Tautological systems and quantum differential equations of homogeneous spaces

• Applicant: Professor Dr. Christian Sevenheck
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 527733662

This renewal proposal concerns primarily the study of differential systems defined by group actions on algebraic varieties. More specifically, we intend to continue the study of so-called tautological systems, both in general and with a view towards applications in algebraic geometry, singularity theory and in mirror symmetry. The latter concerns identification of quantum D-modules for homogeneous spaces with certain (Fourier-Laplace tranforms of) Gauss-Manin systems of Landau-Ginzburg models. These shall occur as dimensional reductions of appropriate tautological systems. One of the subprojects is concerned with identifying these quantum D-modules inside tautological systems. Concerning applications in algebraic geometry, we seek to exploit the relation between tautological systems of cones of projective G-varieties with the local cohomology of these cones. This identification shall be used to study the local cohomological dimension for singular G-varieties, and, on the other hand, to investigate the Bott vanishing property for certain homogeneous spaces. As a preliminary step, which is of independent interest we will determine the Hodge filtration on tautological system. This will allow us to express mirror symmetry as equivalence of non-commutative Hodge structures, thus reaching a similarly satisfactory statement as in the toric case. Another important topic is the (continuation of the) investigation of irregular Hodge theory for D-modules relevant in mirror symmetry of homogeneous space. This is mainly about the so-called Frenkel-Gross connections, and more generally rigid D-modules in arbitrary dimensions. For those, we plan to use recent advances about rescalable twistor D-modules to relate such rigid modules to classical Hodge theory via Fourier-Laplace transformation. A major part of the project shall be dedicated to study toric degenerations of tautological systems. This will involve calculating the weight filtration on the original tautological system, and using it (together with the relative weight filtration of the degeneration) to obtain a functorial description of the limit Hodge-module, which is conjectured to be a tautological system for the automorphism group of the central fibre of the degeneration, in particular, a quotient of a GKZ-system. Classical calculations with quantum differential equations of certain flag varieties shall be reinterpreted using these more general considerations about tautological systems.

• DFG Programme: Research Grants