Mathematics – Algebraic Geometry
Scientific paper
2006-07-03
Ann. Inst. Fourier 59,2 (2009), 811-850
Mathematics
Algebraic Geometry
46 pages, 2 figures, 5 tables, final version
Scientific paper
A complex hypersurface D in complex affine n-space C^n 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 C^n. 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.
Granger Michel
Mond David
Nieto-Reyes Alicia
Schulze Mathias
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