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Zbl 1163.32014
Granger, Michel; Mond, David; Nieto-Reyes, Alicia; Schulze, Mathias
Linear free divisors and the global logarithmic comparison theorem. (English)
[J] Ann. Inst. Fourier 59, No. 2, 811-850 (2009). ISSN 0373-0956; ISSN 1777-5310/e

Summary: A complex hypersurface $D$ in $\Bbb C^{ n }$ 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.\par 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 differential forms computes the complex cohomology of $\Bbb C^{ n }\setminus D$. 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.\par 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 topological proof of a theorem of V. Kac on the number of irreducible components of such discriminants.

MSC 2000:: *32S20 Global theory of singularities (analytic spaces)
14F40 De Rham cohomology
20G10 Cohomology theory of linear algebraic groups
17B66 Lie algebras of vector fields and related algebras

Keywords: free divisor; prehomogeneous vector space; de Rham cohomology; logarithmic comparison theorem; Lie algebra cohomology; quiver representation

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