KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
REZA AKHTAR AND ROY JOSHUA
Abstract. In this paper we discuss Künneth decompositions for finite quotients of several
classes of smooth projective varieties. We establish the strong Künneth decomposition for finite
quotients of projective smooth linear varieties and also Chow Künneth decomposition for certain
finite quotients of abelian varieties.
1. Introduction
Strong Künneth decompositions are known to exist for linear varieties (cf. [8]), a class that
includes projective spaces, flag schemes, toric schemes and spherical schemes (all over a field k).
Previous work of the second author ([9], [10]) shows in particular that for varieties possessing a
strong Künneth decomposition, the cycle map to étale cohomology is an isomorphism. ChowKünneth decompositions, on the other hand, are conjectured (at least by optimists) to exist
for all smooth projective varieties X; so far, they are known to exist (over Q) for curves [13],
surfaces [17], projective spaces [13], abelian varieties ([22], [3], [12]) certain classes of threefolds
[4], [5], and other isolated examples. In each of these cases, though, one must construct the
projectors explicitly and then check the various orthogonality conditions; this process depends
heavily on particular properties of the variety under consideration. In this paper we extend
the strong Künneth decomposition and the Chow-Künneth decomposition to varieties that are
finite quotients of smooth projective varieties that admit the corresponding decomposition,
working of course with rational coefficients throughout.
In the next section, we extend the strong Künneth decomposition to finite quotients of
projective smooth linear varieties. The main result is that if X is a variety possessing a
strong Künneth decomposition and f : X −→ Y is a finite surjective map, then Y possesses
a strong Künneth decomposition which we may describe explicitly in terms of that for X.
The methods used are all elementary and require little more than the definitions and basic
properties of Chow groups. As an application, we describe a strong Künneth decomposition
for symmetric products of projective spaces. We also discuss some formal consequences of such
a strong Künneth decomposition: we show that a strong Künneth decomposition implies a
Chow Künneth decomposition and that the rational higher Chow groups are determined by the
rational higher Chow groups of the base field and the rational (ordinary) Chow groups of the
given variety.
In the following section we establish Chow Künneth decomposition for finite quotients of
abelian varieties for the action of a finite group so that the hypothesis in Theorem ( 3.2)
are satisfied. The techniques are a modification of those of Deninger-Murre [3] and also [Be].
The second author thanks the Miami University, the IHES, the MPI and the NSA for support.
2
REZA AKHTAR AND ROY JOSHUA
This gives Chow-Künneth decompositions for a number of interesting examples of nonabelian
quotients of abelian varieties; examples of these may be found in [7], [15] some of which we
briefly discuss. The existence of such a Chow-Künneth decomposition is guaranteed by the
work of Kimura [11] on finite-dimensionality of motives; however, his results do not yield
explicit formulas for the projectors, which is part of our goal.
We discuss, in an appendix, certain formal consequences of the existence of strong Künneth
decompositions. This discussion is done in a somewhat more general setting so that it applies
to other situations not considered in the body of the paper.
Acknowledgments. The authors would like to thank Claudio Pedrini and Vasudevan Srinivas for discussions which were essential in improving our understanding of the context of this
paper.
1.1. Group Scheme Actions and Quotients. Following the treatment in [16], we review
some definitions and results related to group scheme actions.
Definition 1.1. Let G be a group scheme over a field k with identity section e : Spec k −→ G
and multiplication m : G ×k G −→ G.
An action of G on a scheme X is a morphism µ : G × X −→ X such that:
(1) The composite
µ
e×1X
X∼
= Spec k ×k X −→ G ×k X −→ X
is the identity map.
(2) The diagram below commutes:
G ×k G ×k X
m×1X
µ
1G ×µ
G ×k X
/ G ×k X
µ
/X
In the future, we will identify elements g ∈ G with the morphism µg : X −→ X defined by
µg (x) = µ(g, x) and (by abuse of notation) refer to this morphism simply as g.
If G is a finite group scheme over k acting on a quasi-projective variety X (also over k),
there exists a quasi-projective variety Y together with a finite, surjective G-invariant morphism
f : X −→ Y universal for G-invariant morphisms X −→ Z. The scheme Y is called the quotient
of X by G, and is typically denoted Y = X/G.
Definition 1.2. We say a variety X is pseudo-smooth if it is the quotient of a smooth variety
by the action of a finite group.
1.1.1. Notation and terminology: review of correspondences. In this section we define the category of rational correspondences and rational Chow motives for pseudo-smooth
projective varieties.
KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
3
Let k be a field and Vk the category of schemes pseudo-smooth and projective over k. If X, Y
are objects of Vk and X has pure dimension d, we define the group of degree r correspondences
from X to Y by Corrr (X, Y ) = CH d+r (X ×k Y ) ⊗ Q, the group of codimension d + r (rational)
cycles on X ×k Y modulo rational equivalence. In general, let X1 , . . . , Xn be the irreducible
components of X; we then define Corrr (X, Y ) = ⊕ni=1 Corrr (Xi , Y ). When α ∈ Corrr (X, Y )
and β ∈ Corrs (Y, Z), we define their composition β ◦ α ∈ Corrr+s (X, Z) by the formula
β ◦ α = (p13 )∗ (p∗12 α ◦ p∗23 β);
here pij represents projection of X ×k Y ×k Z on the ith and jth factors.
One then constructs a new category Mk (Q), the category of (rational) Chow motives of
pseudo-smooth projective varieties. The objects of Mk (Q) are pairs (X, π), where X is an
object of Vk of dimension d and π ∈ Corr0 (X, X) is a projector; that is, an element satisfying
π ◦ π = π. For any two Chow motives (X, π) and (Y, ρ), one then defines
HomMk ((X, π), (Y, ρ)) =
M
ρ ◦ Corr0 (X, Y ) ◦ π.
j
If ∆X is the diagonal of X ×k X and [∆X ] its class in CH ∗ (X ×k X) ⊗ Q, a straightforward
computation (cf. ??) shows that ∆X is a projector, and furthermore that ∆X ◦ α = α = α ◦ ∆X
for any pseudo-smooth projective scheme Y and α ∈ Corr∗ (X, Y ). Thus, there is a functor
f
h : Vkopp −→ Mk (Q) defined on objects by h(X) = (X, ∆X ) and on morphisms by h(X → Y ) =
`
Γf , where Γf ∈ HomMk (Q) (h(Y ), h(X)) is the class of the graph of f . Furthermore, letting
denote disjoint union (of schemes), one may define the sum ⊕ and product ⊗ of motives thus:
(X, p) ⊕ (Y, q) = (X
a
Y, p
a
q)
(X, p) ⊗ (Y, q) = (X × Y, p × q)
We denote by I the “trivial” motive h(Spec k), a neutral element for ⊗), and by L the “Lefschetz motive” (P1k , P1k ×k {x}); here x ∈ P1k is any rational point. Finally, if α ∈ Corr∗ (X, Y ) is
any correspondence, we define its “transpose” t α = s∗ (α) ∈ Corr∗ (Y, X), where s : X ×k Y −→
Y ×k X is the exchange of factors. For further discussion of motives, we refer the reader to
[21]. Also see [Ful] Example (8.3.12) and Example (16.1.12) for discussion that shows one can
in fact define a category of Chow motives for pseudo-smooth schemes as we have done. In fact
it is possible to consider the above theory for all smooth Deligne-Mumford stacks over k; some
of our results extend to this situation readily.
1.2. Abelian Varieties. In this section we establish notation and cite a rigidity property for
abelian varieties necessary in the sequel. A comprehensive treatment of abelian varieties may
be found in [16] or [14].
Let k be a field and A an abelian variety over k. Following [14], we denote by m : A×k A −→ A
the morphism representing composition on (the group scheme) A and use additive notation for
4
REZA AKHTAR AND ROY JOSHUA
this (commutative) operation. For any a ∈ A(k), we denote by τa : A −→ A (translation by a)
the map defined by τa (x) = x + a.
A morphism f : A −→ B between abelian varieties is called a homomorphism if for every
a, a0 ∈ A, f (a + a0 ) = f (a) + f (a0 ). When n ∈ Z we define n : A → A by n(a) = na and set
n
A[n] = Ker (A −→ A), the (group scheme of) n-torsion points on A. For clarity of notation,
we write σ instead of −1.
The following important result is a consequence of a general rigidity principle; see [14],
Corollary 2.2 for details:
Proposition 1.3. Let h : A −→ B be a morphism of abelian varieties. Then there exists a
homomorphism h0 : A −→ B and an element a ∈ A(k) such that h = τa ◦ h0 .
We remark that h0 and a are in fact unique. Indeed, one must have a = −h(0); uniqueness
of h0 then follows immediately.
Let  be the dual abelian variety; we will denote by L the Poincaré bundle and ` its class in
1
CHQ
(A ×k Â).
We conclude this section by recalling the definition of strong Künneth and Chow Künneth
decompositions.
Definition 1.4. Let X be any variety of dimension d over a field k.
We say that X possesses a strong Künneth decomposition if there exists n ≥ 0 and
i
elements ai,j , bi,j ∈ CHQ
(X) such that
XX
(1.2.1)
[∆X ] =
ai,j × bd−i,j
i
j
Now suppose X is pseudo-smooth (over k). We say that X has a Chow-Künneth decomposid
tion if there exist elements π0 , . . . , π2d ∈ CHQ
(X ×k X) such that:
P2d
• [∆X ] = i=0 πi
• For every i, πi ◦ πi = πi and for all j 6= i, πi ◦ πj = 0. (Thus, π0 , . . . , π2d form a system
of mutually orthogonal projectors).
• Let H be a Weil cohomology theory H ∗ (cf. [?]) and, for any k-scheme Y , let clY :
∗
CHQ
(Y ) −→ H ∗ (Y ) denote the cycle map. We require that clX×k X (π) = ∆(i), where
∆(i) is the codimension i Künneth component of the class of ∆X in H ∗ (X ×k X).
Observe that if X is projective, X having a Chow-Künneth decomposition is equivalent to
i
i
asserting that h(X) ∼
= ⊕2d
i=0 h (X) where h (X) is the motive (X, πi ).
2. The strong Künneth decomposition for finite quotients
Now suppose X is a pseudo-smooth, projective, equidimensional scheme over a field k and G
a finite group of automorphisms of X. As in [16], we may form the quotient variety Y = X/G
and ask whether an explicit strong Künneth decomposition for X may be used to construct a
strong Künneth decomposition for Y . We answer this question in the affirmative below.
KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
5
First we consider an elementary calculation showing that strong Künneth decompositions
are preserved under finite maps.
Proposition 2.1. Let X and Y be pseudo-smooth proper varieties and f : X −→ Y a finite
surjective map. If X has a strong Künneth decomposition, then Y also has a strong Künneth
decomposition.
Proof.
Let d = dim X, m = deg f . The hypothesis that X has a strong Künneth decomposition
allows us to write
XX
[∆X ] =
ai,j × bd−i,j
i
j
i
where as before ai,j bi,j ∈ CHQ
(X). Furthermore, (f × f )∗ [∆X ] = m[∆Y ], so it suffices to prove
that (f × f )∗ (ai,j × bd−i,j ) = f∗ (ai,j ) × f∗ bd−i,j . This is accomplished by the next lemma.
Lemma 2.2. Let f : X −→ Y be a morphism of pseudo-smooth varieties
∗
(1) If f is proper, then for all α, β ∈ CHQ
(X), f∗ (α × β) = f∗ (α) × f∗ (β).
∗
∗
∗
(2) For all γ, δ ∈ CHQ (Y ), f (γ × δ) = f (γ) × f ∗ (δ).
Proof.
We will prove the first statement, the second being similar. To establish some notation, let
p1 (p2 ) denote the projection of X ×k X onto the first (second) factor; we similarly define qi as
the appropriate projections of Y ×k Y and ri as the appropriate projections of X ×k Y .
Then
(2.0.2)
(f × f )∗ (α × β) = (f × f )∗ (p∗1 α · p∗2 β)
= (f × 1Y )∗ (1X × f )∗ (p∗1 α · p∗2 β)
= (f × 1Y )∗ (1X × f )∗ ((1X × f )∗ r1∗ α · p∗2 β)
= (f × 1Y )∗ (r1∗ α · (1X × f )∗ p∗2 β)
= (f × 1Y )∗ (r1∗ α · r2∗ f∗ β)
= (f × 1Y )∗ (r1∗ α · (f × 1Y )∗ q2∗ f∗ β)
= (f × 1Y )∗ r1∗ α · q2∗ f∗ β
= q1∗ f∗ α · q2∗ f∗ β = f∗ α × f∗ β
We note the following as a special case:
Corollary 2.3. Let X be a pseudo-smooth quasi-projective variety, G a finite group of automorphisms of X. If X possesses a strong Künneth decomposition, so does Y = X/G.
The utility of the previous statements becomes evident from the following easy result:
Proposition 2.4. Let X be a pseudo-smooth projective variety possessing a strong Künneth
decomposition. Then X has a Chow-Künneth decomposition.
6
REZA AKHTAR AND ROY JOSHUA
Proof.
P P
i
(X). For 0 ≤ r ≤ d, set πr =
Suppose [∆X ] = di=0 j ai,j × bd−i,j , where ai , bi ∈ CHQ
P
P2d
j ar,j × bd−r,j and for d + 1 ≤ r ≤ 2d, set πr = 0. Then [∆X ] =
r=0 πr . We decorate
p with subscripts and superscripts to denote the various projectors from and to subfactors of
X ×k X ×k X; for example, p123
13 : X ×k X ×k X −→ X ×k X sends (x, y, z) to (x, z), etc. Finally,
we let σ : X −→ Spec k and τ : X ×k X −→ Spec k denote the respective structure maps. We
claim that πr ◦ πs = 0 when r 6= s and πr ◦ πr = πr for all r, 0 ≤ r ≤ 2d. The first equality is
a consequence of the following more general fact proved in lemma 2.5 (below) To conclude the
proof of Proposition 2.4, we calculate:
πr ◦ πr = (∆X −
X
πs ) ◦ πr = ∆X ◦ πr = πr
s6=r
Lemma 2.5. With notation as above, suppose ar ∈ CH r (X), bd−r ∈ CH d−r (X), as ∈ CH s (X),
bs ∈ CH d−s (X), and set γr = ar × bd−r , γs = as × bd−s . If r 6= s, then γs ◦ γr = 0.
Proof.
∗
∗
123
123
γs ◦ γr = p123
13 ∗ (p12 γr · p23 γs )
X
123 ∗ 12 ∗
12 ∗
123 ∗ 23 ∗
23 ∗
=
p123
13 ∗ (p12 (p1 ar · p2 bd−r ) · p23 (p2 as · p3 bd−s ))
j
=
=
=
123 ∗ 13 ∗
13 ∗
123 ∗
p123
(as
13 ∗ (p13 (p1 ar · p3 bd−s ) · p2
13 ∗
13 ∗
123 123 ∗
p1 ar · p3 bd−s · p13 ∗ p2 (as · bd−r )
∗
13 ∗
∗
p13
1 ar · p3 bd−s · τ σ∗ (as · bd−r )
· bd−r ))
Note that σ∗ (as · bd−r ) ∈ CH s−r (X), so if r 6= s, then σ∗ (as · bd−r ) = 0, and hence γs ◦ γr = 0.
This concludes the proof of Lemma 2.5.
As an application, we compute the strong Künneth decomposition for the nth symmetric
1
m
m
product of projective space Pm
k . Let ` ∈ CHQ (Pk ) be the class of a generic hyperplane in Pk .
It is well-known (cf. [13], p. 455) that Pm
k has a strong Künneth decomposition:
∆Pm
=
k
m
X
`i × `m−i
i=0
Let X =
n
(Pm
k ) .
By the Künneth formula for motives, we have
∆X =
X
fi1 ,...,in
0≤i1 ,...,in ≤m
i1
in
m−i1
where fi1 ,...,in = ` × . . . × ` × `
mn
× . . . `m−in ∈ CHQ
(X ×k X).
n
Now consider the action of the symmetric group on n letters (denoted Sn ) on X = (Pm
k ) by
interchanging of factors. Let Y = X/Sn and q : X −→ Y the quotient map. Note also that for
any σ ∈ Sn , (q × q)∗ fi1 ,...,in = (q × q)∗ fσ(i1 ),...,σ(in ) .
KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
7
Applying (q × q)∗ to the strong Künneth decomposition for ∆X given above, and noting that
deg q = n!, we obtain
X
(n!)∆Y =
(q × q)∗ fi1 ,...,in
0≤i1 ,...,in ≤m
X
=
X
(q × q)∗ fσ(i1 ),...,σ(in )
0≤i1 ≤i2 ≤...≤in ≤m σ∈Sn
=
X
n!(q × q)∗ fi1 ,...,in
0≤i1 ≤i2 ≤...≤in ≤m
Now let `¯i = q∗ (`i ). Then
(2.0.3)
∆Y =
X
(q × q)∗ fi1 ,...,in
X
`¯i1 × . . . × `¯in × `¯m−i1 × . . . × `¯m−in
0≤i1 ≤i2 ≤...≤in ≤m
=
0≤i1 ≤i2 ≤...≤in ≤m
giving a strong Künneth decomposition for Y .
Corollary 2.6. Let Y denote the n-th symmetric product of Pm
k . Then
CH ∗ (Y, Q, r) ∼
= CH ∗ (Y, Q, 0) ⊗ CH ∗ (Spec
k, Q, r)
where CH ∗ (Z, Q, r) = πr (z ∗ (Z, .) ⊗ Q) and z ∗ (Z, .) denotes the higher cycle complex of the
scheme Z.
Proof. This follows readily from the above strong Künneth decomposition for the class ∆Y
and Theorem 4.1.
3. Chow-Künneth decomposition for quotients of abelian varieties
Our goal in this section is to exhibit an explicit Chow-Künneth decomposition for the quotient
of an abelian variety A by the action of a finite group G, assuming only that g(0) is a torsion
point for each g ∈ G. As before, the quotient A/G may be singular. We rely on the following
result, originally due to Shermenev [22], but later proved in a somewhat more functorial setting
by Deninger and Murre ([3], Theorem 3.1); in this latter source the result is proved more
generally for abelian schemes over a smooth quasi-projective base:
Theorem 3.1. Let A be an abelian variety of dimension d over a field k. Then there exists a
Chow-Künneth decomposition for A:
∆A =
2d
X
πi
i=0
Since we need to make explicit use of the projectors πi , we will presently review their construction. First, consider A ×k A as an abelian A-scheme via projection on the first factor;
with respect to this structure, the dual abelian scheme is A ×k Â. Consider then the Fourier
transform (cf. [3], 2.12, [12], 1.3):
8
REZA AKHTAR AND ROY JOSHUA
∗
∗
FCH : CHQ
(A ×k A) −→ CHQ
(A ×k Â)
defined by FCH (α) = p13 ∗ (p12 ∗ α · F ), where
F =
∞
X
1 × `i
i=0
i!
∈ CHQ (A ×k A ×k Â)
and the various pij represent projections from A ×k A ×k  on the ith and jth factor. Note
that the sum defining F is actually finite.
Dualizing this construction, we may define
∗
∗
(A ×k Â) −→ CHQ
(A ×k A)
F̂CH : CHQ
∗
by F̂CH (γ) = q13 ∗ (q12
γ · F̂ ), where
∞
X
1 × t `i
∗
F̂ =
∈ CHQ
(A ×k  ×k A)
i!
i=0
and qij represent the various projections from A ×k  ×k A. By switching the last two factors
and changing notation appropriately, we see that in fact
F̂CH (γ) = p12 ∗ (p∗13 γ · F ).
An argument involving the theorem of the square (cf. [3], Cor. 2.22, also [1], Prop. 3) then
shows that F̂CH (FCH (α)) = (−1)d σ ∗ α for all α ∈ CH ∗ (A ×k A), and similarly for the other
composition.
P
i
Observe that [∆A ] ∈ CH d (A ×k A), and write FCH ([∆A ]) = 2d
i=0 βi , where βi ∈ CHQ (A ×k
Â). It is a fact ([3], p. 214-216) that (1 × n)∗ βi = ni βi . Now define
(3.0.4)
πi = (−1)d σ ∗ F̂CH (βi )
The main result to be proved is:
Theorem 3.2. Let A be an abelian variety of dimension d over a field k and G a finite group
acting on A such that g(0) ∈ A(k) is a torsion point for each g ∈ G. Let f : A −→ A/G
P
be the quotient map. Suppose ∆A = 2d
i=0 πi is a Chow-Künneth decomposition for A and let
1
ηi =
(f × f )∗ πi .
|G|
Then
∆A/G =
2d
X
i=0
is a Chow-Künneth decomposition for A/G.
ηi
KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
9
We remark that the morphisms g : A −→ A associated to the action of G on A are automorphisms of A as a scheme but not necessarily automorphisms of A as a group scheme.
(i.e. we do not require that g(0) = 0). It is because of this semantic ambiguity that we talk
of quotients of A “by the action of G” rather than simply speaking of quotients by a finite
group of automorphisms. Furthermore, the hypothesis that g(0) be a torsion point of A is
not always satisfied. For example, one could choose any non-torsion point a ∈ A(k); then the
automorphism x 7→ −x + a defines an action of Z/2Z on A. Nevertheless, if k is an algebraic
extension of a finite field, this hypothesis is always satisfied.
Our method of proof is based on that of [3], Theorem 3.1; however, there are further technicalities which complicate it somewhat. The content of the proof is, of course, to show that the
1
elements
(f × f )∗ πi , 0 ≤ i ≤ 2d, are mutually orthogonal projectors. Unfortunately, A/G
|G|
is in general not an abelian variety, so we cannot exploit any special properties of this variety.
However, the map f ∗ establishes an isomorphism ([6], Example 1.7.6):
∗
∗
CHQ
(A/G) −→ CHQ
(A)G
1
∗
f∗ . Thus, we will work in the group CHQ
(A)G , constructing mutually or|G|
∗
thogonal G × G-invariant elements which may be descended to elements of CHQ
(A/G) by the
following device:
with inverse
Lemma 3.3. Suppose X is a pseudo-smooth projective variety of dimension d and G a group
of automorphisms of X. Let f : X −→ Y = X/G be the quotient map and suppose
X
∗
(g × h) ∆X =
g,h∈G
2d
X
ρi
i=0
where ρi ◦ ρj = 0 if i 6= j and the ρi are G × G-invariant, i.e. for any g, h ∈ G, (g × h)∗ ρi = ρi .
Then
2d
X
1
∆Y =
(f × f )∗ ρi
3
|G|
i=0
is a Chow-Künneth decomposition for Y .
Proof.
We have
P
g,h∈G (g
× h)∗ = (f × f )∗ (f × f )∗ and (f × f )∗ ∆X = |G|∆Y ,
and therefore:
|G|2 (f × f )∗ ∆X = (f × f )∗
X
i
ρi
10
REZA AKHTAR AND ROY JOSHUA
Hence
∆Y =
1 X
(f × f )∗ ρi
|G|3 i
It remains to show that (f × f )∗ ρi are mutually orthogonal. As in Proposition 2.4, we add
subscripts and superscripts to p (respectively, q) to denote the various projections between
products of X (respectively, Y ), and for convenience of notation set r = (f × f × f ) : X ×k
X ×k X −→ Y ×k Y ×k Y . Now,
(3.0.5)
∗
∗
123
123
123
(f × f )∗ ρi ◦ (f × f )∗ ρj = q13
∗ (q12 (f × f )∗ ρi · q23 (f × f )∗ ρj )
1 123
123 ∗
123 ∗
(f × f )∗ ρj )
=
q
(r∗ r∗ q12
(f × f )∗ ρi · q23
|G|3 13 ∗
1 123
∗
∗
123 ∗
=
q13 ∗ (r∗ p123
12 (f × f ) (f × f )∗ ρi · q23 (f × f )∗ ρj )
3
|G|
The first and last equality above are obvious, whereas the second one follows from the observation r∗ r∗ =multiplication by |G|3 . Because the ρi are G × G-invariant, the last term
1 123
∗
2
123 ∗
=
q
(r∗ p123
12 |G| ρi · q23 (f × f )∗ ρj )
|G|3 13 ∗
1 123
∗
∗ 123 ∗
q13 ∗ r∗ (p123
=
12 ρi · r q23 (f × f )∗ ρj )
|G|
1 123 123 ∗
∗
∗
p13 ∗ (p12 ρi · p123
=
23 (f × f ) (f × f )∗ ρj )
|G|
∗
∗
123
123
= |G|p123
13 ∗ (p12 ρi · p23 ρj )
= |G|(ρi ◦ ρj ) = 0
In [3], the crucial step in the proof of the Chow-Künneth decomposition for abelian varieties
is the following computation, which may be proved using the seesaw theorem ([14], Corollary
5.2):
Proposition 3.4. ([3], 2.15)
For any integer n,
(1 × n)∗ ` = n`
The analogous strategy in our context would seem to be to study the action of (1 × n)∗ on
(g × h)∗ `; however, there is a priori no action of G on Â. (If G acts on A “by isogenies”; that
is, if all of the maps g : A −→ A are in fact homomorphisms of A, then duality gives a natural
action of G on Â, but we are not assuming this). Instead, we rely on the fact ([14], p.119) that
the Poincaré bundle on  ×k A is the transpose of the Poincaré bundle on A ×k Â. Hence:
(3.0.6)
(n × 1)∗ ` = t ((1 × n)∗ t `) = t (nt` ) = n`
and we prove the following:
Proposition 3.5. There is an infinite subset E ⊂ N such that for all n ∈ E,
(n × 1)∗ (g × 1)∗ ` = n(g × 1)∗ `.
KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
11
Proof. For each g ∈ G, write g = ag ◦ g0 as in Proposition 1.3. Let mg be the order of
Q
ag = −g(0); this is guaranteed to be finite by our hypothesis. Next, let m = g∈G mg , and
E = {n ∈ N : n ≡ 1(mod m)}
Note that if n ∈ E, mg divides n − 1 (for any g), so nag = ag .
Now, if n ∈ E, we have
(3.0.7)
(n × 1)∗ (g × 1)∗ ` = (n × 1)∗ (g0 × 1)∗ (τag × 1)∗ `
Since g0 is a homomorphism, n ◦ g0 = g0 ◦ n; therefore the last teuals
(g0 × 1)∗ (n × 1)∗ (τag × 1)∗ `
Since ag = nag , this equals
(g0 × 1)∗ (n × 1)∗ (τnag × 1)∗ ` = (g0 × 1)∗ (τag × 1)∗ (n × 1)∗ `
By (3.0.6) the last term equals,
n(g0 × 1)∗ (τag × 1)∗ ` = n(g × 1)∗ `
The next step in the proof of Theorem 3.2 is to construct the elements ρi appearing in Lemma
3.3; for each i, we simply set
ρi =
X
(g × h)∗ πi
g,h∈G
where πi are the Chow-Künneth components of ∆A from Theorem 3.1. It is clear from the
P
P
∗
formula that the ρi are G × G-invariant and that 2d
i=0 ρi =
g,h∈G (g, h) ∆A ; so it remains
to prove that they are mutually orthogonal. In preparation for this, we study the action of
(1 × n)∗ on ρi :
Proposition 3.6. For n ∈ E, (1 × n)∗ (g × h)∗ πi = ni πi . Hence, (1 × n)∗ ρi = ni ρi .
Proof.
Observe that (1 × n)∗ (g × h)∗ πi = (1 × n)∗ (g × 1)∗ (1 × h)∗ πi = (g × 1)∗ (1 × n)∗ (1 × h)∗ πi ,
so it suffices to consider the case g = 1.
12
REZA AKHTAR AND ROY JOSHUA
We recall the construction of πi from ( 3.0.4):
(3.0.8)
(1 × n)∗ (1 × h)∗ πi = (−1)g σ ∗ (1 × n)∗ (1 × h)∗ F̂CH (βi )
g ∗
∗
∗
∗
= (−1) σ (1 × n) (1 × h) p12 ∗ (p13 βi ·
∞
X
(1 × `i )
i!
i=0
= (−1)g σ ∗ (1 × n)∗ p12 ∗ (1 × h × 1)∗ (p13 ∗ βi ·
∞
X
(1 × `i )
i!
i=0
g ∗
∗
∗
)
∞
X
(1 × `i )
∗
= (−1) σ p12 ∗ (1 × n × 1) (1 × h × 1) (p13 βi ·
i!
i=0
= (−1)g σ ∗ p12 ∗ (p∗13 βi · (1 × n × 1)∗ (1 × h × 1)∗
∞
X
(1 × `i )
i!
i=0
g ∗
= (−1) σ
p12 ∗ (p∗13 βi
)
)
)
∞
X
1
·
(1 × (n × 1)∗ (h × 1)∗ `i ))
i!
i=0
By Proposition 3.5 the last term is given by
∞
∞
X
X
1
(1 × `i )
i
g ∗
∗
∗ i
i
g ∗
∗
∗
n (−1) σ p12 ∗ (p13 βi ·
(1 × (h × 1) ` )) = n (−1) σ p12 ∗ (p13 βi · (1 × h × 1)
)
i!
i!
i=0
i=0
i
g ∗
∗
∗
= n (−1) σ p12 ∗ (1 × h × 1) (p13 βi ·
∞
X
(1 × `i )
i=0
= ni (−1)g σ ∗ (1 × h)∗ p12 ∗ (p13 ∗ βi ·
∞
X
(1 × `i )
i=0
i
i!
i!
)
)
∗
= n (1 × h) πi
To prove orthogonality of the ρi , we need a version of Liebermann’s trick (cf. [3], Proof of
Theorem 3.1); first we prove the following simple lemma:
Lemma 3.7. For every g, h ∈ G, ρj ◦ (g × h)∗ ∆A = ρj .
Proof.
Certainly the lemma is true if g = h = 1. In the general case,
(3.0.9)
ρj ◦ (g × h)∗ ∆A = p13 ∗ (p∗12 (g × h)∗ ∆A · p∗23 ρj ) = p13 ∗ ((g × h × 1)∗ p∗12 ∆A · p∗23 ρj )
= p13 ∗ (g × h × 1)∗ (p∗12 ∆A · (g −1 × h−1 × 1)∗ p∗23 ρj )
= (g × 1)∗ p13 ∗ (p∗12 ∆A · p∗23 (h−1 × 1)∗ ρj )
Since ρj is G × G-invariant the last term equals
(g × 1)∗ p13 ∗ (p∗12 ∆A · p∗23 ρj ) = (g × 1)∗ (ρj ◦ ∆A ) = (g × 1)∗ (ρj ) = ρj
KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
13
Proposition 3.8. (Liebermann’s trick) For every i, j, i 6= j, ρi ◦ ρj = 0.
Proof.
Suppose n ∈ E. By Proposition 3.6,
nj ρj = (1 × n)∗ ρj
= (1 × n)∗ (ρj ◦ ∆A )
By Lemma 3.7, the last term equals
2g
X
X
1
1
∗
∗
∗
(g
×
h)
∆
)
=
(1
×
n)
(ρ
◦
(1
×
n)
(ρ
◦
ρi )
j
A
j
2
|G|2
|G|
i=0
g,h
2g
1 X
=
(1 × n)∗ p13 ∗ (p∗12 ρj · p∗23 ρi )
|G|2 i=0
=
2g
1 X
p13 ∗ (1 × 1 × n)∗ (p∗12 ρj · p∗23 ρi )
|G|2 i=0
2g
1 X
=
p13 ∗ (p∗12 ρj · p∗23 (1 × n)∗ ρi )
2
|G| i=0
2g
1 X i
=
n (ρj ◦ ρi )
|G|2 i=0
Hence
nj ((ρj ◦ ρj ) − |G|2 ρj ) +
X
ni (ρi ◦ ρj ) = 0
i6=j
for all n ∈ E. Since E is infinite, this forces ρi ◦ ρj = 0 for all i 6= j, and also ρj ◦ ρj = |G|2 ρj .
This final step in the proof of Theorem 3.2 is to show that the images of the ηi under the
cycle map clA/G×k A/G : CH ∗ (A/G ×k A/G) −→ H ∗ (X/G ×k X/G) to any Weil cohomology
theory are in fact the Künneth components of the class of the diagonal. This follows easily
from the analogous fact for the variety A and commutativity of the following diagram:
clA×k A
∗
/ H ∗ (A ×k A)
CHQ
(A ×k A)
(f ×k f )∗
∗
CHQ
(A/G ×k A/G)
(f ×k f )∗
clA/G×k A/G
/ H ∗ (A/G ×k A/G)
This concludes the proof of Theorem 3.2.
14
REZA AKHTAR AND ROY JOSHUA
Among the formulas proved by Künnemann is the so-called Poincaré duality for abelian
varieties ([12], Theorem 3.1.1 (iii)); in our notation, this reads π2d−i = t πi for each i. This fact
immediately implies the analogue for quotients:
Corollary 3.9. (Poincaré duality for quotients) The Chow-Künneth decomposition for A/G of
Theorem 3.2 satisfies Poincaré duality; that is, for any i, η2d−i = t ηi ,
3.1. Examples. 1. Symmetric products of abelian varieties Let X denote an abelian
variety. We let X n /Σn denote the n-fold symmetric power of X. Observe that for every σεΣn ,
σ(0, · · · , 0) = (0, · · · , 0). Therefore the hypotheses of Theorem ( 3.2) are satisfied irrespective
of the base field k. Therefore, we obtain a Chow Kunneth decomposition for X n /Σn . (Observe
that the action of Σn is not free so that the quotient X n /Σn is only pseudo-smooth and not
smooth.)
2. Example of Mehta and Srinivas (See [15] .) Let X be an elliptic curve (or more generally
any abelian variety) over k, with char(k) 6= 2. Let t denote a point of order 2 on X. Let the
group Z/2Z act on X × X by : (x, y) 7→ (x + t, −y). Let Y denote the quotient variety.
Now one may see easily that the action is free so that Y is smooth. Nevertheless, in positive
characteristics, Y need not be an abelian variety as is shown in [15]. Theorem ( 3.2) provides
a Chow Kunneth decomposition for Y .
KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
15
4. Appendix:The strong relative Künneth decomposition of cohomology
In this appendix, we consider, with a view to applications elsewhere, a relative form of
the strong Künneth decomposition in arbitrary cohomology theories satisfying certain mild
conditions. It should be remarked that in this section, it suffices to assume the cohomology
theory is, at least in principle, part of a twisted duality theory in the sense of Bloch-Ogus.
(See [2].) It should be added that the arguments are modifications of the ones in [20] where he
considered the case of K-theory.
Accordingly we will denote H s (X, Γ(r)) by H s (X, (r)). Throughout this section we will make
the following additional hypotheses on our cohomology theories. (Observe these hypotheses are
not identical to the ones in [2], but are implied by them.)
(H.1): for every flat map f : X → Y , there is an induced map f ∗ : H s (Y, (r)) → H s (X, (r))
and this is natural in f .
(H.2): for every proper smooth map f : X → Y of relative dimension d, there is a pushforward f∗ : H i (X; j) → H i−2d (Y ; j − d) so that if g : Y → Z is another proper smooth map of
relative dimension d0 , one obtains g∗ ◦ f∗ = (g ◦ f )∗ . In this case the obvious projection formula
f∗ (x ◦ f ∗ (y)) = f∗ (x) ◦ y, xεH s (X, (r)), yεH s (Y, (r)) holds.
(H.3) : for each smooth scheme X and closed smooth sub-scheme Y of pure codimension
c, there exists a canonical class [Y ]εH 2c (X; c). Moreover the last class lifts to a canonical
class [Y ]εHY2c (X; c). (The latter has the obvious meaning in the setting Bloch-Ogus twisted
duality theories. In case the cohomology theory is defined as hyper-cohomology with respect
to a complex, we let HY (X; c) = the canonical homotopy fiber of the obvious map H(X; c) →
H(X − Y ; c); now HY2c (X, c) = H 2c (HY (X; Γ(c))).) The cycle classes are required to pull-back
under flat pull-back and push-forward under proper push-forwards.
(H.4): if X is a smooth scheme, there exists the structure of a graded commutative ring on
0
0
H (X; .) = ⊕H r (X; s). i.e. ◦ : H r (X; s) ⊗ H r (X; s0 ) → H r+r (X; s + s0 ). In addition to this,
∗
r,s
0
0
there exists an external product H r (X; s) ⊗ H r (X; s0 ) → H r+r (X × X; s + s0 ) so that the
internal product is obtained from the latter by pull-back with the diagonal.
For the purposes of this section it is also convenient to consider only cohomology theories
that are singly graded or non-weighted. Given a bigraded cohomology theory H s (X; (r)), we
will re-index it as follows: we let
(4.0.1)
hr (X; 2r − s) = H s (X; (r)) and h∗ (X; n) = ⊕hr (X; n)
r
We view {h∗ (X; n)|n} as a singly graded cohomology theory. Observe that if f : X → Y is a
proper smooth map of relative dimension d, the induced map f∗ sends h∗ (X; n) to h∗ (Y ; n).
Similarly if f : X → Y is a flat map, the induced map f ∗ sends h∗ (Y ; n) to h∗ (X; n).
16
REZA AKHTAR AND ROY JOSHUA
Theorem 4.1. Let f : X → Y denote a proper smooth map of smooth schemes of relative
dimension d and let [∆]εH 2d (X×X; d) denote the class of the diagonal. Assume that [∆] =
Y
Σi,j ai,j × bd−i,j , with each ai,j εH 2i (X; i), bd−i,j εH 2d−2i (X; d − i). Then for every fixed integer n
one obtains the isomorphism:
h∗ (X; n) ∼
= h∗ (X; 0) ⊗ h∗ (Y ; n)
(4.0.2)
h∗ (Y ;0)
Proof. We will first prove that the classes {ai,j |i} generate h∗ (X; n) as a module over h∗ (Y ; .)
i.e. the obvious map from the right hand side to the left hand side of 4.0.2 (which we will denote
by ρ) is surjective.
Let pi : X×X → X denote the projection to the i-th factor. For each xεh∗ (X; n) we will
Y
first observe the equality :
x = p1∗ (∆ ◦ p2∗ (x))
(4.0.3)
(To see this observe that [∆] = ∆∗ (1), 1εH ∗ (X; Γ(.)). Therefore, ∆ ◦ p2∗ (x) = ∆∗ (∆∗ p∗2 (x))
and hence p1∗ (∆ ◦ p∗2 (x)) = p1∗ ∆∗ (∆∗ p∗2 (x)) = (p1 ◦ ∆)∗ ((p2 ◦ ∆)∗ (x)) = x.)
Now we substitute [∆] = Σi,j p∗1 (ai,j ) ◦ p∗2 (bd−i,j ) into the above formula to obtain:
(4.0.1)
x = p1∗ (Σi,j p∗1 (ai,j ) ◦ p∗2 (bd−i,j ◦ p∗2 (x)))
= p1∗ (Σi,j p∗1 (ai,j ) ◦ p∗2 (bd−i,j ◦ x))
= Σi,j ai,j ◦ p1∗ p∗2 (bd−i,j ◦ x)
= Σi,j ai,j ◦ f ∗ (f∗ (bd−i,j ◦ x))
This proves the assertion that the classes {ai,j |i} generate h∗ (X; .) i.e. the map ρ is surjective.
The rest of the proof is to show that the map ρ is injective. The key is the following diagram:
∗
∗
o
ρ h (X, 0) ∗ ⊗ h (Y ; n)
h∗ (X; n) SSS
h (Y,0)
SSS µ
SSS
SSS
SSS
α
SS)
Homh∗ (Y ;0) (h∗ (X, 0), h∗ (Y ; n))
where the map α (µ(x), xεh∗ (X, 0)) is defined by α(x ⊗ y) = the map x0 7→ f∗ (x0 ◦ x) ◦ y (the
map x0 7→ f∗ (x0 ◦ x), respectively). The commutativity of the above diagram is an immediate
consequence of the projection formula: observe ρ(x ⊗ y) = x ◦ f ∗ (y). Therefore, to show the
map ρ is injective, it suffices to show the map α is injective. For this we define a map β to be a
splitting for α as follows: if φεHomh∗ (Y ;0) (h∗ (X, 0), h∗ (Y ; n)), we let β(φ) = Σi,j ai,j ⊗(φ(bd−i,j )).
Observe that β(α(x ⊗ y)) = β(the map x0 → f∗ (x0 ◦ x) ⊗ y) = (Σi,j ai,j ⊗ f∗ (bd−i,j ◦ x)) ⊗ y.
Now observe that f∗ (bd−i,j ◦ x)εh∗ (Y ; 0) so that we may write the last term as = (Σi,j ai,j ◦
f ∗ f∗ (bd−i,j ◦ x)) ⊗ y. By ( 4.0.1), the last term = x ⊗ y. This proves that α is injective and
hence that so is ρ.
We end this section by considering some explicit examples of what the last theorem implies
for various cohomology theories.
KUNNETH DECOMPOSITIONS FOR QUOTIENT VARIETIES
17
Examples 4.2. 1. K-theory. In this case the theorem takes on the form:
K i (X) ∼
= K 0 (X) ⊗ K i (Y )
(4.0.2)
K 0 (Y )
See [20] for a proof of this. The last part of the proof of the above theorem is clearly an
adaptation of this argument.
2. The theorem takes on the following simple form in the case of any of the following bigraded
cohomology theories: absolute, motivic or Deligne cohomology.
(4.0.3)
H n (X; t) = (⊕H 2a (X; a))
a
⊗
(⊕H 2a (Y
a
(⊕H n−2a (Y ; t − a))
;a)) a
Remarks 4.3. 1. Taking n = 0, the argument in the last part of the proof of the theorem
shows that one obtains a non-degenerate pairing < , >: h∗ (X, 0) ⊗ h∗ (X, 0) → h∗ (Y, 0)
h∗ (Y,0)
by α ⊗ β 7→ f∗ (α ⊗ β).
2. As an example of the usefulness of the last theorem or the formula in 2, we consider the
following result. Let H denote motivic cohomology. Recall that the Beilinson-Soulé vanishing
conjecture for the motivic cohomology of a scheme is the following statement: H s (X, r; Q) = 0
if s < 0 or if (s = 0 and r 6= 0) while H 0 (X, 0; Q) = Q. Now we leave it as an easy exercise to
prove from 2 the following proposition.
Proposition 4.4. If X → Y satisfies the hypotheses of Theorem 4.1 and X and Y are both
connected, then the Beilinson-Soulé vanishing conjecture holds for X if it holds for Y .
Corollary 4.5. The following class of varieties over a number field k satisfy the Beilison-Soulé
vanishing conjecture.
• All toric varieties over k
• All spherical varieties over k. (A variety X is spherical if there exists a reductive group
G defined over k acting on X so that there exists a Borel subgroup scheme S having a
dense orbit. The orbits are products of tori and affine spaces (over k).)
• Any variety over k on which a connected solvable group scheme defined over k acts with
finitely many orbits. (For example projective spaces and flag varieties over k.)
• Any variety over k that has a stratification into strata each of which is the product of a
torus with an affine space.
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Department of Mathematics, Miami University, Oxford, Ohio, 45056, USA.
E-mail address: [email protected]
Department of Mathematics, Ohio State University, Columbus, Ohio, 43210, USA
E-mail address: [email protected]