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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
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##### 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
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