínguez Alberto; Reichelt, Thomas & Sevenheck, Christian
  
• Gauß–Manin systems of families of Laurent polynomials and A-hypergeometric systems. Communications in Algebra, 47(6), 2503-2524.
  Reichelt, Thomas & Walther, Uli
  
• Hypergeometric Hodge modules. Algebraic Geometry, 263-345.
  Sevenheck, Christian
  
• Dependence of Lyubeznik numbers of cones of projective schemes on projective embeddings. Selecta Mathematica, 27(1).
  Reichelt, Thomas; Saito, Morihiko & Walther, Uli
  
• On Lyubeznik type invariants. Topology and its Applications, 313, 107983.
  Reichelt, Thomas; Walther, Uli & Zhang, Wenliang
  
• Weight filtrations on GKZ-systems. American Journal of Mathematics, 144(5), 1437-1484.
  Reichelt, Thomas & Walther, Uli
  
• Topological calculation of local cohomological dimension. Journal of Singularities, 26.
  Reichelt, Thomas; Saito, Morihiko & Walther, Uli
  
• Log degenerations of LG models for toric orbifolds and tt∗ geometry. Kyoto Journal of Mathematics, 65(2).
  Mann, Etienne & Reichelt, Thomas