On Hard Lefschetz Conjecture on Lawson Homology

0.13 MB·English

by  Ze Xu

∗
On Hard Lefschetz Conjecture on Lawson Homology
Ze Xu
Institute of Mathematics, Academy of Mathematics and System Science,
Chinese Academy of Sciences, Beijing 100190, China
1
1 Email: [email protected]
0
2 January 26, 2011

n
Abstract: Friedlander and Mazur proposed a conjecture of hard Lefschetz type on Lawson
a
J homology. We shall relate this conjecture to Suslin conjecture on Lawson homology. For abelian
varieties, this conjecture is shown to be equivalent to a vanishing conjecture of Beauville type on
6
2 Lawson homology. For symmetric products of curves, we show that this conjecture amounts to
the vanishing conjecture of Beauville type for the Jacobians of the corresponding curves. As a

] consequence, Suslin conjecture holds for all symmetric products of curves with genus at most 2.
G
A 1. Introduction
.
h In this note, all varieties are integral schemes of finite type over the complex numbers. If X is
at a quasi-projective variety of dimension n, then the Lawson homology of algebraic p-cycles on X is
m defined as
L H (X):=π (Z (X)) for k ≥2p≥0,
[ p k k−2p p

wherethe groupofp-cyclesZ (X)isgiventhe Chow topology([14]). Naturally,onecanalsodefine
1 p
the negative Lawson homology by the same formula in which case Z (X) := Z (X × A−p) :=
v p 0
0 Z0(X×P−p)/Z0(X×P−p−1)forp<0([7,11,16]). Likeotherhomologytheories,Lawsonhomology
9 has a cohomological counterpart, morphic cohomology, which will be discussed in Section 4. From
9 nowon,weassumethatLawsonhomologyandmorphiccohomologyareallwithrationalcoefficients
4
unless otherwise stated.
.
1 Now let X be a smooth projective variety of dimension n and D an ample divisor on X. By
0 abuse of notation, D still denotes its class in Ln−1H2n−2(X)∼=NS(X)⊗ZQ ([5]). Friedlander and
1
Mazur ([9, 1.3]) proposed the following hard Lefschetz conjecture on Lawson homology.
1
:
v Conjecture 1 The multiplication map Dk−n·:L H (X)−→L H (X)is injective for
p k p+n−k 2n−k
i n<k ≤2n (and k ≥2p).
X
r
InspiredbyFu’sstudyonhardLefschetzconjecturesonChowgroups([10]),inthisnotewestudy
a
theaboveconjectureandexplaintherelationwithotherconjecturesonLawsonhomology. Usingan
observation due to Beilinson([3]), we first show that part of this conjecture is equivalent to Suslin
conjectureinSection2. InSection3,weprovethatforabelianvarieties,thisconjectureisequivalent
to an analogue of Beauville’s vanishing conjecture on Chow groups with rational coefficients ([1]).
The idea of the proof goes back to [2, Prop.5.11]. In Section 4, we show that for symmetric
products of curves, this conjecture is equivalent to the vanishing conjecture of Beauville type for
the Jacobiansofthe correspondingcurvesby the method in [13]. As a corollary,we get the validity
of Suslin conjecture for symmetric products of curves with genus at most 2.
Acknowledgements I would like to thank Professor Baohua Fu for encouraging me to write
this note, and for the discussions and suggestions. I thank also Professor Wenchuan Hu for useful
comments.
∗
The author is partially supported byNational Natural Foundation of China (10871106).
1
2
2. Lawson Homology
Let us first recall some basic properties of Lawson homology.
It is known that there is a canonical isomorphism L H (X) ∼= H (X) for p < 0 ([7,11,16]).
p k k
Note also that L H (X) = 0, if p > n or k < 2p. For latter use, denote LH(X) := L H (X) =:
p k ∗ ∗
L H (X). By a theorem of Friedlander ([5, Th.4.6]), L H (X)=π (Z (X)) is isomor-
k≥2p,p∈Z p k p 2p 0 p
Lphic to the group of algebraic p-cycles modulo algebraic equivalence. In this context, if X is a
smooth projective variety of dimension n and p,q ∈Z, then there is an intersection pairing
•:L H (X)⊗L H (X)→L H (X)
p k q l p+q−n k+l−2n
which at the level of π induces the usual intersection pairing on algebraic cycles modulo algebraic
0
equivalence ([6]). This intersection product is graded-commutative and associative ([11, Prop.2.4]).
There exists a canonical cycle map Φ : L H (X)→ H (X) which is compatible with pull back,
p,k p k k
push forward and intersection product. In general, it is very hard to compute Lawson homology
explicitly. Only a few of cases are known.
NowfortwosmoothprojectivevarietiesX,Y ofdimensionn,mrespectively,denotebyCorr (X,Y):=
d
CH (X×Y) the Chow groupof algebraiccycles of dimensionn+d onX×Y. By [11],eachΓ∈
n+d
Corr (X ×Y) determines a homomorphism of Lawson homology Γ : L H (X) → L H (Y)
d ∗ p k p+d k+2d
for k ≥2p and p∈Z. This will play an essential role in this note.
Turning to Conjecture 1, we have the following easy proposition.
Proposition 2.1. Conjecture 1 holds for p≤0 and for p≥n−1. In particular, Conjecture 1 holds
for all smooth projective varieties of dimension at most 2.
Proof. ThiscomesdirectlyfromthehardLefschetztheoremforsingularhomologyandthefollowing
known results on Lawson homology: if p < 0, then L H (X) ∼= H (X) ([7,11,16]). If p = 0,
p k k
LpHk(X) = Hk(X). By [5, Th.4.6], Ln−1Hk(X) ∼= Q (resp. H2n−1(X,Q),NS(X)⊗Z Q and 0)
if k = 2n (resp. 2n−1,2n−2 and k > 2n). If p = n, the only nontrivial Lawson homology is
L H (X))∼=Q ([5]). (cid:3)
n 2n
Suslin conjecture on Lawson homology (see for example [7]) asserts that for a smooth projective
variety X of dimension n, the cycle map Φ : L H (X) → H (X) is bijective if k ≥ p+n. Note
p,k p k k
that the above statement is a weak form of the original form of Suslin conjecture ([7, 7.9]). In
relation with Conjecture 1, we have the following
Proposition 2.2. For a smooth projective variety X of dimension n, Conjecture 1 implies Suslin
conjecture. Conversely, Suslin conjecture implies Conjecture 1 for k ≥p+n.
Proof. Assume Conjecture 1 holds. We have the following commutative diagram
Dk−n· //
L H (X) L H (X)
p k p+n−k 2n−k
Φp,k Φp+n−k,2n−k
(cid:15)(cid:15) (cid:15)(cid:15)
Dk−n· //
H (X) H (X).
k 2n−k
The bottom arrow is an isomorphism by hard Lefschetz theorem for singular homology. Note that
the cycle map Φ is an isomorphism if p < 0. Now it is easy to see the injectivity of Φ for
p,k p,k
k ≥ p+n. Since Suslin conjecture with finite coefficients is true by Milnor-Bloch-Kato conjecture
(for a proof, see ([3, p.5]), Suslin conjecture with Z-coefficients is equivalent to the assertion that
the Lawson homology L H (X,Z) is finitely generated for k ≥p+n. Therefore, Suslin conjecture
p k
with Z-coefficients amounts to that with Q-coefficients, which is true by the injectivity of the cycle
map Φ for k≥p+n.
p,k
The second statement follows easily from the above commutative diagram. (cid:3)
3
3. Abelian Varieties
Now let A be an abelian variety of dimension g and ℓ ∈ CH1(A×A) the class of the Poincar´e
bundle. Similarly to [1], we have the Fourier transform on Lawson homology F =eℓ :L H (A)→
b ∗ p k
[ig=+pk2]Lp+g−iHk+2g−2i(A). Fors∈Z,defineLpHk(A)(s) ={α∈LpHk(A)|m∗α=m2g−k−sαfor allm∈
LZ}. Then LpHk(A) = b [sg=−pk−2]kLpHk(A)(s) ([12]). Similarly to the situation of Chow groups,
LpHk(A)(s) =(π2g−k−s)∗LLpHk(A)(s) where πi are the canonical Chow-Ku¨nneth projectors ([4]) of
A,sinceπ | isequaltotheidentityifi=2g−k−sandto0otherwise([15]). Notethatwe
i LpHk(A)(s)
have an intersection pairing • : L H (A) ⊗L H (A) → L H (A) . The Fourier
p k (s) q l (t) p+q−g k+l−2g (s+t)
transform gives an isomorphism
F :L H (A) ≃L H (A)
p k (s) p+g−k−s 2g−k−2s (s)
if p−k ≤s≤[g− k]. b
2
In [12], Hu proposed the following conjecture which is an analogue of Beauville’s vanishing con-
jecture on Chow groups ([1]).
Conjecture 2 For an abelian variety A, if s<0, then L H (A) =0.
p k (s)
Immediately, we have the following
Corollary 1 For an abelian variety A, Conjecture 2 implies Friedlander-Mazur conjecture ([9,
Chapter 7] or [16, 2.7]), i.e., L H (A)=0 for all k>2g.
p k
We will prove that Conjecture 1 and Conjecture 2 are equivalent for abelian varieties following
Beauville’s idea ([2]). First, let us fix some notation. Let A be an abelian variety of dimension
g with a polarization θ of degree d. The Pontryagin product ∗ : L H (A) × L H (A) →
p k (s) q m (t)
L H (A) on LH(A) is defined by the formula α∗β =µ (p∗α·p∗β), where µ is the
p+q−g k+m−2g (s+t) ∗ 1 2
multiplication law of A and p the projection to the i-th factor of A×A.
i
By [2], there exists a morphism of Q-groups ϕ : SL → Corr∗(A). The homomorphism
2
Corr(A,A) → End (LH(A)) defines a morphism of Q-groups Corr∗(A) → GL(LH(A)). This
Q
gives a representation of SL on LH(A). In sum, similarly to [2, Th. 4.2], we get the following
2
Lemma 3.1. There is a representation of SL on LH(X), which is a sum of finite dimensional
2
representation such that for n∈Z−{0},t∈Q,α∈LH(A),
n 0 0−1
·α=n−gn∗α, ·α=F(α)
(cid:18)0n−1(cid:19) (cid:18)1 0 (cid:19)
1t 10
·α=etθα, ·α=d−1tgeθ/t∗α.
(cid:18)01(cid:19) (cid:18)t1(cid:19)
The corresponding action of the Lie algebra sl (Q), in the standard basis
2
01 0 0 1 0
(X = ,Y = ,H = ), is given by
(cid:18)00(cid:19) (cid:18)−10(cid:19) (cid:18)0−1(cid:19)
θg−1
Xα=θα, Yα=d−1 ∗α,
(g−1)!
Hα=(g−k−s)α, for α∈L H (X) .
p k (s)
4
The interesting point for us is that Hα= (i−g)π α=(g−k−s)α for α∈L H (A) .
i i∗ p k (s)
An explicit description of the action of SLP2 on LH(A) is in order. An element α ∈LpHk(A)(s)
is called primitive if θg−1 ∗α = 0. The primitive elements are just the lowest weight elements for
the action of SL on LH(A). If α ∈ L H (A) , then the subspace Q·θjα of LH(A) is an
2 p k (s) j
irreducible representationof SL2. This space can be identified with thePlinear space of polynomials
inonevariableofdegree≤k+s−g withthe standardactionby themapf 7→f(θ)α. Inparticular,
one has the following
Proposition 3.2. If α∈L H (A) is primitive, then k+s−g≥0. The set {α,θα,...,θk+s−gα}
p k (s)
forms a basis of an irreducible subrepresentation of LH(A) and LH(A) is a direct sum of subrepre-
sentations of this type. If Pp,k ⊆ L H (A) denotes the subspace of primitive elements, then we
(s) p k (s)
get
g−p
L H (A) = θjPp+j,k+2j.
p k (s) (s)
Mj=0
Now we can prove the main theorem in this section.
Theorem 3.3. For an abelian variety A, Conjecture 1 is equivalent to Conjecture 2.
Proof. First,supposeConjecture2holds. AnampledivisorD ∈L H (A)hasadecomposition
g−1 2g−2
D =θ+θ whereθ = D+(−1)∗D ∈L H (A) andθ = D−(−1)∗D ∈L H (A) . Thenθ
1 2 g−1 2g−2 (0) 1 2 g−1 2g−2 (1)
isample andsymmetric. Therefore,Acanbe consideredasanabelianvarietywiththe polarization
θ. Since the injectivity of θk−g· : L H (A) → L H (A) for g < k ≤ 2g implies the
p k p+g−k 2g−k
injectivity of Dk−g·:L H (A)→L H (A) for g <k ≤2g,it suffices to show that if s≥0,
p k p−g−k 2g−k
then the multiplication map θk−g·: L H (A) →L H (A) is injective. Now assume that
p k (s) p−j k−2j (s)
α∈L H (A) suchthatθk−gα=0.ByProposition3.2,α= g−pθrα withα ∈Pp+r,k+2r.
p k (s) r=0 p+r p+r (s)
Then θk−g+rα =0 for each 0≤r ≤g−p. Since k−g+r ≤Pk+s−g+r≤(k+2r)+s−g, by
p+r
Proposition 3.2 again, α =0 for each 0≤r ≤g−p. Hence, α=0.
p+r
Now assume that Conjecture 1 holds. Suppose that there is a nonzero primitive element α ∈
L H (A) forsomes<0.LetθbethecomponentofDinL H (A) .Sincek−g >k+s−g,
p k (s) g−1 2g−2 (0)
by Proposition 3.2, θk−gα = 0. Note that D = T∗θ for some a ∈ A, where T is the translation
a a
of A by a. Since (T −id)∗α = α∗([−a]−[o]) ∈ L H (A) , we must have T∗α 6= 0. But
a p k (s+1) a
Dk−g·T∗α=0, a contradiction. (cid:3)
a
4. Symmetric Products of Curves
For the convenience of statement, in this section we will mainly consider morphic cohomology,
whichisthecohomologicalcounterpartofLawsonhomologyandithassimilarpropertiesasthoseof
Lawson homology ([8]). Furthermore, there is a canonical duality homomorphism D : LqHl(X)→
L H (X)whichiscompatiblewiththecyclemaps. Forsmoothprojectivevarieties,theduality
n−q 2n−l
homomorphismisanisomorphism([8])andL∗H∗(X):= LqHl(X)isabigradedringwith
l≤2q,q∈Z
the intersection product as multiplication and being antiL-commutative for the second grading ([11,
Prop.2.4]). Since we are always interested in smooth projective varieties, every statement about
Lawson homology has an equivalent dual statement about morphic cohomology.
Now let C be a smooth projective curve with genus g and P ∈ C a fixed point. The n-th
0
symmetric product of C is denoted by C(n). The Jacobian of C is denoted by J := J(C). The
morphism φ :C(n−1) →C(n) is the addition of the point P . Note that C(0) ={P } and φ is the
n 0 0 0
inclusion of the point P in C. We will need the following well-known fact ([13, Prop.2.7]). Let P
0
be the Poincar´e line bundle on J ×C, p : J ×C → J and q : J ×C → C the projections. Define
E =p (E ⊗q∗O(nP )). Thenthe projectivizationP(E )is canonicallyisomorphicto C(n) andthe
n ∗ n 0 n
5
natural homomorphism E →E induces the morphism φ . Therefore, φ∗O (1)= O (1). If
n−1 n n n En En−1
n>2g−2, E is a locally free sheaf, C(n) is a projective bundle over J via the Albanese map and
n
there is an exact sequence of sheaves on J: 0 → E → E → O → 0. Moreover, the divisor
n n+1 J
φ ([C(n)])∈CH1(C(n)) is the first Chern class of the line bundle O (1). Now define the infinite
n∗ En
symmetric product of C, denoted by C(∞), to be the direct system (C(n),φ ) ([13]). Then we can
n
regardallthe O (1)asaline bundle O(1)onC(∞) (viaφ∗). Theclassofthe divisorφ (C(n−1))∈
En n n
CH1(C(n)) is denoted by z which will also be used to denote its image in L1H2(C(n)). Define the
n
correspondence Ψ : C(n) ⊢ C(n−1) to be n−1 (−1)n−1−i(P ,...,P ,π ,...,π ).
n i=0 1≤a1<...<ai≤n 0 0 a1 ai
Then as correspondences, Ψn◦φn∗ = ∆C(nP−1) ([1P3]). Transposing the above identity, we get φ∗n◦
(tΨn)=∆C(n−1). Hence, φn∗ :L∗H∗(C(n−1))→L∗H∗(C(n)) is injective, while φ∗n :L∗H∗(C(n))→
L∗H∗(C(n−1)) is surjective. Define L∗H∗(C(∞)) := lim L∗H∗(C(n)) via φ∗ and L∗H∗(C(+)) :=
←−n n
lim L∗H∗(C(n)) via tΨ . Exactly the same argument as that in [13, Prop.1.17] shows that there
−→n n∗
is a canonical injective map L∗H∗(C(+)) → L∗H∗(C(∞)) defined by sending α ∈ L∗H∗(C(n)) to
(...,(φ2)∗α,φ∗α,α,(tΨ ) α,(tΨ )2α,...),whichmakesL∗H∗(C(+))intoasubringofL∗H∗(C(∞)).
n n n ∗ n ∗
Then we have the following
Lemma 4.1. As bigraded rings, L∗H∗(C(∞)) ∼= L∗H∗(J)[H] where H is the image of the Chern
class of O(1) c (O(1)) in L1H2(C(∞)).
1
Proof. First,notethatc (O(1))∈CH1(C(∞))makessense. Fromtheabovediscussion,ifn>2g−2,
1
the Albanese mapC(n) →J makesC(n) into aprojectivebundle overJ. By [5,Prop.2.5],itis easy
to see that there is a canonical bigraded ring isomorphism L∗H∗(C(n)) ∼= L∗H∗(J)[H], where H is
(Fn(H))
the image of c (O(1)) in L1H2(C(∞)) and F is a monic polynomial of degree n−g+1. Assume
1 n
nown>2g−1.Thenφ∗ :LqHl(C(n))→LqHl(C(n−1))isbijectiveforq <(n−1)−g+1=n−g.If
n
q ≥g−1, then LqHl(C(∞))∼=LqHl(C(q+g)) which is isomorphic to the (q,l)-part of L∗H∗(J)[H].
If q < g − 1, then LqHl(C(∞)) ∼= LqHl(C(2g−1)) which is also isomorphic to the (q,l)-part of
L∗H∗(J)[H]. Therefore, L∗H∗(C(∞))∼=L∗H∗(J)[H]. (cid:3)
Let Γ := (π ,...,π ) : Cm → C(n). Then for α ∈ LH∗(C(n)) with φ∗α = 0,
n,m 1≤a1,...,an≤m a1 an n
as in [13, Prop.P2.1], φ∗m+1(tΓn,m+1)∗α = (tΓn,m)∗α. Then we can define a map Kn : kerφ∗n →
L∗H∗(C(∞)) by sending α to ((tΓ ) α,...,(tΓ ) α,...). The same proof as that in [13, Th.2.3]
n,0 ∗ n,m ∗
gives the following
Lemma 4.2. The homomorphism K is injective and L∗H∗(C(+))= ∞ K (kerφ∗). Denote by
n n=0 n n
NC(∞) : C(∞) → C(∞) the multiplication by N (defined by the N-tupLle diagonal embedding), then
kerφ∗ has eigenvalue Nn for the action of N∗ .
n C(∞)
Note that for morphic cohomology, if A is an abelian variety of dimension g, then LqHl(A) =
[2l] LqHl(A) .ThecohomologicalversionofConjecture2isthatifs<0,thenLqHl(A) =
s=l−g−q (s) (s)
0L.
Now we can prove the main theorem of this section.
Theorem 4.3. Given a smooth projective curve C with genus g, the following three statements are
equivalent:
(i) For any integer n≥1, Conjecture 1 holds for (C(n),z ).
n
(ii) For any integer n ≥ 1, the homomorphism φ∗ : LqHl(C(n)) → LqHl(C(n−1)) is an isomor-
n
phism for 0≤l <n and l ≤2q.
(iii) Conjecture 2 holds for J =J(C).
6
Proof. (i)⇔(ii). Note thatthe homomorphismz ·:LqHl(C(n))→LqHl(C(n)) is equalto φ∗ ◦φ .
n n n∗
Since φ∗ is surjectiveandφ is injective,the injectivity ofz · isequivalentto the injectivity ofφ∗.
n n∗ n n
Now the equivalence follows.
(iii)⇒(ii). Suppose that Conjecture 2 holds for J = J(C). Sine φ∗ is surjective, it suffices to
n
prove the injectivity. By Lemma 4.1, LqHl(C(∞)) = q Lq−jHl−2j(J)·Hj. Note that H has
j=0
eigenvalueN forthe multiplicationbyN ([13,Lem.2.10L]). Thenforeachj, the possibleeigenvalues
of Hl−2j(J)·Hj for the multiplication by N are N[2l],...,Nl−j. Hence the possible eigenvalues
of LqHl(C(∞)) for the multiplication by N are Nl−q,...,Nl. On the other hand, by Lemma 4.2,
L∗H∗(C(+))= ∞ K (kerφ∗), whereK (kerφ∗)haseigenvalueNn forthe multiplicationby N.
n=0 n n n n
Then Kn(kerφ∗nL)=0 if l <n. Since Kn is injective, φ∗n is injective if l<n.
(ii)⇒(iii). Now assume that for any integer n≥1, φ∗ is bijective for 0≤l<n and l≤2q. Since
n
L∗H∗(J) is a subring of L∗H∗(C(∞)) as bigraded rings, then LqHl(J) ⊆ LqHl(C(∞)). Therefore
the possible eigenvalues of LqHl(J) for the multiplication by N are N0,...,Nl. This shows that
LqHl(J) =0 if s<0. (cid:3)
(s)
Combining Proposition 2.1 and Theorem 4.3, we get the following
Corollary 4.4. For any integer n≥1, Conjecture 1 holds for (C(n),z ) if g(C)≤2.
n
Now we can obtain the validity of Suslin’s conjecture for C(n) if g(C) ≤ 2, which can also be
deduced from [7].
Corollary 4.5. Suslin conjecture holds for all symmetric products of curves with genus at most 2,
i.e., for any integer n≥ 1, the cycle map Φ : L H (C(n))→H (C(n)) is bijective for k ≥p+n
p,k p k k
if g(C)≤2.
Proof. This follows from Proposition 2.2, Corollary 4.4 and the proof of Proposition 2.2. (cid:3)
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