Linear free divisors and the global logarithmic comparison theorem

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

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

A free divisor D in C is linear if its module of logarithmic vector fields has a basis of global vector fields of degree 0. It is then defined by a homogeneous polynomial of degree n and its complement is an open orbit of an algebraic subgroup GD in Gln(C). The best known example is the normal crossing divisor. Many other such divisors arise, for instance, from quiver representations. We give a...