Logarithmic Comparison Theorem and some Euler homogeneous free divisors

Logarithmic Comparison Theorem and Some Euler Homogeneous Free Divisors

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 un...

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.

Linear Free Divisors and the Global Logarithmic Comparison Theorem

A complex hypersurface D in Cn is a linear free divisor (LFD) if its module of logarithmic vector elds has a global basis of linear vector elds. We classify all LFDs for n at most 4. By analogy with Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic di erential forms computes the complex cohomology of ...

Testing the Logarithmic Comparison Theorem for Free Divisors

Chern Classes of Logarithmic Vector Fields for Locally Quasi-homogeneous Free Divisors

Let X be a nonsingular complex projective variety and D a locally quasihomogeneous free divisor in X. In this paper we study a numerical relation between the Chern class of the sheaf of logarithmic derivations on X with respect to D, and the Chern-Schwartz-MacPherson class of the complement of D in X. Our result confirms a conjectural formula for these classes, at least after push-forward to pr...