Beilinson's Hodge Conjecture for K_1 revisited

0.19 MB·

by  Su-Jeong Kang

BEILINSON’S HODGE CONJECTURE FOR K REVISITED
1
Su-Jeong Kang and James D. Lewis
8
0 Abstract. Let U/C be a smooth quasiprojective variety and CHr(U,1) a special instance
0 of Bloch’s higher Chow groups ([Blo]). Jannsen was the first to show that the cycle class
2 map clr,1 : CHr(U,1)⊗Q → homMHS`Q(0),H2r−1(U,Q(r))´ is not in general surjective,
contradictinganearlierconjectureofBeilinson. InthispaperwegivearefinementofJannsen’s
n
counterexample, andfurthershowthattheaforementionedcycleclassmapbecomessurjective
a
atthegenericpoint.
J

1
2

§0. Introduction
]
G
Let U/C be a smooth quasiprojective variety, CHr(U,m;Q) := CHr(U,m)⊗Q, where
A CHr(U,m) is Bloch’s higher Chow group, and
.
h
clU :CHr(U,m;Q)→Γ H2r−m(U,Q(r)) :=hom Q(0),H2r−m(U,Q(r)) ,
t r,m MHS
a
m (cid:0) (cid:1) (cid:0) (cid:1)
the Betti cycle class map. If m = 0, then the Hodge conjecture (classical form) implies
[ that clU is surjective. Beilinson ([Be]) once conjectured that clU is always surjective. It
r,0 r,m
1 was Jannsen ([J3]) who was the first to find a counterexample, in the case m = 1, where
v the complex numbers C are used in an essential way. In contrast to this, one expects the
6 surjectivityofclU inthecasewhereU isobtainedviabaseextensionfromavarietydefined
r,m
3 overanumberfield. ItturnsoutthatJannsen’scounterexampleisindeedveryspecial,being
2
the complement of a closed subscheme of codimension r in a projective algebraic manifold
3
X. In contrast to this, we show rather easily that the corresponding limit map
.
1
0 cl :CHr(Spec(C(X)),1;Q)→Γ H2r−1(C(X),Q(r)) ,
8 r,1
0
(cid:0) (cid:1)
: is onto, provided that a certain reasonable conjectural type statement in [J1] holds, and
v
unconditionally in the case r = dimX. We review this in Example 2.7 below. Broadly
i
X speaking, besides providing a closer examination of this cycle class map, we want to also
r consider the relative situation as well. Consider a proper morphism ρ : X → S of smooth
a
complex quasiprojective varieties. Let η be the generic point of S.
Question 0.1. Is the induced map
clη :CHr(X ,m;Q)→Γ H2r−m(X ,Q(r)) ,
r,m η η
(cid:0) (cid:1)
surjective? [Here H2r−m(Xη,Q(r))=lim → H2r−m(ρ−1(U)(C),Q(r)).]
U⊂S
Keywords and phrases. Hodgeconjecture,Chowgroup,mixedHodgestructures.
First author supported by a PIMS postdoctoral fellowship. Second author is partially supported by a
grantfromtheNaturalSciences andEngineeringResearchCouncilofCanada.
TypesetbyAMS-TEX
1
2 S-J. KANG & J. D. LEWIS
Note that 0.1 in the case m = 0 is equivalent to the classical Hodge conjecture, using
a standard localization argument ([J3]). Based on our limited results in §2 for the case
m = 1, we feel that the answer to this question is yes. The main reason for considering
the relative situation is as follows. As a formal consequence of M. Saito’s theory of mixed
Hodge modules (see [As], [KL]), there is an exact sequence:
Germs of
higher order
0→Ext1 Q(0),Hν−1(η,R2r−ν−mρ Q(r)) →
PMHS ∗ normal functions
(cid:0) (cid:1)  at η 
→hom Q(0),Hν(η,R2r−m−νρ Q(r)) →0 
MHS ∗
Here PMHS stands for the catego(cid:0)ry of graded polarizable MHS(cid:1)(mixed Hodge structures).
Thekeypointis,istherelurkingageneralizedPoincar´eexistencetheoremforhighernormal
functions? (Namely, are these normal functions cycle induced?) An affirmative answer to
0.1,wouldimplyageneralizedPoincar´eexistencetheorem,andconverselysuchanexistence
theorem should lead to an affirmative answer to 0.1.
In §2 we discuss a relative version of the cycle class map on “the generic fiber”, as well
as examples pointing to an affirmative answer to 0.1 in the case m=1. In §3 we study the
relativesituationmorecarefully,bylookingatthoseopensubsetsofacompletebasevariety
S, obtained by deleting codimension p subvarieties from S. The cokernel of a resulting
limiting cycle class map admits a rather explicit description in terms of two filtrations on
a given Chow group. The main results of §3 are summarized in Corollary 3.6. For the
remainderofthepaper,werestricttothecaseX =S,withρ=identity,andwhereX :=X
is a projective algebraic manifold. We then describe the aforementioned cokernel in terms
of the complexity of the Chow ring CH∗(X;Q). This is accomplished with the aid of our
main technical result in Theorem 4.2, with subsequence corollaryconsequences given in §5.
Finally in §6 we present some key examples.
§1. Notation
The following notation will be used throughout the paper. Let X be a smooth complex
projective variety of dimension d.
(i) Γ(−)=hom (Q(0),−)
MHS
(ii) J(−)=Ext1 (Q(0),−)
MHS
(iii) Jr(X)=Ext1 Q(0),H2r−1(X,Q(r))
MHS
(iv)CHr(X;Q)=CHd(cid:0)−r(X;Q)istheChowg(cid:1)roupofcyclesofcodimensionr(dimd−r)inX
modulo rational equivalence, tensored with Q, CHr (X;Q) ⊂ CHr (X;Q) ⊂ CHr(X;Q)
alg hom
are the subgroups of cycles algebraically equivalent to zero (resp. nullhomologous), and
Griffr(X) = CHr (X;Q)/CHr (X;Q) is the Griffiths group (⊗Q). More generally, one
Q hom alg
has the higher Chow groups CHr(X,m) due to Bloch, and defined in [Blo].
(v) AJ :CHr (X;Q)→Jr(X) the Abel-Jacobi map
hom
(vi) CHr(X;Q) =ker(AJ)
AJ
(vii) HC = Hodge conjecture (classical form)
(viii) GHC = (Grothendieck amended) general Hodge conjecture ([G], [L]).
BEILINSON-HODGE CONJECTURE 3
§2. The cycle map on the generic fiber
The setting is the following diagram
X ֒→ X
(2.1) ρ ρ
 
 
S ֒→ S
y y
whereX andSarenonsingularcomplexprojectivevarieties,ρisadominatingflatmorphism,
D ⊂ S a divisor, Y := ρ−1(D), S := S\D, X := X\Y and ρ := ρ . For notational
X
simplicity, we will identify H2r−1(X,Q(r)) with its image in H2r−1(X,Q(r)). There is a
Y (cid:12)
short exact sequence (cid:12)
H2r−1(X,Q(r))
(2.2) 0→ →H2r−1(X,Q(r))→H2r(X,Q(r))◦ →0,
H2r−1(X,Q(r)) Y
Y
and corresponding diagram:
CHr(X,1;Q) → CHr(X;Q)◦ −α−→Y CHr (X;Q)
Y hom
(2.3) clXr,1 βY AJX
  
  
Γ H2r−1(X,Q(r)) ֒→ Γ H2r(X,Q(r))◦ → J H2r−1(X,Q(r))
y Y y H2r−y1(X,Q(r))
(cid:18) Y (cid:19)
(cid:0) (cid:1) (cid:0) (cid:1)
whereAJX isacorrespondingreducedAbel-Jacobimap. Letusassumethatβ issurjective
Y
(such is the case if the Hodge conjecture holds.) If we apply the snake lemma, we arrive at
ker AJX :Im(α )→J H2r−1(X,Q(r))
(2.4) coker(clX )≃ (cid:20) Im(αY) Y (cid:18)HY2r−1(X,Q(r))(cid:19)(cid:21).
r,1 (cid:12) α ker(β )
(cid:12) Y Y
(cid:0) (cid:1)
Now take the limit over all D ⊂S to arrive at an induced cycle map:
(2.4.1) clη :CHr(X ,1;Q)→Γ H2r−1(X ,Q(r)) .
r,1 η η
(cid:0) (cid:1)
where η is the generic point of S. We arrive at:
ker K−A−→J J H2r−1(X,Q(r))
(2.5) Γ H2r−1(Xη,Q(r)) ≃ (cid:20) (cid:18)NS1H2r−1(X,Q(r))(cid:19)(cid:21),
clη CHr(X ,1;Q) N1CHr(X;Q)
(cid:0)r,1 η (cid:1) S
(cid:0) (cid:1)
where K := ker[CHr (X;Q) → CHr(X ;Q)], and NqCHr(X) ⊂ CHr (X;Q) is the
hom η S hom
subgroup generated by cycles which are homologous to zero on some codimension q sub-
scheme of X obtained from a (pure) codimension q subscheme of S via ρ−1, and where
4 S-J. KANG & J. D. LEWIS
NqH2r−1(X,Q(r)) is the subspace of the coniveau NqH2r−1(X,Q(r)) arising from q codi-
S
mensional subschemes of S via ρ−1. In §3 we show that under the assumption of the HC,
(2.5) becomes:
Γ H2r−1(X ,Q(r)) N1CHr(X;Q) + ker K−A−→J J H2r−1(X,Q(r))
(2.6) η ≃ S
cl(cid:0)ηr,1 CHr(Xη,1;Q)(cid:1) NS1C(cid:2)Hr(X;Q)(cid:0) (cid:1)(cid:3)
(cid:0) (cid:1)
Example 2.7. Suppose that X =S with ρ the identity. In this case (2.6) becomes:
Γ H2r−1(C(X),Q(r)) N1CHr(X;Q) + CHr(X;Q)
AJ
≃ ,
cl CHr(Spec(C(X)),1;Q) N1CHr(X;Q)
r,1(cid:0) (cid:1)
(cid:0) (cid:1)
where N1CHr(X;Q)is the subgroupofcycles, thatare homologousto zerooncodimension
1 subschemes of X. According to Jannsen ([J1], p. 227), there is reason to believe that
the righthand side of 2.7 should be conjecturally zero. In particular,since Spec(C(X)) is a
point, this implies that Γ H2r−1(C(X),Q(r)) = 0 for r > 1. The reader can easily check
that
(cid:0) (cid:1)
cl CHr(Spec(C(X)),1;Q) =Γ H2r−1(C(X),Q(r)) ,
r,1
holds unconditionally(cid:0)in the case r =dimX. (cid:1) (cid:0) (cid:1)
Example 2.8. Here we give some evidence that the RHS (hence LHS) of (2.6) is zero.
Suppose X =X×S, and let us assume the condition
r
CHr(X;Q)= CHr−ℓ(X;Q)⊗CHℓ(S;Q).
ℓ=0
M
An example situation is when S is a flag variety, such as a projective space; however con-
jecturally speaking, this condition is expected to hold for a much broaderclass of examples
(see [CL]). Thus
r r
ξ ∈CHr(X;Q) ⇒ ξ = ξ ∈ CHr−ℓ(X;Q)⊗CHℓ(S;Q).
ℓ
ℓ=0 ℓ=0
X M
Fix an ℓ and write
Nℓ,Mℓ
ξ = γℓ⊗βℓ.
ℓ i j
i=1,j=1
X
We can assume that {[γℓ],... ,[γℓ ]} is a basis for Q[γℓ]+···+Q[γℓ ] ⊂ H2r−2ℓ(X,Q),
1 N1,ℓ 1 Nℓ
and that {[βℓ],... ,[βℓ ]} is a basis for Q[βℓ]+···+Q[βℓ ] ⊂ H2ℓ(S,Q). Therefore we
1 M1,ℓ 1 Mℓ
can write
N1,ℓ,M1,ℓ N1,ℓ,M1,ℓ N1,ℓ,M1,ℓ
ξ = γℓ⊗βℓ + γℓ⊗(βℓ)′ + (γℓ)′⊗βℓ + ξ′,
ℓ i j i j i j ℓ
i=1,j=1 i=1,j=1 i=1,j=1
X X X
BEILINSON-HODGE CONJECTURE 5
where ξ′ ∈ CHr−ℓ(X;Q)⊗CHℓ (S;Q), (γℓ)′ ∈ CHr−ℓ(X;Q), (βℓ)′ ∈ CHℓ (S;Q). It
ℓ hom hom i hom j hom
is obvious that ξ ∈ CHr (X;Q) ⇒ N1,ℓ,M1,ℓγℓ ⊗ βℓ = 0 for each ℓ. Furthermore,
hom i=1,j=1 i j
AJ(ξ′)=0, where ξ′ = r ξ′. This is easily seen from the description
ℓ=0 ℓ P
P r Ext1 Q(0),H2r−2ℓ−1(X,Q)⊗H2ℓ(S,Q))(r)
MHS
Ext1 Q(0),H2r−1(X,Q(r) = .
MHS  (cid:0) (cid:1)
ℓ=0 Ext1 Q(0),H2r−2ℓ(X,Q)⊗H2ℓ−1(S,Q))(r)
(cid:0) (cid:1) M MHS L
 
Note that for ℓ ≥ 1, N1ℓ,M1ℓ(γℓ)′ ⊗ βℓ +(cid:0) ξ′ ∈ N1CHr(X;Q). Recall that K(cid:1) =
i=1,j=1 i j ℓ S
ker[CHr (X;Q)→CHr(X ;Q)]. Then:
hom P η
K⊂ CHr−ℓ(X;Q)⊗CHℓ(S;Q),
ℓ≥1
M
and
ξ ∈K∩CHr(X;Q) ⇒AJ (βℓ)′ =0.
AJ S j
Thus if we assume that ker(AJ )⊂N1CHℓ(S;Q), then(cid:0) we a(cid:1)rrive at ξ ∈N1CHr(X;Q).
S S
Conjecture 2.9. The map clη in (2.4.1) is surjective.
r,1
§3. A (Relative) Filtration on Chow Groups
Werecallthe settingin(2.1)andthe definitionsofK, NqCHr(X;Q),NqH2r−1(X,Q(r))
S S
in (2.5) above, and introduce the following descending filtration {Lp CHr(X;Q)} :
X/S p≥0
(3.1)
H2r−1(X,Q(r))
ker AJp :K →J for 0≤p≤r
LXp/SCHr(X;Q)=  " Kp p NSpH2r−1(S,Q(r))!#
0 (cid:12)(cid:12)
for p≥r+1

whereK =ker CHr (X;Q)→ lim CHr(X \ ρ−1(D) ;Q) ,Zp(S)isthefreeabelian
p  hom → 
D∈Zp(S)
(cid:12) (cid:12)
group generatedby codimension p subvarieties of S.(cid:12) (cid:12) 
Example 3.2. Suppose that X := X = S where ρ is the identity. In this case put
{LpCHr(X;Q)} ={Lp CHr(X;Q)} . We then have:
p≥0 X/X p≥0
H2r−1(X,Q(r))
ker AJp :CHr (X;Q)→J for 0≤p≤r
LpCHr(X;Q)= hom NpH2r−1(X,Q(r))
 (cid:20) (cid:18) (cid:19)(cid:21) for p≥r+1
0
where AJp : CHr (X;Q) → J H2r−1(X,Q(r)) is the reduced Abel-Jacobi map. Note
hom NpH2r−1(X,Q(r))
that NrH2r−1(X;Q)=0 and he(cid:16)nce AJr =AJ w(cid:17)ith:
LrCHr(X;Q)=ker[AJ :CHr (X;Q)→Jr(X)]=CHr(X;Q)
hom AJ
The coniveau filtration {NpCHr(X;Q)} on Chow groups introduced above appears in
p≥0
[J1]. Recall its p−th level NpCHr(X;Q) is defined to be the subgroup generated by cycles
whicharehomologoustozeroonsomep−codimensionalpossiblyreduciblesubvarietyofX.
We have the following relations:
6 S-J. KANG & J. D. LEWIS
Proposition 3.3. For each p, we have
NpCHr(X;Q)⊆Lp CHr(X;Q)⊆CHr (X;Q)
S X/S hom
Proof. The first inclusion is a consequence of functoriality of the Abel-Jacobimap, and the
second inclusion is by definition. (cid:3)
For r≥1 we consider
cl (Y):CHr(X\Y,1;Q)→Γ H2r−1(X\Y,Q(r))
r,1
(cid:0) (cid:1)
for a closed subscheme Y of X. Set
clp := lim cl ( ρ−1(D) ):
r,1 → r,1
D∈Zp(S)
(cid:12) (cid:12)
(cid:12) (cid:12)
lim CHr(X\ ρ−1(D) ,1;Q)→ lim Γ H2r−1(X\ ρ−1(D) ,Q(r)) .
→ →
D∈Zp(S) D∈Zp(S)
(cid:12) (cid:12) (cid:0) (cid:12) (cid:12) (cid:1)
(cid:12) (cid:12) (cid:12) (cid:12)
Proposition 3.4. Assume the HC. Then,
Lp CHr(X;Q)
coker(clp )∼= X/S , for 0≤p≤r.
r,1 NpCHr(X;Q)
S
Further, one does not need the HC assumption if p≥min{dimX −3,r−1}.
Proof. Theproofproceedsexactlythesamewayasin§2inobtainingtheexpressionin(2.5),
with D now a closed subscheme of S of (pure) codimension p, and Y = ρ−1(D) being a
closed subscheme of X of codimension p, and where N1 in (2.5) is replaced by Np. As in
S (cid:12) (cid:12)S
§2, the surjectivity of β in diagram (2.3) is only guaranteed if the homo(cid:12)logical v(cid:12)ersion of
Y
the HC ([J3]) holds for Y. For p ≥ dimX −3, that is clearly the case since dimY ≤ 3. A
similar story holds for p≥r−1, by the Lefschetz (1,1) theorem. (cid:3)
Proposition 3.5. Assume the HC. For any p such that 0≤p≤r, we have
Lp CHr(X;Q)=NpCHr(X;Q) + CHr(X;Q) K .
X/S S AJ p
\
Further, one does not needtheHC assumption when both conditions hold: p≥min{dimX−
3,r−1} and NpH2r−1(X,Q(r))=0.
S
Proof. Let ξ ∈Lp CHr(X;Q)). Then by definition, there is Y = ρ−1(D) ∈ Zp(X) such
X/S
that if j :Y ֒→X is the inclusion, then ξ ∈Im(αY) representsan ele(cid:12)ment in(cid:12)the numerator
(cid:12) (cid:12)
of the right hand side of the following expression derived similarly to that in (2.4):
ker AJ :Im(α )→J H2r−1(X,Q(r))
coker(cl (Y))≃ (cid:20) Im(αY) Y (cid:18)j∗HY2r−1(X,Q(r))(cid:19)(cid:21).
r,1 (cid:12) α ker(β )
(cid:12) Y Y
By the HC, the inclusion j H2r−1(X,Q(r)) ֒→ H(cid:0)2r−1(X,(cid:1)Q(r)) has a cycle induced left
∗ Y
inverse
[w] :H2r−1(X,Q(r))։j H2r−1(X,Q(r))⊂H2r−1(X,Q(r)),
∗ ∗ Y
BEILINSON-HODGE CONJECTURE 7
where|w|⊂X×Y. Usingfunctorialityofthe Abel-Jacobimapandthe factthatIm(α )⊂
Y
K , we have ξ−α (w (ξ))∈ker(AJ) K , and w (ξ)∈ker(β ). Thus
p Y ∗ p ∗ Y
ξ ∈ker(AJ) K + α (keTr(β ))⊂ker(AJ) K + NpCHr(X;Q).
p Y Y p
S
Hence this shows that\ \
Lp CHr(X;Q)⊂ker(AJ) K +NpCHr(X;Q).
X/S p S
Also by using Proposition 3.3, we have \
Lp CHr(X;Q)⊂ker(AJ) K + NpCHr(X;Q)⊂Lp CHr(X;Q)), for 1≤p≤r.
X/S p S X/S
(cid:3) \
Corollary 3.6. Assume the HC. Then for any p such that 1≤p≤r,
Lp CHr(X;Q) CHr(X;Q) K
coker(clp )∼= X/S ∼= AJ p
r,1 NpCHr(X;Q) CHr(X;Q) kerK NpCHr(X;Q)
S AJ p T S
Further, theHC assumption is not needed if both conditionTs hold: pT≥min{dimX−3,r−1}
and NpH2r−1(X,Q(r))=0.
S
For the remainder of this section, and indeed for the rest of this paper, we are going to
restricttothespecialsituationinExample3.2withX :=X =S andwhereρistheidentity.
Remarks. 3.7. (i) Note that Corollary 3.6 implies that
coker(clr )∼=CHr(X;Q) .
r,1 AJ
Since the right hand side need not be zero, we recover the counterexample in [J3].
(ii) Assuming the HC, together with Nr−1CHr(X;Q) = CHr (X;Q) ([J1] (Lemma 5.7)),
alg
and J(Nr−1H2r−1(X,Q(r))) = Jr(X):= AJ(CHr (X;Q)) (see [L]), the reader can check
a alg
the existence of the commutative diagram:
0 0 0
↓ ↓ ↓
0 → CHr (X;Q) → CHr(X;Q) → Lr−1CHr(X;Q) → 0
alg AJ AJ Nr−1CHr(X;Q)
↓ ↓ ↓
(3.8) 0 → CHr (X;Q) → CHr (X;Q) → Griffr(X) → 0.
alg hom Q
↓AJr−1 ↓AJ ցAJr−1 ↓
a
0 → Jr(X) → Jr(X) → J H2r−1(X,Q(r)) → 0
a Nr−1H2r−1(X,Q(r))
(cid:16) (cid:17)
↓
0
where AJr−1 =AJ , CHr (X;Q) =ker(AJr−1).
a CHr (X;Q) alg AJ a
alg
(cid:12)
(cid:12)
8 S-J. KANG & J. D. LEWIS
§4. Level of Chow groups
We recall the following notion introduced in [L], that measures the complexity of Chow
groups.
Definition 4.1. We define
Level(H∗(X))=max |p−q| Hp,q(X)6=0, p, q ≥0 ,
(cid:8) (cid:12)| CHr(X\Y;Q)=0(cid:9)
Level(CHr(X;Q))=min ℓ≥0 (cid:12)| over all closed Y ⊂X ,
 
| with codim Y =r−ℓ
 X 
Level(CH∗(X;Q))=max Level(CHr(X;Q)) r ≥0 .
 
(cid:8) (cid:12) (cid:9)
(cid:12)
ThenotionofaconjectureddescendingBloch-Beilinson(B-B)filtration{FνCHr(X;Q)}r
ν=0
iswidelyintroducedintheliterature. Wereferthereaderto[J1]forthedefinition,aswellas
the factthatifitexists,thenitisunique. Amongthemanypropertiesofthis filtration,one
hasF0CHr(X;Q)=CHr(X;Q), F1CHr(X;Q)=CHr (X;Q), F2CHr(X;Q)⊂ker(AJ),
hom
and that the induced action of a correspondence on Grν depends only on the cohomology
F
class of that correspondence. Various candidate B-B filtrations have been introduced in
the literature, such as by Griffiths/Green, J. P. Murre, M. Saito/M. Asakura, S. Saito, W.
Raskind, J.D. Lewis,U. Jannsen, et al. Under the aforementioned uniqueness property, we
will make the blanket assumption that all such candidates define the same filtration.
Theorem 4.2. Assume GHC and existence of the B-B filtration. Then for ℓ≥1,
Level(CH∗(X;Q))≤ℓ ⇒ CHr (X;Q)⊂Nr−ℓCHr(X;Q)
hom
andhenceCHr (X;Q)=LpCHr(X;Q)=NpCHr(X;Q)foranypsatisfying1≤p≤r−ℓ.
hom
Proof. First note that Level(H∗(X))=Level(CH∗(X;Q))≤ℓ ([L], Corollary 15.64), hence
NpH2r(X,Q(r)) = H2r(X,Q(r)) for any p ≤ (2r −ℓ)/2. By definition of level and by
the GHC, there is a closed Y ⊂ X possibly reducible of codimension r−ℓ such that both
of CHr (X;Q) ։ CHr(X;Q) and H2r(X,Q(r)) ։ H2r(X,Q(r)) = Nr−ℓH2r(X,Q(r)) are
Y Y
surjective. LetY˜ →Y be a desingularizationofY andσ :Y˜ →Y ֒→X be the composition
of desingularizationfollowed by an inclusion. Then we have the following diagram:
CHℓ(Y˜;Q) −σ→∗ CHr(X;Q)
↓clY˜ ↓clX
ℓ r
H2ℓ(Y˜,Q(ℓ)) −[−σ→]∗ H2r(X,Q(r)) = Nr−ℓH2r(X,Q(r))
Note that [σ] is surjective by Deligne’s mixed Hodge theory ([D] Proposition 8.2.8). Then
∗
by the HC, there is a cycle induced map
[Γ] :H2r(X,Q(r))→H2ℓ(Y˜,Q(ℓ))
∗
such that [σ] ◦ [Γ] = id on H2r(X,Q(r)). We now assume our given B-B filtration
∗ ∗
F•CHr(−;Q), let ξ ∈ FpCHr(X;Q), and set ξ := Γ (ξ ) ∈ CHℓ(Y˜;Q). Then ξ ∈
1 0 ∗ 1 0
BEILINSON-HODGE CONJECTURE 9
FpCHℓ(Y˜;Q) since the B-B filtration respects correspondences. Consider the following
commutative diagram:
qp
0 → Fp+1CHℓ(Y˜;Q) → FpCHℓ(Y˜;Q) −→Y˜ GrpCHℓ(Y˜;Q) → 0
F
σ ↓ ↑Γ σ ↓ ↑Γ σ ↓ ↑Γ
∗ ∗ ∗ ∗ ∗ ∗
0 → Fp+1CHr(X;Q) → FpCHr(X;Q) −q−→Xp GrpCHr(X;Q) → 0
F
Claim 1: qp (ξ −σ (ξ ))=0 :
X 1 ∗ 0
NotethatongradedlevelGr•,the actionofcorrespondencesdepends onlyonthe cohomol-
F
ogy class of such correspondences, so
qp (ξ −σ (ξ ))=qp (ξ )−qp (σ (Γ (ξ ))=qp (ξ )−σ ◦qp(Γ (ξ ))
X 1 ∗ 0 X 1 X ∗ ∗ 1 X 1 ∗ Y˜ ∗ 1
=qp (ξ )−σ (Γ ◦qp (ξ ))=qp (ξ )−([σ] ◦[Γ] )(qp (ξ ))=0.
X 1 ∗ ∗ X 1 X 1 ∗ ∗ X 1
Claim 2: For any ξ ∈ CHr (X;Q) = F1CHr(X;Q), there is ξ ∈ F1CHℓ(Y˜;Q) and
hom Y
ζ ∈Fℓ+1CHr(X;Q) such that ξ =σ (ξ )+ζ :
ℓ 1 ∗ Y ℓ
Let ξ ∈CHr (X;Q)=F1CHr(X;Q). Set ξ =Γ (ξ)∈F1CHℓ(Y˜;Q). Then by Claim 1,
hom 1 ∗
ζ :=ξ−σ (ξ )∈ker(q1 )=F2CHr(X;Q)
1 ∗ 1 X
Set ξ =Γ (ζ )∈F2CHℓ(Y˜;Q). By applying Claim 1 to ζ ∈F2CHr(X;Q), we get
2 ∗ 1 1
ζ :=ζ −σ (ξ )∈ker(q2 )=F3CHr(X;Q)
2 1 ∗ 2 X
By repeating this process ℓ times, we arrive at
ξ =σ (ξ )+σ (ξ )+···+σ (ξ )+ζ ,
∗ 1 ∗ 2 ∗ ℓ ℓ
where ξ ∈FiCHℓ(Y˜;Q) and ζ ∈Fℓ+1CHr(X;Q). Set ξ = ℓ ξ ∈F1CHℓ(Y˜;Q).
i ℓ Y i=1 i
Claim 3: Fℓ+1CHr(X;Q)=0 : P
Note that
GrpCHr(X;Q)=∆ (2d−2r+p,2r−p) CHr(X;Q).
F X ∗
where d=dimX and ∆ ∈CHd(X ×X;Q) is the diagonal class. Then,
X
[∆ (2d−2r+p,2r−p)]∈H2d−2r+p(X,Q)⊗H2r−p(X,Q),
X
andbytheGHCtogetherwithLevel(H∗(X))≤ℓwecanassumethat[∆ (2d−2r+p,2r−p)]
X
is supportedon some W ×W ⊂X×X with codim(W ,X)≥(2d−2r+p−ℓ)/2 and
1,p p,2 1,p
codim(W ,X)≥(2r−p−ℓ)/2,whereonecaneasilycheckthat∆ (2d−2r+p,2r−p) =0
p,2 X ∗
on CHr(X;Q) for p≥ℓ+1. Hence Fℓ+1CHr(X;Q)=0.
From Claims 1, 2 and 3, we get, for any ξ ∈CHr (X;Q),
hom
ℓ
ξ =σ ξ ∈σ F1CHℓ(Y˜;Q)
∗ i ∗
!
Xi=1 (cid:16) (cid:17)
since F•CHℓ(Y˜;Q) is a descending filtration.
10 S-J. KANG & J. D. LEWIS
Claim 4: σ (F1CHℓ(Y˜;Q))⊂Nr−ℓCHr(X;Q) :
∗
This follows easily from the fact that F1CHℓ(Y˜;Q)=CHℓ (Y˜;Q).
hom
By the above claims, we have now completed the proof of the lemma. (cid:3)
Recall that Nr−1CHr(X;Q)=CHr (X;Q). We deduce:
alg
Corollary 4.3. Assume the GHC and the existence of the B-B filtration. Then
Level(CH∗(X;Q))≤1⇒Griff∗(X) =0.
Q
Remark 4.4. The converse of Corollary 4.3 is false, as can be seen by taking X to be any
surface of positive geometric genus.
§5. Summary Consequences
Proposition 5.1. Assume the HC and a given p. Then the following statements are equiv-
alent:
(i) clp is surjective,
r,1
(ii) CHr(X;Q) ⊂NpCHr(X;Q),
AJ
(iii) clν is surjective for all 1≤ν ≤p.
r,1
Proof. Use Corollary 3.6. (cid:3)
Corollary 5.2. Assume the GHC and existence of the B-B filtration. Then for ℓ ≥ 1, if
Level(CH∗(X;Q))≤ℓ, then clp is surjective for all p satisfying 1≤p≤r−ℓ.
r,1
Proof. Inordertoshowthesurjectivityofclp for1≤p≤r−ℓ,itisenoughbyProposition
r,1
5.1 to show that clr−ℓ is surjective, equivalently
r,1
Lr−ℓCHr(X;Q)
coker(clr−ℓ)∼= =0
r,1 Nr−ℓCHr(X;Q)
First note that Level(H∗(X)) ≤ ℓ implies Nr−ℓH2r−1(X,Q(r)) = H2r−1(X,Q(r)) and in
turn
H2r−1(X,Q)
Lr−ℓCHr(X;Q)=ker CHr (X;Q)→J
hom Nr−ℓH2r−1(X,Q(r))
(cid:20) (cid:18) (cid:19)(cid:21)
=CHr (X;Q)⊂Nr−ℓCHr(X;Q),
hom
where the last inclusion follows from Theorem 4.2. Hence we are done. (cid:3)
In case when ℓ=1, Corollary 5.2 gives the surjectivity of clp up to p=r−1. However
r,1
we can extend this to the surjectivity of clr by the result of S. Saito F2CHr (X;Q) =
r,1 alg
CHr(X;Q) ∩CHr (X;Q) ([S] Corollary 3.7). In fact, we have following:
AJ alg
Corollary 5.3. Assume the GHC and existence of the B-B filtration. Then the following
statements are equivalent:
(i) Level(CH∗(X;Q))≤1,
(ii) clp is surjective for any r, p,
r,1