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Logarithmic Comparison Theorem and some Euler homogeneous free divisors. (English) Zbl 1077.32012
Let $$D$$, $$x$$ be a free divisor germ in a complex manifold of dimension $$n>2$$. If the complex of holomorphic differential forms with logarithmic poles along $$D$$ calculates the cohomology of the complement of $$D$$ in $$X$$ we say that $$D$$, $$x$$ satisfy the ‘logarithmic comparison theorem’ (LCT). It is known that locally quasi-homogeneous free divisors satisfy the LCT. The authors present a family of Euler-homogeneous free divisors that do not satisfy the LCT. The method they use here is different from the computational approach they proposed in an earlier paper.
It remains an open question which Euler homogeneous free divisors do satisfy the LCT.

##### MSC:
 32S20 Global theory of complex singularities; cohomological properties 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects)
##### Keywords:
logarithmic differential form; free divisors
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##### References:
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