Skip to main content
SpringerLink
Log in

A sharp lower bound for the geometric genus and Zariski multiplicity question

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

It is well known that the geometric genus and multiplicity are two important invariants for isolated singularities. In this paper we give a sharp lower estimate of the geometric genus in terms of the multiplicity for isolated hypersurface singularities. In 1971, Zariski asked whether the multiplicity of an isolated hypersurface singularity depends only on its embedded topological type. This problem remains unsettled. In this paper we answer positively Zariski’s multiplicity question for isolated hypersurface singularity if Milnor number or geometric genus is small.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A’Campo, N.: Le nombre de Lefschetz d’une monodromie. Nederl. Akad. Wetensch. Proc. Ser. A 76 = Indag. Math. 35, 113–118 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of differentiable maps, vol. 2. Birkhäuser, Boston (1988)

    Book  MATH  Google Scholar 

  3. Artin, M.: isolated rational singularities of surfaces. Am. J. Math. 88, 129–136 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chen, I., Lin, K.-P., Yau, S.S.-T., Zuo, H.Q.: Coordinate-free characterization of homogeneous polynomials with isolated singularities. Commun. Anal. Geom. 19(4), 661–704 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. Durfee, A.H.: The signature of smoothings of complex surface singularities. Math. Ann. 232(1), 85–98 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eyral, C.: Zariskis multiplicity question—a survey. N. Zeal. J. Math. 36, 253–276 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Ishii, S.: On isolated Gorenstein singularities. Math. Ann. 270, 541–554 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ishii, S.: Small deformation of normal singularities. Math. Ann. 275, 139–148 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  9. Laufer, H.B.: On rational singularities. Am. J. Math. 94, 597–608 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  10. Laufer, H.B.: On \(\mu \) for surface singularities. Proc. Symp. Pure Math. Am. Math. Soc. Providence, R.I. 30, 45–49 (1977)

  11. Lê, D.T.: Calcul du nombre de cycles évanouissants d’une hypersurface complexe. Ann. Inst. Fourier (Grenoble) 23, 261–270 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lehmer, D.H.: The lattice points of an n-dimensional tetrahedron. Duke Math. J. 7, 341–353 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  13. Liang, A., Yau, S.S.-T., Zuo, H.Q.: A sharp estimate of positive integral points in 6-dimensional polyhedra and a sharp estimate of smooth numbers. Sci. China Math. 59(3), 425–444 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lin, K.P., Luo, X., Yau, S.S.-T., Zuo, H.Q.: On a number theoretic conjecture on positive integral points in a 5-dimensional tetrahedron and a sharp estimate of the Dickman-De Bruijn function. J. Eur. Math. Soc. 16(9), 1937–1966 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Luo, X., Yau, S.S.-T., Zuo, H.Q.: A sharp estimate of Dickman-De Bruijn function and a sharp polynomial estimate of positive integral points in 4-dimension tetrahedron. Math. Nachr. 288(1), 61–75 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  16. Merle, M., Teissier, B.: Conditions d’adjonction d’aprěs Du Val, Sĕminaire sur les singularités des surfaces (Center de Math. de l’Ecole Polytechniqe, 1976–1977), Lecture Notes in Math., vol. 777, pp. 229–245. Springer, Berlin (1980)

  17. Milnor, J., Orlik, P.: Isolated singularities defined by weighted homogeneous polynomials. Topology 9, 385–393 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  18. Navarro Aznar, V.: Sobre la invariància topològica de la multiplicitat. Pub. Sec. Mat. Univ. Autònoma Barcelona 20, 261–262 (1980)

    MathSciNet  Google Scholar 

  19. Neumann, W.: A calculus for plumbing applied to the topology of complex surface singularities and degenerating complex curves. Trans. AMS 268, 299–344 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  20. Okuma, T.: The plurigenera of Gorenstein surface singularities. Manuscripta. Math. 94, 187–194 (1997)

  21. Perron, B.: Conjugaison topologique des germes de fonctions holomorphes à singularité isolée en dimension trois. Invent. Math. 82, 27–35 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Steenbrink, JHM.: Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math. 40(Part 2), 513–536 (1983)

  23. Teissier, B.: Déformations à type toplogique constant. Séminaire Douady-Verdier 1971-72, Astérisque (Société Mathématique de France) 16, 215–249 (1974)

  24. Watanabe, K.: On plurigenera of normal isolated singularities I. Math. Ann. 250, 65–94 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xu, Y.J., Yau, S.S.-T.: Classification of topological types of isolated quasi-homogeneous two dimensional hypersurfaces singularities. Manuscripta Math. 64, 445–469 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yau, S.S.-T.: Two theorems on higher dimensional singularities. Math. Ann. 231, 55–59 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yau, S.S.-T.: Topological types and multiplicities of isolated quasi-homogeneous surfaces singularities. Bull. Am. Math. Soc. 19, 447–454 (1988)

    Article  MATH  Google Scholar 

  28. Yau, S.S.-T.: The multiplicity of isolated twodimensional hypersurface singularities: Zariski problem. Am. J. Math. 111, 753–767 (1989)

    Article  MATH  Google Scholar 

  29. Yau, S.S.-T., Yuan, B.H., Zuo, H.Q.: On the polynomial sharp upper estimate conjecture in 7-dimensional simplex. J. Numb. Theory 160, 254–286 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yau S. S.-T., Zhao L., Zuo H.Q.: Biggest sharp polynomial estimate of integral points in right-angled simplices. Topology of algebraic varieties and singularities, vol. 538, pp. 433–467. Contemp. Math., Am. Math. Soc., Providence, RI (2011)

  31. Yau, S.S.-T., Zuo, H.Q.: Lower estimate of Milnor number and characterization of isolated homogeneous hypersurface singularities. Pacif. J. Math. 260(1), 245–255 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  32. Yau, S.S.-T., Zuo, H.Q.: Characterization of isolated complete intersection singularities with \(\mathbb{C}^*\)-action of dimension \(n\ge 2\) by means of geometric genus and irregularity. Commun. Anal. Geom. 21(3), 509–526 (2013)

    Article  MATH  Google Scholar 

  33. Yau, S.S.-T., Zuo, H.Q.: Complete characterization of isolated homogeneous hypersurface singularities. Pacif. J. Math. 273(1), 213–224 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zariski, O.: Open questions in the theory of singularities. Bull. Am. Math. Soc. 77, 481–491 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zariski, O.: On the topology of algebroid singularities. Am. J. Math. 54, 453–465 (1932)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Stephen S.-T. Yau

  2. Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, People’s Republic of China

    Huaiqing Zuo

Authors
  1. Stephen S.-T. Yau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Huaiqing Zuo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stephen S.-T. Yau.

Additional information

Research partially supported by NSFC (Grant nos. 11531007, 11771231, 11401335), Tsinghua University Initiative Scientific Research Program and start-up fund from Tsinghua University.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yau, S.ST., Zuo, H. A sharp lower bound for the geometric genus and Zariski multiplicity question. Math. Z. 289, 1299–1310 (2018). https://doi.org/10.1007/s00209-017-1999-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-017-1999-4

Keywords

Mathematics Subject Classification

Search

Navigation