Mirror Symmetry and Irregular Singularities coming from Physics
Final Report Abstract
Systems of meromorphic differential equations arise naturally in algebraic geometry and in mathematical physics, and especially in mirror symmetry, which connects these two domains. The differential equations which one encounters in this way have special properties which have to be considered from different angles. First to be mentioned is the monodromy, which is integer in many concrete cases, but which is defined only over Q or an algebraic number field or R in more abstract cases. Considering the meromorphic system as a D-module leads to the more general question to determine whether its (derived) solutions can be defined over a proper subfield of the complex numbers (which is referred to as “Betti structure” of this D-module). Other points of view involve the study of irregular singularities or of (mixed) Hodge structures, which exist is many cases of geometric origin. In the project, the integral monodromy of the Gauß-Manin connection of invertible polynomials has been studied, and an old conjecture of Orlik on this monodromy was proved. Together with an additional result on the Orlik blocks (occurring as building blocks), this allows one to understand the automorphisms of the system. Another main line of research that was pursued in this project concerns systems of hypergeometric differential equations (GKZ-systems) and generalizations thereof. Both invertible polynomials and GKZ-systems arise prominently in mirror symmetry. For certain onedimensional hypergeometric systems, criteria for the existence of Betti structures have been established. Tautological systems (which generalize GKZ-systems beyond the case of torus actions) have been studied and it has been shown that they underly mixed Hodge modules. As an application, the holonomic rank problem (as stated and studied by Yau et al.) for such systems has been solved in full generality. Finally, we have studied a natural generalization of GKZ-systems involving character groups with torsion that play a role in mirror symmetry for orbifolds.
Publications
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Algebraic aspects of hypergeometric differential equations. Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, 62(1), 137-203.
Reichelt, Thomas; Schulze, Mathias; Sevenheck, Christian & Walther, Uli
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Betti Structures of Hypergeometric Equations. International Mathematics Research Notices, 2023(12), 10641-10701.
Barco, Davide; Hien, Marco; Hohl, Andreas & Sevenheck, Christian
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The combinatorics of weight systems and characteristic polynomials of isolated quasihomogeneous singularities. Journal of Algebraic Combinatorics, 56(3), 929-954.
Hertling, Claus & Mase, Makiko
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The integral monodromy of isolated quasihomogeneous singularities. Algebra & Number Theory, 16(4), 955-1024.
Hertling, Claus & Mase, Makiko
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The integral monodromy of the cycle type singularities. Journal of Singularities, 25(2022).
Hertling, Claus & Mase, Makiko
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“Tautological systems, homogeneous spaces and the holonomic rank problem”, arXiv:2211.05356, 57 pages
Gorlach, Paul; Reichelt, Thomas; Steiner, Avi; Sevenheck, Christian & Uli, Walther
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A note on simple K3 singularities and families of weighted K3 surfaces. Rendiconti del Circolo Matematico di Palermo Series 2, 73(1), 19-44.
Mase, Makiko
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“Hypergeometric systems from groups with torsion”, arXiv:2402.00762, 16 pages
Reichelt, Thomas; Sevenheck, Christian & Walther, Uli
