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 Cn \ D. We develop a general criterion for the GLCT for LFDs and prove that it is ful lled whenever the Lie algebra of linear logarithmic vector elds is reductive. For n at most 4, we show that the GLCT holds for all LFDs. We show that LFDs arising naturally as discriminants in quiver representation spaces (of real Schur roots) ful ll the GLCT. As a by-product we obtain a topological proof of a theorem of V. Kac on the number of irreducible components of such discriminants. Diviseurs linéairement libres et le théorème de comparaison logarithmique global