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Linear free divisors and Frobenius manifolds

Part of: Representation theory of rings and algebras Singularities Local theory Theory of singularities and catastrophe theory

Published online by Cambridge University Press:  23 October 2009

Ignacio de Gregorio ,
David Mond  and
Christian Sevenheck
Ignacio de Gregorio
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom (email: degregorio@gmail.com)
David Mond
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom (email: D.M.Q.Mond@warwick.ac.uk)
Christian Sevenheck
Affiliation:
Lehrstuhl VI für Mathematik, Universität Mannheim, A6 5, 68131 Mannheim, Germany (email: Christian.Sevenheck@uni-mannheim.de)
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Abstract

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We study linear functions on fibrations whose central fibre is a linear free divisor. We analyse the Gauß–Manin system associated to these functions, and prove the existence of a primitive and homogenous form. As a consequence, we show that the base space of the semi-universal unfolding of such a function carries a Frobenius manifold structure.

Keywords

linear free divisorsprehomogenous vector spacesquiver representationsGauß–Manin systemBrieskorn latticeBirkhoff problemspectral numbersFrobenius manifolds

MSC classification

Secondary: 16G20: Representations of quivers and partially ordered sets 32S40: Monodromy; relations with differential equations and $D$-modules 14B07: Deformations of singularities 58K60: Deformation of singularities
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 5 , September 2009 , pp. 1305 - 1350
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Barannikov, S., Semi-infinite Hodge structures and mirror symmetry for projective spaces, Preprint (2000), math.AG/0010157.Google Scholar
[2]Buchweitz, R.-O. and Mond, D., Linear free divisors and quiver representations, in Singularities and computer algebra, London Mathematical Society Lecture Note Series, vol. 324 (Cambridge University Press, Cambridge, 2006), 41–77.CrossRefGoogle Scholar
[3]Broughton, S. A., Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math. 92 (1988), 217–241.Google Scholar
[4]Coates, T., Corti, A., Lee, Y.-P. and Tseng, H.-H., The Quantum Orbifold Cohomology of weighted projective spaces, Acta Math. 202 (2009), 139–193.Google Scholar
[5]Damon, J., On the legacy of free divisors. III. Functions and divisors on complete intersections, Q. J. Math. 57 (2006), 49–79.Google Scholar
[6]de Gregorio, I., Some examples of non-massive Frobenius manifolds in singularity theory, J. Geom. Phys. 57 (2007), 1829–1841.CrossRefGoogle Scholar
[7]de Gregorio, I. and Sevenheck, C., Good bases for some linear free divisors associated to quiver representations, (2009), work in progress.Google Scholar
[8]Dimca, A., Sheaves in topology (Springer, Berlin, 2004).Google Scholar
[9]Douai, A., Construction de variétés de Frobenius via les polynômes de Laurent: une autre approche, Singularités, Publications de l’Institut Élie Cartan, vol. 18 (Université de Nancy, 2005), 105–123, updated version available under math.AG/0510437.Google Scholar
[10]Douai, A., Examples of limits of Frobenius (type) structures: the singularity case, Preprint (2008), math.AG/0806.2011.Google Scholar
[11]Douai, A. and Sabbah, C., Gauss–Manin systems, Brieskorn lattices and Frobenius structures. I, Ann. Inst. Fourier (Grenoble) 53 (2003), 1055–1116.Google Scholar
[12]Douai, A. and Sabbah, C., Gauss Manin systems, Brieskorn lattices and Frobenius structures. II, in Frobenius manifolds, eds C. Hertling and M. Marcolli, Aspects of Mathematics, vol. E36 (Vieweg, Wiesbaden, 2004), 1–18.Google Scholar
[13]Dubrovin, B., Geometry of 2D topological field theories, in Integrable systems and quantum groups. Lectures given at the 1st session of the Centro Internazionale Matematico Estivo (CIME) held in Montecatini Terme, Italy, June 14–22, 1993, Lecture Notes in Mathematics, vol. 1620, eds M. Francaviglia and S. Greco (Springer, Berlin, 1996), 488.Google Scholar
[14]Givental, A., Homological geometry and mirror symmetry, Proceedings of the International Congress of Mathematicians, vols 1, 2 (Birkhäuser, Basel, 1995), 472–480.Google Scholar
[15]Givental, A., A mirror theorem for toric complete intersections, in Topological field theory, primitive forms and related topics (Kyoto, 1996), Progress in Mathematics, vol. 160 (Birkhäuser Boston, Boston, MA, 1998), 141–175.Google Scholar
[16]Granger, M., Mond, D., Nieto, A. and Schulze, M., Linear free divisors and the global logarithmic comparison theorem, Ann. Inst. Fourier (Grenoble) 59 (2009), 811–850.Google Scholar
[17]Greuel, G.-M., Pfister, G. and Schönemann, H., Singular 3.0, A computer algebra system for polynomial computations, Centre for Computer Algebra, University of Kaiserslautern, 2005, http://www.singular.uni-kl.de.Google Scholar
[18]Guzzetti, D., Stokes matrices and monodromy of the quantum cohomology of projective spaces, Comm. Math. Phys. 207 (1999), 341–383.Google Scholar
[19]Hertling, C., Frobenius manifolds and moduli spaces for singularities, Cambridge Tracts in Mathematics, vol. 151 (Cambridge University Press, Cambridge, 2002).CrossRefGoogle Scholar
[20]Hertling, C., tt *-geometry, Frobenius manifolds, their connections, and the construction for singularities, J. Reine Angew. Math. 555 (2003), 77–161.Google Scholar
[21]Hertling, C. and Manin, Y., Unfoldings of meromorphic connections and a construction of Frobenius manifolds, in Frobenius manifolds, eds C. Hertling and M. Marcolli, Aspects of Mathematics, vol. E36 (Vieweg, Wiesbaden, 2004), 113–144.Google Scholar
[22]Hochschild, G. and Serre, J.-P., Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603.Google Scholar
[23]Hertling, C. and Sevenheck, C., Nilpotent orbits of a generalization of Hodge structures, J. Reine Angew. Math. 609 (2007), 23–80.Google Scholar
[24]Malgrange, B., Deformations of differential systems. II, J. Ramanujan Math. Soc. 1 (1986), 3–15.Google Scholar
[25]Mann, E., Orbifold quantum cohomology of weighted projective spaces, J. Algebraic Geom. 17 (2008), 137–166.Google Scholar
[26]Mather, J. N., Stability of C ∞ mappings. IV. Classification of stable germs by R-algebras, Publ. Math. Inst. Hautes Études Sci. 37 (1969), 223–248.CrossRefGoogle Scholar
[27]Mochizuki, T., Asymptotic behaviour of tame harmonic bundles and an application to pure twistor -modules, Part 1, Mem. Amer. Math. Soc. 185 (2007), xi+324Google Scholar
[28]Némethi, A. and Sabbah, C., Semicontinuity of the spectrum at infinity, Abh. Math. Sem. Univ. Hamburg 69 (1999), 25–35.Google Scholar
[29]Reichelt, Th., A construction of Frobenius manifolds with logarithmic poles and applications, Comm. Math. Phys. 287 (2009), 1145–1187.Google Scholar
[30]Sabbah, C., Hypergeometric periods for a tame polynomial, Port. Math. (N.S.) 63 (2006), 173–226, written in 1998.Google Scholar
[31]Sabbah, C., Isomonodromic deformations and Frobenius manifolds, Universitext (Springer, 2007).Google Scholar
[32]Sabbah, C., Fourier-Laplace transform of a variation of polarized complex Hodge structure, J. Reine Angew. Math. 621 (2008), 123–158.Google Scholar
[33]Saito, K., Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980), 265–291.Google Scholar
[34]Saito, M., On the structure of Brieskorn lattice, Ann. Inst. Fourier (Grenoble) 39 (1989), 27–72.CrossRefGoogle Scholar
[35]Sato, M. and Kimura, T., A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.Google Scholar
[36]Teissier, B., Variétés polaires. II. Multiplicités polaires, sections planes, et conditions de Whitney, in Algebraic geometry (La Rábida, 1981), Lecture Notes in Mathematics, vol. 961, eds Aroca José Manuel, Buchweitz Ragnar, Giusti Marc and Merle Michel (Springer, Berlin, 1982), 314–491.Google Scholar
[37]Tráng, L. D. and Teissier, B., Cycles evanescents, sections planes et conditions de Whitney. II, Singularities, Part 2 (Arcata, Calif., 1981), Proceedings Symposia in Pure Mathematics, vol. 40, ed. Orlik Peter (American Mathematical Society, Providence, RI, 1983), 65–103.Google Scholar