Granger, Michel, Mond, David, Nieto-Reyes, Alicia and Schulze, Mathias (2009) Linear free divisors and the global logarithmic comparison theorem. Annales de l'institut Fourier, Vol.59 (No.2). pp. 811-850. doi:10.5802/aif.2448

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Abstract

A complex hypersurface D in 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 C-n \ 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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Annales de l'institut Fourier
Publisher:	Association des Annales de l'Institut Fourier
ISSN:	0373-0956
Official Date:	2009
Dates:	Date Event 2009 Published
Volume:	Vol.59
Number:	No.2
Number of Pages:	40
Page Range:	pp. 811-850
DOI:	10.5802/aif.2448
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	EGIDE, Humboldt Foundation

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