Relative étale slices and cohomology of moduli spaces
[Slices étales relatives et cohomologie des espaces de modules]
de Cataldo, Mark Andrea1  ; Fernandez Herrero, Andres2  ; Ibáñez Núñez, Andrés3
1Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651 (USA)
2Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104 (USA)
3Mathematics Department, Columbia University, New York, NY 10027 (USA)
Annales de l'Institut Fourier, Online first, 34 p.

We use techniques of Alper–Hall–Rydh to prove a local structure theorem for smooth morphisms between smooth stacks around points with linearly reductive stabilizers. This implies that the good moduli space of a smooth stack over a base has equisingular fibers. As an application, we show that any two fibers have isomorphic $\ell $-adic cohomology rings and intersection cohomology groups. If we work over the complex numbers, we show that the family is topologically locally trivial on the base, and that the intersection cohomology groups of the fibers fit into a polarizable variation of pure Hodge structures. We apply these results to derive some consequences for the moduli spaces of $G$-bundles on smooth projective curves, and for certain moduli spaces of sheaves on del Pezzo surfaces.

On utilise des techniques d’Alper–Hall–Rydh pour démontrer un théorème de structure locale pour des morphismes lisses entre deux champs lisses autour de points ayant des stabilisateurs linéairement réductifs. Cela implique que le bon espace de modules d’un champ lisse sur une base a des fibres équisingulières. Comme application, nous montrons que les anneaux de cohomologie $\ell $-adique et les groupes de cohomologie d’intersection de deux fibres quelconques sont isomorphes. Si l’on travaille sur le corps des nombres complexes, nous montrons que la famille est localement triviale sur la base dans le sens topologique, et que les groupes de cohomologie d’intersection des fibres s’inscrivent dans une variation polarisable de structure de Hodge pure. Nous appliquons ces résultats pour en déduire certaines conséquences pour les espaces de modules de fibrés principaux sur des courbes projectives lisses, ainsi que pour certains espaces de modules de faisceaux sur des surfaces de del Pezzo.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3794
Classification : 14D23
Keywords: algebraic stacks, cohomology, moduli spaces
Mots-clés : champs algébriques, cohomologie, espaces de modules

de Cataldo, Mark Andrea  1   ; Fernandez Herrero, Andres  2   ; Ibáñez Núñez, Andrés  3

1 Department of Mathematics, Stony Brook University, Stony Brook, NY 11794-3651 (USA)
2 Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104 (USA)
3 Mathematics Department, Columbia University, New York, NY 10027 (USA) 
de Cataldo, Mark Andrea; Fernandez Herrero, Andres; Ibáñez Núñez, Andrés. Relative étale slices and cohomology of moduli spaces. Annales de l'Institut Fourier, Online first, 34 p.
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