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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)].

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
Full Text: DOI EuDML
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