Logarithmic Comparison Theorem and Some Euler Homogeneous Free Divisors
نویسنده
چکیده
Let D, x be a free divisor germ in a complex manifold X of dimension n > 2. It is an open problem to find out which are the properties required for D, x to satisfy the so-called Logarithmic Comparison Theorem (LCT), that is, when the complex of logarithmic differential forms computes the cohomology of the complement of D, x. We give a family of Euler homogeneous free divisors which, somewhat unexpectedly, does not satisfy the LCT.
منابع مشابه
Logarithmic Comparison Theorem and Euler Homogeneity for Free Divisors
We prove that if the Logarithmic Comparison Theorem holds for a free divisor in a complex manifold then this divisor is Euler homogeneous. F.J. Calderón–Moreno et al. have conjectured this statement and have proved it for reduced plane curves.
متن کاملON b-FUNCTIONS OF LINEAR FREE DIVISORS
We describe b-functions of linear free divisors and use these results to prove that the logarithmic comparison theorem holds for Koszul free reductive linear free divisors exactly if they are (strongly) Euler homogeneous.
متن کاملar X iv : m at h / 06 07 04 5 v 3 [ m at h . A G ] 3 0 Ju l 2 00 7 LINEAR FREE DIVISORS
A hypersurface D in C n is a linear free divisor if the module of logarithmic vector fields along D has a basis of global linear vector fields. It is then defined by a homogeneous polynomial of degree n and its complement is an open orbit of an algebraic subgroup G D of Gln(C) whose Lie algebra g D can be identified with that of linear logarithmic vector fields along D. We classify all linear f...
متن کاملFree Divisors in Prehomogeneous Vector Spaces
We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric and representation theoretic irreducible components coinci...
متن کاملOn the symmetry of b-functions of linear free divisors
We introduce the concept of a prehomogeneous determinant as a possibly nonreduced version of a linear free divisor. Both are special cases of prehomogeneous vector spaces. We show that the roots of the b-function are symmetric about −1 for reductive prehomogeneous determinants and for regular special linear free divisors. For general prehomogeneous determinants, we describe conditions under whi...
متن کامل