Abstract
The Brieskorn lattice of an isolated hypersurface singularity gives rise to an invariant of the right equivalence class of the singularity. It is finer than the mixed Hodge structure of the singularity, and it is a good candidate for Torelli type questions. Here we prove a generic Torelli type result for semiquasihomogeneous singularities f(x 0, …, x n ) with weights (w 0,…, w n ) with n + − Σ i w i ≥ 4.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Andreotti, On a theorem of Torelli, Am. J. Math. 80 (1958), 801–828.
V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, Singularities of differentiable maps, volume I, Birkhäuser, Boston 1985.
V.I. Arnold, S.M. Gusein-Zade, A.N. Varchenko, Singularities of differentiable maps, volume II, Birkhäuser, Boston 1988.
E. Brieskorn, Die Monodromie der isolierten Singularitäten von Hyperflächen, Manuscripta Math. 2 (1970), 103–161.
J. Carlson, P. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem, In: Journées de géométrie algébrique d’Angers. Sijthoff and Nordhoff 1980,51–76.
J. Carlson, M. Green, P. Griffiths, J. Harris, Infinitesimal variations of Hodge structure (I),Comp. Math. 50 (1983), 109–205.
D. Cox, R. Donagi, L. Tu, Variational Torelli implies generic Torelli, Invent. Math. 88 (1987), 439–446.
R. Donagi, Generic Torelli for projective hypersurfaces, Comp. Math 50 (1983), 325–353.
R. Donagi, L. Tu, Generic Torelli for weighted hypersurfaces, Math. Ann. 276 (1987), 399–413.
G.-M. Greuel, C. Hertling, G. Pfister, Moduli spaces of semiquasihomogeneous singularities with fixed principal part, J. Algebraic Geometry 6 (1997), 169–199.
P. Griffiths, Periods of integrals on algebraic manifolds,III (some global differential geometric properties of the period mapping), Publ. Math. IHES 38 (1970), 125–180.
P. Griffiths, J. Harris, Principles of algebraic geometry. John Wiley and sons, New York 1978.
C. Hertling, Analytische Invarianten bei den unimodularen und bimodularen Hyperflächensingularitäten, Dissertation. Bonner Math. Schriften 250, Bonn 1992.
C. Hertling, Ein Torellisatz fii,r die unimodalen und bimodularen Hyperflächensingularitäten, Math. Ann. 302 (1995), 359–394.
C. Hertling, Brieskorn lattices and Torelli type theorems for cubics in IP 3 and for Brieskorn -Pham singularities with coprime exponents, In: Singularities, the Brieskorn anniversary volume. Progress in Mathematics 162. Birkhäuser Verlag, Basel-Boston-Berlin 1998, 167–194.
C. Hertling, Classifying spaces and moduli spaces for polarized mixed Hodge structures and for Brieskorn lattices, Compositio Math. 116 (1999), 1–37.
C. Hertling, Multiplication on the tangent bundle, First part of the habilitation, 1999 (also math.AG/9910116).
C. Hertling, Frobenius manifolds and moduli spaces for hypersurface singulari- ties, Second part of the habilitation, 2000.
Va.S. Kulikov, Mixed Hodge structures and singularities, Cambridge tracts in mathematics 132, Cambridge University Press, 1998.
B. Malgrange, Intégrales asymptotiques et monodromie,,Ann. Sci. École Norm. Sup. 7 (1974), 405–430.
J. Mather, Stability of C°°-maps IV. Classification of stable germs by RI-algebras, Publ. Math. I.H.E.S. 37 (1969), 223–248.
F. Pham, Structures de Hodge mixtes associées à un germe de fonction à point critique isolé, Astérisque 101–102 (1983), 268–285.
K. Saito, Quasihomogene isolierte Singularitäten von Hyperflächen, Invent. Math. 14 (1971), 123–142.
K. Saito, Period mapping associated to a primitive form, Publ. RIMS, Kyoto Univ. 19 (1983), 1231–1264.
M. Saito, On the structure of Brieskorn lattices, Ann. Inst. Fourier Grenoble 39 (1989), 27–72.
M. Saito, Period mapping via Brieskorn modules, Bull. Soc. math. France 119 (1991), 141–171.
J. Scherk, A propos d’un théorème de Mather et Yau, C. R. Acad. Sci. Paris, Série I, 296 (1983), 513–515.
J. Scherk, J.H.M. Steenbrink, On the mixed Hodge structure on the cohomology of the Milnor fibre, Math. Ann. 271 (1985), 641–665.
O.P. Sherbak, Conditions for the existence of a nondegenerate mapping with a given support, Func. Anal. Appl. 13 (1974), 154–155.
J.H.M. Steenbrink, Mixed Hodge structure on the vanishing cohomology, In: Real and complex singularities, Oslo 1976, P. Holm (ed.). Alphen aan den Rijn: Sijthoff and Noordhoff 1977, 525–562.
J.H.M. Steenbrink, Intersection form for quasihomogeneous singularities, Comp. Math. 34 (1977), 211–223.
B.L. van der Waerden, Algebra II. 3. Auflage, Springer, Berlin Heidelberg Göttingen 1955.
A.N. Varchenko, The asymptotics of holomorphic forms determine a mixed Hodge structure, Sov. Math. Dokl. 22 (1980), 772–775.
A.N. Varchenko, A lower bound for the codimension of the stratum µ =constantin terms of the mixed Hodge structure, Moscow Univ. Math. Bull. 37 (1982), 30–33.
C.T.C. Wall, Weighted homogeneous complete intersections, In: Algebraic geometry and singularities, La Rábida 1991. Progress in Math 134. Birkhäuser, Basel 1996, 277–300.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer Basel AG
About this chapter
Cite this chapter
Hertling, C. (2002). Generic Torelli for Semiquasihomogeneous Singularities. In: Libgober, A., Tibăr, M. (eds) Trends in Singularities. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8161-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8161-6_5
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9461-6
Online ISBN: 978-3-0348-8161-6
eBook Packages: Springer Book Archive