Frobenius manifolds, twistor structures and singularity theory
Final Report Abstract
The overall aim of the project was the study of holomorphic resp. algebraic functions and their singularities which arise as the so-called B-model in mirror symmetry. In particular, we have looked at the case of (families of) Laurent polynomials, which occur as mirror partner of toric varieties resp. toric orbifolds. We have shown a mirror statement for nef toric varieties in terms of Frobenius manifolds with logarithmic poles along a certain boundary divisor. For that purpose, a detailed analysis of the Gauß-Manin systems of the functions in question have been done, with a number of side results on their duality theory and their Hodge theoretic properties. This part of the project has been carried out in collaboration with Th. Reichelt, who could independently show a statement on the existence of mixed Hodge module structures on certain hypergeometric differential systems. His result has been applied to study mirror symmetry for nef complete intersections in toric varieties, and we were able to construct a so-called non-affine Landau-Ginzburg model for them. It consists of a family of (singular) projective varieties. The intersection complex of this family models the quantum D-module of the complete intersection subvariety. This approach also unifies the a priori distinct Fano resp. nef-case and the more classical Calabi-Yau case. In another direction, potential B-model functions constructed from quiver representations have been studied. We could show a duality statement concerning their Gauß-Manin systems, this is an analogue of the classical higher residue pairings from the theory of isolated hypersurface singularities. Many examples of such functions have been investigated. Their cohomological invariants, such as Hodge-spectra, have been calculated. These calculations will serve as a basis for further studies of quiver discriminants and their hyperplane sections, and to advance on an understanding of their occurrence in mirror symmetry.
Publications
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Mirror symmetry, singularity theory and non-commutative Hodge structures, ”Jahresbericht der DMV”, volume 114, number 3, p. 131-162, 2012
Christian Sevenheck
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Non-affine Landau-Ginzburg models and intersection cohomology, 58 pages
Thomas Reichelt and Christian Sevenheck
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Direct images of mixed Hodge modules under open embeddings, 25 pages, proceedings of a 2013 Clay Workshop
Thomas Reichelt and Christian Sevenheck
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Duality of Gauß-Manin systems associated to linear free divisors, “Mathematische Zeitschrift”, volume 274, issue 1, p. 249-261, 2013
Christian Sevenheck
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Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules,“Journal of Algebraic Geometry”, 81 pages
Thomas Reichelt and Christian Sevenheck
