Mathematical Research Letters

Volume 25 (2018)

Number 4

On the existence of ghost classes in the cohomology of the Shimura variety associated to $GU(2, 2)$

Pages: 1227 – 1249

DOI: https://dx.doi.org/10.4310/MRL.2018.v25.n4.a9

Author

Matías Victor Moya Giusti (Centro de Investigación y Estudios de Matemática, Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, Argentina)

Abstract

In this paper we study the existence of ghost classes in the cohomology of the Shimura variety attached to the group of unitary similitudes of signature $(2, 2)$, denoted by $GU(2, 2)$. We use considerations on the weights of the mixed Hodge structures attached to the cohomology spaces involved in their definition. The non-existence of ghost classes is known in the cases in which the highest weight of the irreducible representation is regular. We prove that for most irreducible representations with irregular highest weight there are no ghost classes and for the other cases we show that the possible weights in the mixed Hodge structure on the space of ghost classes belong always to the set consisting of the middle weight and the middle weight plus one.

Keywords

Shimura varieties, ghost classes, mixed Hodge structures

2010 Mathematics Subject Classification

14D07, 14G35

Full Text (PDF format)

Received 17 June 2016

Accepted 25 March 2017

Published 16 November 2018