×

zbMATH — the first resource for mathematics

Milnor number and Tjurina number of a complete intersection. (English) Zbl 0539.14002
Let (X,x) be an isolated complete intersection singularity of dimension \(n\geq 2\). The main result of this note is a formula for the difference of the Milnor number \(\mu\) (X,x) and dim \(T^ 1_{X,x}\) (the dimension of the base of a miniversal deformation of (X,x)). It is of the form: \(\mu(X,x)-\dim T^ 1_{X,x}=\sum^{n-1}_{p=0}h^{p,0}(X,x)+a_ 1+a_ 2+a_ 3,\) where \(h^{p,q}(X,x)\) denotes the (p,q)-Hodge number of the mixed Hodge structure which is naturally defined on the local cohomology group \(H^ n(X,X-\{x\})\) and the \(a_ i's\) are dimensions of vector spaces associated to (X,x). In particular, \(\mu(X,x)\geq \dim T^ 1_{X,x},\) which had been conjectured by G.-M. Greuel [Math. Ann. 250, 157-173 (1980; Zbl 0417.14003)].

MSC:
14B05 Singularities in algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14D15 Formal methods and deformations in algebraic geometry
58A14 Hodge theory in global analysis
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
14B15 Local cohomology and algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Deligne, P.: Théorie de Hodge II, III. Publ. Math. IHES40, 5-75 (1972);44, 6-77 (1975)
[2] Greuel, G.-M.: Der Gaus-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten. Math. Ann214, 235-266 (1975) · Zbl 0292.14007
[3] Greuel, G.-M.: Dualität in der lokalen Kohomologie isolierter Singularitäten. Math. Ann.250, 157-173 (1980) · Zbl 0417.14003
[4] Looijenga, E.J.N.: Milnor number and Tjurina number in the surface case. Catholic University Nijmegen, Report 8216, July 1982
[5] Steenbrink, J.H.M.: Mixed Hodge structures associated with isolated singularities. Proc. Symp. Pure Math.40, 513-536 (1983) · Zbl 0515.14003
[6] Steenbrink, J.H.M.: Vanishing theorems for singular spaces. Systèmes différentiels et singularités. Luminy 1983 (Astérisque) (to appear)
[7] Wahl, J.M.: A characterization of quasihomogeneous Gorenstein singularities. Compositio Math. (to appear) · Zbl 0587.14024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.