Mathematische Zeitschrift

, Volume 274, Issue 1–2, pp 249–261 | Cite as

Duality of Gauß–Manin systems associated to linear free divisors



We investigate differential systems occurring in the study of particular non-isolated singularities, the so-called linear free divisors. We obtain a duality theorem for these \({\mathcal{D}}\) -modules taking into account filtrations, and deduce degeneration properties of certain Frobenius manifolds associated to linear sections of the Milnor fibres of the divisor.


Frobenius manifold Linear free divisors Spectral numbers Holonomic dual Brieskorn lattice Birkhoff problem 

Mathematics Subject Classification (2000)

32S40 34M35 


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  1. 1.
    Buchweitz, R.-O., Mond, D.: Linear free divisors and quiver representations, Singularities and computer algebra (Cambridge). In: Lossen, C., Pfister, G. (eds.) London Math. Soc. Lecture Note Ser., vol. 324. Cambridge University Press, Cambridge; 2006. Papers from the conference held at the University of Kaiserslautern, Kaiserslautern, pp. 41–77 (2004)Google Scholar
  2. 2.
    Bridgeland, T.: Spaces of stability conditions. Algebraic geometry—Seattle 2005. Part 1. In: Abramovich, D., Bertram, A., Katzarkov, L., Pandharipande, R., Thaddeus, M. (eds.) Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence; 2009. Papers from the AMS Summer Research Institute held at the University of Washington, Seattle, pp. 1–21 (2005)Google Scholar
  3. 3.
    Antoine, D., Etienne, M.: The small quantum cohomology of a weighted projective space, a mirror \({\mathcal{D}}\) -module and their classical limits. Geom. Dedicata (2012). doi:10.1007/s10711-012-9768-3
  4. 4.
    Douai A.: A canonical Frobenius structure. Math. Z. 261(3), 625–648 (2009)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Douai A., Sabbah C.: Gauss–Manin systems, Brieskorn lattices and Frobenius structures I. Ann. Inst. Fourier (Grenoble) 53(4), 1055–1116 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Michel G., Mond D., Nieto A., Schulze M.: free divisors and the global logarithmic comparison theorem. Ann. Inst. Fourier (Grenoble) 59(1), 811–850 (2009)MATHMathSciNetGoogle Scholar
  7. 7.
    Gregorio I., Mond D., Sevenheck C.: Linear free divisors and Frobenius manifolds. Compos. Math. 145(5), 1305– (2009)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Granger M., Schulze M.: On the symmetry of b-functions of linear free divisors. Publ. Res. Inst. Math. Sci. 46(3), 479–506 (2010)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Reichelt T.: A construction of Frobenius manifolds with logarithmic poles and applications. Commun. Math. Phys. 287(3), 1145–1187 (2009)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Saito M.: Modules de Hodge polarisables. Publ. Res. Inst. Math. Sci. 24(6), 849–995 (1989)CrossRefGoogle Scholar
  11. 11.
    Saito M.: On the structure of Brieskorn lattice. Ann. Inst. Fourier (Grenoble) 39(1), 27–72 (1989)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Schapira P.: Microdifferential systems in the complex domain, Grundlehren der Mathematischen Wissenschaften. Fundamental Principles of Mathematical Sciences, vol. 269.. Springer, Berlin (1985)Google Scholar
  13. 13.
    Sevenheck C.: Bernstein polynomials and spectral numbers for linear free divisors. Ann. Inst. Fourier (Grenoble) 61(1), 379–400 (2011)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Sato, M., Kawai, T., Kashiwara Masaki.: Microfunctions and pseudo-differential equations, Hyperfunctions and pseudo-differential equations. In: Komatsu, H. (ed.) Proc. Conf., Katata, 1971. Lecture Notes in Mathematics, vol. 287. Springer, Berlin; 1973 (dedicated to the memory of André Martineau)Google Scholar
  15. 15.
    Saito M., Sturmfels B., Takayama N.: Gröbner deformations of hypergeometric differential equations. Algorithms and Computation in Mathematics, vol. 6. Springer, Berlin (2000)CrossRefGoogle Scholar
  16. 16.
    Takahashi, A.: Matrix factorizations and representations of quivers I. math.AG/0506347 (2005, preprint)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Lehrstuhl für Mathematik VI, Institut für MathematikUniversität MannheimMannheimGermany

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