 
Summary: K ˜
UNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
REZA AKHTAR AND ROY JOSHUA
Abstract. In this paper we discuss K˜unneth decompositions for finite quotients of several classes of smooth
projective varieties. The main result is the existence of an explicit (and readily computable) ChowK˜unneth
decomposition in the sense of Murre with several pleasant properties for finite quotients of abelian varieties. This
applies in particular to symmetric products of abelian varieties and also to certain smooth quotients in positive
characteristics which are known to be not abelian varieties, examples of which were considered by Enriques and
Igusa. We also consider briefly a strong K˜unneth decomposition for finite quotients of projective smooth linear
varieties.
1. Introduction
ChowK˜unneth decompositions are conjectured (by optimists) to exist over Q for all smooth projective
varieties X ; at present, they are known to exist for curves [18], surfaces [21], projective spaces [18], abelian
varieties ([5]) and other special types of varieties (see for example [6], [7]). Igusa [11] gives a construction
(possibly discovered earlier by Enriques) of a finite group action on an abelian variety such that the quotient
variety is smooth, but not an abelian variety. It is natural to ask whether varieties of this sort  or more generally,
quotients of abelian varieties by a finite group action  admit a ChowK˜unneth decomposition. Indeed, even
when the quotient variety is singular, one may still speak of motives and correspondences (and hence Chow
K˜unneth decompositions) using the reasoning of [8], Examples 8.3.12 and 16.1.13.
In this paper we construct a ChowK˜unneth decomposition for quotients of abelian varieties by a finite
