Stokes matrices of isolated hypersurface singularities

Balnojan, Sven

phdthesis-as-of-2018-04-14.pdf - Published

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URN: urn:nbn:de:bsz:180-madoc-451477
Document Type: Doctoral dissertation
Year of publication: 2018
Place of publication: Mannheim
University: Universität Mannheim
Evaluator: Hertling , Claus
Date of oral examination: 9 March 2018
Publication language: English
Institution: School of Business Informatics and Mathematics > Algebraische Geometrie (Hertling 2005-)
License: CC BY 4.0 Creative Commons Attribution 4.0 International (CC BY 4.0)
Subject: 510 Mathematics
Subject headings (SWD): singularity theory
Keywords (English): singularity theory , stokes matrices
Abstract: This thesis provides a classification of Stokes matrices of isolated hypersurface singularities (ihs), based on their spectrum in the case that the intersection form on the middle homology is positive (semi-) definite. We also provide results for the case of an indefinite intersection form, namely we show that certain matrices, so-called HOR matrices, carry a form of a polarized mixed Hodge structure which enables the definition of a spectrum for them. The first part of thesis deals with the positive (semi-) definite case. To investigate the root lattices that arise from any upper triangular matrix with ones on the diagonal, we introduce the language of p.n. root lattices. Extending old results, we establish their classification and compute important numbers for (lattice, sublattice) pairs. We prove an interesting theorem which is that any root generating system of a root lattice as Z-module also contains a Z-basis. Using those results we are able to prove that an upper triangular matrix S, where S + St is positive (semi-) definite, equipped with an ad-hoc spectrum in this case, can be classified as belonging to an ihs if the spectrum of their variance fulfills the Hertling variance inequality, their Coxeter Dynkin diagram is connected and a trace is correct. The second part of the thesis deals with the indefinite case. We recall the classification of isometric triples and thereby of real Seifert pairs, based on works by Milnor and Némethi. We use those abstract results to provide a Thom-Sebastiani formula for TEZP structures. Using the classification, we are able to prove that any so-called HOR matrix carries a Steenbrink polarized mixed Hodge structure. This in turn enables us to define a way of assigning a spectrum, only based on the HOR matrix and the monodromy eigenvalues, which agrees with the Steenbrink spectrum, in the case the matrix represents a distinguished basis of an ihs. This work is based the physicists Cecotti & Vafa and conjectures they made. Building upon this spectrum, we are able to prove some results for chain type singularities, which serve as ample evidence, that their conjecture is true, and a natural way of assigning spectral numbers should exist.

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