Geodesic
Mixed Hodge complexes and $L^2$-cohomology for local systems on ball quotients
Documenta mathematica, Tome 17 (2012), pp. 517-543.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We study the $L^2$--cohomology of certain local systems on non-compact arithmetic ball quotients $X=\\Gamma \\B_n$. In the case of a ball quotient surface $X$ we show that vanishing theorems for $L^2$--cohomology are intimately related to vanishing theorems of the type $$ H^0(\overline{X}, S^n \Omega^1_{\overline{X}}(\log D) \otimes{\mathcal O}_{\overline{X}}(-D) \otimes (K_{\overline{X}}+D)^{-m/3})=0 $$ for $m \ge n \ge 1$ on the toroidal compactification $(\overline{X},D)$. We also give generalizations to higher dimensional ball quotients and study the mixed Hodge structure on the sheaf cohomology of a local system in general with the $L^2$-cohomology contributing to the lowest weight part.
Classification : 14G35, 14F17, 32M15, 32Q30
Keywords: Shimura variety, uniformization, ball quotient, Higgs bundle, mixed Hodge theory, monodromy representation, abelian variety 
@article{DOCMA_2012__17__a14,
     author = {M\"uller-Stach, Stefan and Ye, Xuanming and Zuo, Kang},
     title = {Mixed {Hodge} complexes and $L^2$-cohomology for local systems on ball quotients},
     journal = {Documenta mathematica},
     pages = {517--543},
     publisher = {mathdoc},
     volume = {17},
     year = {2012},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2012__17__a14/}
}
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Müller-Stach, Stefan; Ye, Xuanming; Zuo, Kang. Mixed Hodge complexes and $L^2$-cohomology for local systems on ball quotients. Documenta mathematica, Tome 17 (2012), pp. 517-543. http://geodesic.mathdoc.fr/item/DOCMA_2012__17__a14/