2006/07/31 by Michel Granger, David Mond, Alicia Nieto-Reyes +1 · 2 citations
Mathematics · #math.AG #math.RT #msc:32S20 #msc:14F40 #msc:20G10 #msc:17B66
published as Ann. Inst. Fourier 59,2 (2009), 811-850 · 46 pages, 2 figures, 5 tables, final version
arxiv created 2009/02/16 · arxiv updated 2009/12/01
A complex hypersurface D in complex affine n-space Cn is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4. Analogous to Grothendieck's comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of the complement of D in Cn. We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the Lie algebra of linear logarithmic vector fields 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) fulfill the GLCT. As a by-product we obtain a simplified proof of a theorem of V. Kac on the number of irreducible components of such discriminants.