Linear free divisors and the global logarithmic comparison theorem

Granger, Michel; Mond, David; Nieto-Reyes, Alicia; Schulze, Mathias

Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 811-850.

Une hypersurface complexe de ${ℂ}^n$ 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 $n$ au plus égal à $4$.

Par analogie avec le théorème de comparaison de Grothendieck, on dit que le théorème de comparaison logarithmique global (ou TCLG) est vrai pour $D$ si le complexe des formes différentielles logarithmiques globales permet de calculer la cohomologie de ${ℂ}^n\setminus D$ 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 $n$ inférieur ou égal à $4$, 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 $D$ 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 ${ℂ}^n\setminus D$. 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.

Classification : 32S20, 14F40, 20G10, 17B66
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

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