Summary: LEFSCHETZ DECOMPOSITIONS FOR QUOTIENT VARIETIES
REZA AKHTAR AND ROY JOSHUA
Abstract. In an earlier paper, the authors constructed an explicit Chow Kunneth decomposition for quotient
varieties of Abelian varieties by actions of finite groups. In the present paper, the authors extend the techniques
there to obtain an explicit Lefschetz decomposition for such quotient varieties for the Chow­Kunneth projectors
constructed there.
1. Introduction
It has been conjectured that every smooth projective variety X over a field k has a Chow­K˜unneth de­
composition. Currently, Chow­K˜unneth decompositions are known to exist for curves and projective spaces
[14], surfaces [16], abelian varieties ([5], [19]), varieties with ``finite­dimensional'' motives [10], and several other
special classes. In an earlier paper [1], the authors proved that the quotient A/G of an abelian variety A by
the action of a finite group G has a Chow­K˜unneth decomposition, the projectors of which can be described
explicitly by pushing forward the Chow­K˜unneth projectors of A (as constructed by Deninger and Murre [5])
via the quotient map A×A # A/G×A/G. Although A/G is not in general smooth, the finiteness of G ensures
that the machinery of intersection theory and Chow motives can be extended to varieties of this sort, which
we term pseudo­smooth. Moreover, there are quotient varieties of the above form which are smooth, but not
abelian varieties; Igusa [9] gives such a construction, possibly due earlier to Enriques.
In [13], K˜unnemann proves the existence of a Lefschetz decomposition for Chow motives of abelian schemes;
it seems natural to ask whether such a decomposition can be given for the quotient of an abelian variety,
and, if so, it this can be given explicitly. Kahn, Murre, and Pedrini [10] have shown the existence of such

Source: Akhtar, Reza - Department of Mathematics and Statistics, Miami University (Ohio)

Collections: Mathematics