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Linear free divisors and the global logarithmic comparison theorem
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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 ISSN 0373-0956.
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Official URL: http://dx.doi.org/10.5802/aif.2448
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 | ||||
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Subjects: | Q Science > QA Mathematics | ||||
Divisions: | Faculty of Science, Engineering and Medicine > 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: |
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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 |
Data sourced from Thomson Reuters' Web of Knowledge
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