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Generalized Tjurina numbers of an isolated complete intersection singularity

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Abstract.

We define a series of non-negative integer valued invariants the generalized Tjurina numbers, of an analytic n-dimensional isolated complete intersection singularity. The Tjurina number τ equals τ(1). We show that all these numbers are bounded by the Milnor number μ by establishing explicit formulas for the differences μ−τ(p) involving the mixed Hodge numbers of the cohomology of the link and a new range of analytic invariants for (X,x). This generalizes earlier results of Looijenga and Steenbrink on the relation between the Milnor number and the Tjurina number. We show that singularities satisfying μ=τ(1)=⋯=τ(n−1) behave cohomologically like weighted homogeneous singularities, and we expect this to be a criterion for weighted homogeneity.

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  1. Department of Mathematics, Aarhus University, Ny Munkegade, DK-8000, Aarhus C

    Henrik Vosegaard

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  1. Henrik Vosegaard
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Correspondence to Henrik Vosegaard.

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Mathematics Subject Classification (2000): 32S10, 32S35, 32C36, 14E15

The author gratefully acknowledges financial support from Carlsbergfondet.

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Vosegaard, H. Generalized Tjurina numbers of an isolated complete intersection singularity. Math. Ann. 329, 197–224 (2004). https://doi.org/10.1007/s00208-003-0179-5

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  • DOI: https://doi.org/10.1007/s00208-003-0179-5

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