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Linear free divisors and the global logarithmic comparison theorem. (English) Zbl 1163.32014
Summary: A complex hypersurface \(D\) in \(\mathbb C^{ n }\) 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.
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 \(\mathbb C^{ n }\setminus D\). 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.
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:
32S20 Global theory of complex singularities; cohomological properties
14F40 de Rham cohomology and algebraic geometry
20G10 Cohomology theory for linear algebraic groups
17B66 Lie algebras of vector fields and related (super) algebras
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