[Diviseurs linéairement libres et le théorème de comparaison logarithmique global]
Une hypersurface complexe de est appelée un diviseur linéairement libre (ou DLL) si son module de champs de vecteur logarithmiques a une base globale formée de champs de vecteurs linéaires. Nous classifions tous les DLL pour au plus égal à .
Par analogie avec le théorème de comparaison de Grothendieck, on dit que le théorème de comparaison logarithmique global (ou TCLG) est vrai pour si le complexe des formes différentielles logarithmiques globales permet de calculer la cohomologie de à coefficients complexes. Nous mettons en évidence un critère général pour qu’un DLL ait la propriété TCLG, et nous démontrons que ce critère s’applique lorsque l’algèbre de Lie des champs de vecteurs logarithmiques linéaires est réductive. Pour inférieur ou égal à , nous montrons que le TCLG est vrai pour tous les DLL.
Nous montrons que les DLL qui apparaissent naturellement comme discriminants dans les espaces de représentations de carquois pour des racines de Schur réelles satisfont au TCLG. Comme corollaire nous obtenons une démonstration topologique d’un résultat de V. Kac sur le nombre de composantes irréductibles de tels discriminants.
A complex hypersurface in 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 at most .
By analogy with Grothendieck’s comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for if the complex of global logarithmic differential forms computes the complex cohomology of . 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 at most , 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.
Keywords: Free divisor, prehomogeneous vector space, De Rham cohomology, logarithmic comparison theorem, Lie algebra cohomology, quiver representation
Mot clés : diviseur linéairement libre, espace vectorielle préhomogène, cohomologie de De Rham, théorème de comparaison logarithmique, cohomologie des algèbres de Lie, représentation des quivers
Granger, Michel 1 ; Mond, David 2 ; Nieto-Reyes, Alicia 3 ; Schulze, Mathias 4
@article{AIF_2009__59_2_811_0,
author = {Granger, Michel and Mond, David and Nieto-Reyes, Alicia and Schulze, Mathias},
title = {Linear free divisors and the global logarithmic comparison theorem},
journal = {Annales de l'Institut Fourier},
pages = {811--850},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {59},
number = {2},
year = {2009},
doi = {10.5802/aif.2448},
mrnumber = {2521436},
zbl = {1163.32014},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2448/}
}
TY - JOUR AU - Granger, Michel AU - Mond, David AU - Nieto-Reyes, Alicia AU - Schulze, Mathias TI - Linear free divisors and the global logarithmic comparison theorem JO - Annales de l'Institut Fourier PY - 2009 SP - 811 EP - 850 VL - 59 IS - 2 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2448/ DO - 10.5802/aif.2448 LA - en ID - AIF_2009__59_2_811_0 ER -
%0 Journal Article %A Granger, Michel %A Mond, David %A Nieto-Reyes, Alicia %A Schulze, Mathias %T Linear free divisors and the global logarithmic comparison theorem %J Annales de l'Institut Fourier %D 2009 %P 811-850 %V 59 %N 2 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2448/ %R 10.5802/aif.2448 %G en %F AIF_2009__59_2_811_0
Granger, Michel; Mond, David; Nieto-Reyes, Alicia; Schulze, Mathias. Linear free divisors and the global logarithmic comparison theorem. Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 811-850. doi : 10.5802/aif.2448. https://aif.centre-mersenne.org/articles/10.5802/aif.2448/
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