The stable cohomology of the Satake compactification of 𝒜g

Chen, Jiaming¹ ; Looijenga, Eduard²

¹ Institut de mathématiques de Jussieu-Paris Rive Gauche, Université Paris 7 Diderot, Campus de Grands Moulins, Bureau 745, Bâtiment Sophie Germain, 75205 Paris Cedex 13, France
² Yau Mathematical Sciences Center, Tsinghua University, Jin Chun Yuan West Building, Haidian District, Beijing, 100084, China

Geometry & topology, Tome 21 (2017) no. 4, pp. 2231-2241.

Résumé

Charney and Lee have shown that the rational cohomology of the Satake–Baily–Borel compactification ${\mathcal{A}}_g^{bb}$ of ${\mathcal{A}}_g$ stabilizes as $g\to \infty$ and they computed this stable cohomology as a Hopf algebra. We give a relatively simple algebrogeometric proof of their theorem and show that this stable cohomology comes with a mixed Hodge structure of which we determine the Hodge numbers. We find that the mixed Hodge structure on the primitive cohomology in degrees $4r+2$ with $r\ge 1$ is an extension of $\mathbb{Q}\left(-2r-1\right)$ by $\mathbb{Q}\left(0\right)$; in particular, it is not pure.

Détail

DOI : 10.2140/gt.2017.21.2231

Classification : 14G35, 32S35
Keywords: Satake compactification, stable cohomology, mixed Hodge structure

Bibliographie

[1] A Ash, D Mumford, M Rapoport, Y S Tai, Smooth compactifications of locally symmetric varieties, Cambridge Univ. Press (2010) | DOI

[2] A Borel, Stable real cohomology of arithmetic groups, Ann. Sci. École Norm. Sup. 7 (1974) 235 | DOI

[3] A Borel, Stable real cohomology of arithmetic groups, II, from: "Manifolds and Lie groups" (editors S Murakami, J Hano, K Okamoto, A Morimoto, H Ozeki), Progr. Math. 14, Birkhäuser (1981) 21 | DOI

[4] A Borel, J P Serre, Corners and arithmetic groups, Comment. Math. Helv. 48 (1973) 436 | DOI

[5] R Charney, R Lee, Cohomology of the Satake compactification, Topology 22 (1983) 389 | DOI

[6] J Chen, E Looijenga, The homotopy type of the Baily–Borel and allied compactifications, preprint (2015)

[7] P Deligne, Théorie de Hodge, III, Inst. Hautes Études Sci. Publ. Math. 44 (1974) 5

[8] G Faltings, C L Chai, Degeneration of abelian varieties, 22, Springer (1990) | DOI

[9] W Fulton, J Harris, Representation theory, 129, Springer (1991) | DOI

[10] M Goresky, W Pardon, Chern classes of automorphic vector bundles, Invent. Math. 147 (2002) 561 | DOI

[11] E Looijenga, Discrete automorphism groups of convex cones of finite type, Compos. Math. 150 (2014) 1939 | DOI

[12] E Looijenga, Goresky–Pardon lifts of Chern classes and associated Tate extensions, Compos. Math. 153 (2017) 1349 | DOI

[13] M Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988) 849 | DOI