Communications in Mathematical Physics

, Volume 287, Issue 3, pp 1145–1187 | Cite as

A Construction of Frobenius Manifolds with Logarithmic Poles and Applications

Article

Abstract

A construction theorem for Frobenius manifolds with logarithmic poles is established. This is a generalization of a theorem of Hertling and Manin. As an application we prove a partial generalization of the reconstruction theorem of Kontsevich and Manin for projective smooth varieties with convergent Gromov-Witten potential. A second application is a construction of Frobenius manifolds out of a variation of polarized Hodge structures which degenerates along a normal crossing divisor when certain generation conditions are fulfilled.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. CaK.
    Cattani E., Kaplan A.: Polarized mixed Hodge structures and the local monodromy of a variation of Hodge structures. Invent. Math. 67, 101–115 (1982)MATHCrossRefADSMathSciNetGoogle Scholar
  2. CaKSch.
    Cattani E., Kaplan A., Schmid W.: Degeneration of Hodge structures. Ann. Math. 123, 457–535 (1986)CrossRefMathSciNetGoogle Scholar
  3. CaFe.
    Cattani E., Fernandez J.: Frobenius modules and Hodge asymptotics. Commun. Math. Phys. 238, 489–504 (2003)MATHCrossRefADSMathSciNetGoogle Scholar
  4. CK.
    Cox, D.A., Katz, S.: Mirror symmetry and Algebraic Geometry. Mathematical Surveys and monographs, vol. 68, Providence, RI: Amer. Math. Soc., 1999Google Scholar
  5. De1.
    Deligne P.: Théorie de Hodge. II. Publ. Math. I.H.E.S. 40, 5–57 (1971)MATHMathSciNetGoogle Scholar
  6. De2.
    Deligne, P.: Local behaviour of Hodge structures at infinity. In: Mirror Symmetry II (eds. Greene, B., Yau, S.-T.), Studies in Advanced Mathematics, volume 1, Providence, RI: AMS/IP, 1997, pp 683–699Google Scholar
  7. Du.
    Dubrovin, B.: Geometry of 2D topological field theories. In: Integrable Systems and quantum groups (Montecatini, Terme 1993), (Francoviglia, M., Greco, S., eds.). Lecture Notes in Math. 1620, Berlin-Heidelberg-New York: Springer Verlag 1996, pp 120–348Google Scholar
  8. EV.
    Esnault H., Viehweg E.: Logarithmic de Rham complexes and vanishing theorems. Invent. Math. 86, 161–194 (1986)MATHCrossRefADSMathSciNetGoogle Scholar
  9. FePe.
    Fernandez, J., Pearlstein, G.: Opposite filtrations, variations of Hodge structure, and Frobenius modules. In: Frobenius manifolds, eds. Hertling, C., Marcolli, M., Aspects of Mathematics E 36, Wiesbaden: Vieweg, 2004, pp 19–34Google Scholar
  10. Fo.
    Folland G.: Introduction to partial differential equations 2nd ed. Princeton University Press, Princeton, NJ (1995)MATHGoogle Scholar
  11. He1.
    Hertling, C.: Frobenius manifolds and moduli spaces for singularities. Cambridge Tracts Math. 151, Cambridge: Cambridge University Press, 2002Google Scholar
  12. He2.
    Hertling C.: tt* geometry, Frobenius manifolds, their connections, and the construction for singularities. J. Reine Angew. Math. 555, 77–161 (2003)MATHMathSciNetGoogle Scholar
  13. He3.
    Hertling C.: Classifying spaces and moduli spaces for polarized mixed Hodge structures and for Brieskorn lattices. Compos. Math. 116, 1–37 (1999)MATHCrossRefMathSciNetGoogle Scholar
  14. HM.
    Hertling, C., Manin, Yu.: Unfoldings of meromorphic connections and a construction of Frobenius manifolds. In: Frobenius manifolds, eds. Hertling, C., Marcolli, M., Aspects of Mathematics E36, Wiesbaden: Vieweg, 2004, pp 113–144Google Scholar
  15. KM.
    Kontsevich M., Manin Yu.: Gromov-Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994)MATHCrossRefADSMathSciNetGoogle Scholar
  16. Mal.
    Malgrange B.: Deformations of differential systems, II. J. Ramanujan Math. Soc. 1, 3–15 (1986)MATHMathSciNetGoogle Scholar
  17. Ma.
    Manin Yu.: Three constructions of Frobenius manifolds: a comparitive study. Asian J. Math 3, 179–220 (1999)MATHMathSciNetGoogle Scholar
  18. Sab.
    Sabbah, C.: Déformation isomonodromique et variétés de Frobenius, une introduction. Centre des Mathematiques, Ecole Polytechnique, U.M.R. 7640 du C.N.R.S., no. 2000-05, 251 pagesGoogle Scholar
  19. Sch.
    Schmid W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)MATHCrossRefADSMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institut für MathematikUniversität MannheimMannheimGermany

Personalised recommendations