Algebraic cycles on abelian varieties and their decomposition

B OLLETTINO

U NIONE M ATEMATICA I TALIANA

Giambattista Marini

Algebraic cycles on abelian varieties and their decomposition

Bollettino dell’Unione Matematica Italiana, serie 8, volume 7-B (2004), n. 1, p. 231-240.

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Algebraic Cycles on Abelian Varieties and their Decomposition.

GIAMBATTISTA MARINI

Sunto. – In questo lavoro consideriamo una varietà abeliana X ed il suo anello di Chow CH^l(X) dei cicli algebrici modulo equivalenza razionale. Tramite la decom- posizione di Künneth della diagonale D % X 3X è possibile ottenere delle formule esplicite per i proiettori associati alla decomposizione di Beauville (1) di CH^l(X), tali formule sono espresse in termini delle immagini dirette e inverse dei morfismi di moltiplicazione per un intero m . Il teorema (4) fornisce delle drastiche semplifi- cazioni di tali formule, la Proposizione (9) ed il Corollario (10) forniscono alcuni risultati ad esse correlati.

Summary. – For an Abelian Variety X , the Künneth decomposition of the rational equivalence class of the diagonal D % X 3X gives rise to explicit formulas for the projectors associated to Beauville’s decomposition (1) of the Chow ring CH^l(X), in terms of push-forward and pull-back of m-multiplication. We obtain a few simplifi- cations of such formulas, see theorem (4) below, and some related results, see propo- sition (9) below.

0. – Introduction.

Let X be an abelian variety of dimension n and denote by CH_l(X) its Chow group of algebraic cycles modulo rational equivalence. In our notation, CH_d(X) is the subgroup of d-dimensional cycles and CH^p(X) »4 CH_{n 2d}(X) is the subgroup of p-codimensional cycles. For m
Z , let mult (m) denote the multiplication map X KX, x O mx. By the use of Fourier-Mukai transform for abelian varieties (see [M] and [Be]), Beauville has established a decomposition

CH_d(X)_Q4_{s 42d}^{n 2d}

5

^[CHd(X)Q]s

(1)

where, by definition, CH_d(X)_Q4 CH^d(X) 7Q is the Chow group with Q-

coefficients and the right-hand-side subgroups are defined as follows:

[CH_d(X)_Q]_s»4 4

]W
CH^d(X)_QN mult (m)xW 4m^{2 d 1s}W , (m
Z(

]W
CH^p(X)_QN mult (m)^xW 4m^{2 p 2s}W , (m
Z( , (2)

where p 4n2d is the codimension of W.

This decomposition is a tool to understand cycles and rational equivalence on abelian varieties and it would give a beautiful answer to many questions concerning the Chow groups of abelian varieties (see [Be], [Bl], [J], [Ku] and [S]), provided that Beauville’s vanishing conjecture [Be] holds. This conjec- ture states that the factors of CHd(X) with s E0 vanish (see B.C. below). As pointed out in the abstract, by the use of Deninger-Murre projectors d_i, (see [DM], [Ku]), the projections CH_d(X) K [CHd(X) ]_s with respect to Beauville’s decomposition (1) can be written as linear forms of mult (m)_x and mult (m)^x. Theorem (4) simplifies such explicit descriptions. A further simplification is given for the case where one works modulo a piece of the decomposition, see proposition (9); see corollary (10) for a reformulation of Beauville’s conjec- ture.

1. – The algebraic set up.

We denote by v(z) the series expansion of log(z 11). Namely,

v(z) »4z2 1 2z²1 1

3z³R.

Furthermore, for k and j non-negative integers we define constants a_{k , j} via the formal equality

j 40

!

Q

a_{k , j}z^j4 1 k!v(z)^k

Let A_r
M_{r11, r11}(Q) be the matrix (a_{k , j}), where k and j run in [ 0 , R , r].

Let B_r
M_{r11, r11}(Z) be the matrix (b_{j , h}), where j and h run in [ 0 , R , r] and where, by definition, b_{j , h}4 (21 )^{j 2h}

g

h^j

h

. It is understood that

g

h^j

h

4 0 provided that h Dj. For k40, 1, R , r we define linear forms Lk^(r)(x₀, R , x_r) by the following equality:

u

^L0QQQ(r)

L_r^(r)

v

^{4 ArBr}

u

^xQQQ0

x_r

v

^,

ALGEBRAIC CYCLES ON ABELIAN VARIETIES ETC. 233

namely we define (observe that a_{k , j}4 0 , if j E k and b^{j , h}4 0 , if h D j)

L_k^(r)(x₀, R , x_r) 4

!

j 4k

r

!

h 40 j

a_{k , j}(21)^{j 2h}

g

h^j

h

^xh,

and for k Dr we define L^k(r)4 0 .

We now introduce a numerical lemma, the proof of which is very straight- forward (and omitted).

LEMMA 3. – Let j F1 and sF0 be integers. Then

h 40

!

j

(21)^{j 2h}

g

h^j

h

^hs4./_´⁰s!

if s Ej if s 4j .

2. – Projections of cycles.

Next, using linear forms L_k^(r), we give a criterium to identify components (with respect to Beauville’s decomposition 1) of the algebraic cycles. In the se- quel, X denotes an abelian variety of dimension n ; W
CH_d(X)_Qdenotes a ra- tional algebraic cycle of dimension d and p 4n2d its codimension; further- more, W_sdenotes a component of W with respect to Beauville’s decomposition (1), in particular s is an integer in the range [2d, n2d]. We also consider lin- ear forms L_k^(r) as introduced in the previous section. The interpretation, in terms of push-forward and pull-back of multiplication maps, of the decomposi- tion of the diagonal D
CHn(X 3X) (see [DM], [Ku]) gives

W_s4

(

[ log (D) ]^{yrel2 d1s}iW

)

O( 2 d1s) !4

(

^t[ log (D) ]^{yrel2 n22d2s}iW

)

O( 2 n22 d2s) ! , where yrel denotes the relative Pontryagin product on CH_l(X 3X) with re- spect to projection on the first factor and where, for a
CH_l(X 3X),^ta de- notes its transpose. This equality in turn, in terms of our L_k^(r) gives

W_s4 L2 d 1s^(r) (mult ( 0 )_x, R , mult (r)_x) W

4 L2 p 2s^(r)

(

mult ( 0 )^x, R , mult (r)^x

)

W , (r F 2 n .

It is worthwhile to stress that the linear forms L_k^(r)enter in a natural way (for r 42n) as an explicit version of Deninger-Murre-Künnemann projectors in terms of push-forward and pull-back of multiplication maps. The following theorem (4) goes further, it says that such equalities hold for r that takes smaller values (see ( 4_a) and ( 4_b) below). We also want to stress that linear forms L_k^(r)have an increasing length with respect to r (see the list at the next page).

THEOREM 4. – Let X , W and W_s be as above. Then

W_s4 L2 d 1s^(r) (mult ( 0 )_x, R , mult (r)_x) W , (r F n 1 d ; (4_a)

Ws4 L2 p 2s^(r) (mult ( 0 )^x, R , mult (r)^x) W , (r F n 1 p . (4_b)

Formulas ( 4_a) and ( 4_b) are obtained by using lemma (7) below. We shall also see that ( 4_b) can be refined: the equality there also holds for r Fn1p2 min ]d, 2(. A similar achievement does not hold for (4a). As an explicit example we want to point out that for a 4-dimensional abelian variety and a 2-cycle W the known formula for projectors would give

W₁4 8 W 2 14 mult ( 2 )^xW 1 56

3 mult ( 3 )^xW 2 35

2 mult ( 4 )^xW 1 56

5 mult ( 5 )^xW 2 14

3 mult ( 6 )^xW 1 8

7 mult ( 7 )^xW 2 1

8 mult ( 8 )^xW meanwhile, by theorem (4), or better by remark (8), one has the simpler ex- pression W₁4 4 W 2 3 mult ( 2 )^xW 1⁴

3mult ( 3 )^xW 2¹

4mult ( 4 )^xW . REMARK. – Beauville’s conjecture (see [Be]) states that

[CH_d(X)_Q]_s4 0 , if s E 0 . (B.C.)

As a consequence of theorem (4), proving the conjecture is equivalent to proving that either

L_{2 d 1s}^{(n 1d)}(mult ( 0 )_x, R , mult (n 1d)x) or L_{2 p 2s}^{(n 1p)}(mult ( 0 )^x, R , mult (n 1p)^x) acts trivially on CH_d(X)_Q, for s E0. Another equivalent formulation for Beauville’s conjecture (B.C.) is that the property ( 4_b) holds also for r F2p (this is trivial: since L_{2 p 2s}^{( 2 p)} 4 0 for s E 0 , if ( 4^b) holds for r 42p, B.C. holds as well; it is straightforward to check that the converse implication follows from the proof of theorem 4).

REMARK. – Let us look at ( 4_a) and ( 4_b). The operators L_{2 d 1s}^(r) (mult ( 0 )_x, R , mult (r)_x)

are non-trivial for r Fn1d and the operators L2p2s^(r) (mult ( 0 )^x, R , mult (r)^x) are non-trivial for r Fn1p. Infact, since 2dGsGn2d, then 2d1sGn1d as well as 2 p 2sGn1p.

ALGEBRAIC CYCLES ON ABELIAN VARIETIES ETC. 235

Clearly, one has

mult ( 0 )_xW 4. /´ 0 deg W Q o

if d 4dimWD0 ;

if W is a 0 -cycle , where o is the origin of X . mult ( 1 )_xW 4W

For n 1d that takes the indicated value, the operators Lk4 L_k^{(n 1d)}(R , mult (i)_x, R ) act as follows.

n+d=1

L₀W 4 mult(0)xW L1W 42mult(0)xW 1W n+d=2

L0W 4 mult(0)xW L1W 42³

2mult ( 0 )_xW 12W2¹

2mult ( 2 )_xW L₂W 4¹

2mult ( 0 )_xW 2W1¹

2mult ( 2 )_xW n+d=3

L₀W 4 mult(0)xW L₁W 42¹¹

6 mult ( 0 )_xW 13W2³

2mult ( 2 )_xW 1¹

3mult ( 3 )_xW L2W 4 mult(0)xW 2⁵

2W 12 mult(2)xW 2¹

2mult ( 3 )_xW L₃W 42¹

6mult ( 0 )_xW 1¹

2W 2¹

2mult ( 2 )_xW 1¹

6mult ( 3 )_xW n+d=4

L0W 4 mult(0)xW L1W 42²⁵

12mult ( 0 )_xW 14W23 mult(2)xW 1⁴

3mult ( 3 )_xW 2¹

4mult ( 4 )_xW L₂W4³⁵

24mult ( 0 )_xW2¹³

3 W119

4 mult ( 2 )_xW2⁷

3mult ( 3 )_xW1¹¹

24mult ( 4 )_xW L₃W 42 ⁵

12mult ( 0 )_xW 1³

2W 22 mult(2)xW 1⁷

6mult ( 3 )_xW 2¹

4mult ( 4 )_xW L₄W 4 ¹

24mult ( 0 )_xW 2¹

6W 1¹

4mult ( 2 )_xW 2¹

6mult ( 3 )_xW 1 ¹

24mult ( 4 )_xW From Beauville’s conjecture point of view the first interesting case is W₂₁4 L^{5( 8 )}(R , mult (i)_x, R ) 4L^{5( 7 )}

(

^R, mult (i)^x, R

)

, for W
CH²(X)_Q and dim X 45, see [Be]. Indeed, we have also W214 L5(r)

(

^R, mult (i)^x, R

)

, for r F5 4n1p2min ]d, 2(.

Next we prove theorem (4) and some related results. First, we recall that

the Chow group of an abelian variety has two ring structures: the first one is given by the intersection product, the second one is given by the Pontryagin product, which we shall always denote by y. Consider the ring CHl(X 3X) with the natural sum of cycles and the relative Pontryagin product with re- spect to projection on the first factor X 3XKX (in other terms, we consider Pontryagin product on X 3X regarded as an abelian scheme over X via the first-factor-projection). Let D
CHn(X 3X) be the diagonal and let E4X3 ]o(
CHn(X 3X) be the unit of CHl(X 3X) with respect to the product above, where o is the origin of X . The projectors d₀, R , d_{2 n}are defined by (see [Ku], pag. 200)

d_j4 1

( 2 n 2j)![ log (D) ]^{yrel2 n 2j}

4 1

( 2 n 2j)!

k

^{(D 2E)2 1}2(D 2E)^yrel21 1

3(D 2E)^yrel3R

l

^{yrel2 n2j.}

Since (D 2E)^{yrel2 n 11}4 0 (see [Ku]), the series above are infact finite sums.

Now let D_m denote the graph of mult (m). By Deninger, Murre and Künne- mann theorem (see [DM], [Ku]) we have

[^tD_m]id_j4 m^jd_j, (m
Z , 0 G j G 2 n ;

td_j4 d2 n 2j, ( 0 G j G 2 n ; (5)

where the composition above is the composition of correspondences and where, for s
Corr (A , B),^ts
Corr (B , A) denotes its transpose. As a conse- quence, for W
CH_d(X)_Q and 0 GjG2n, one has

mult (m)^x(d_jiW) 4 [^tDm]i(d_jiW)

4 m^j(d_jiW) , (m
Z .

Clearly, one identifies CH_l(X) with Corr (Spec C , X) 4CHl(Spec C 3X).

Thus, by the definition (2) one has

djiW
[CHd(X)_Q]_s, s »42n22d2j.

(58)

Since

!

^dj4 D acts as the identity map, (5) and (58) give W_s4 d2 n 22d2sⁱW 4^td_{2 d 1s}iW (59)

where, as usual, W_s denotes the component of W with respect to Beauville’s decomposition (1).

For the proof of theorem (4) we need the following.

ALGEBRAIC CYCLES ON ABELIAN VARIETIES ETC. 237

LEMMA 6. – Let W be as in the theorem. Then

[ (D 2E)^yrelj]_iW 4

!

h 40 j

(21)^{j 2h}

g

h^j

h

^{mult (h)xW}

t[ (D 2E)^yrelj]iW 4

!

h 40 j

(21)^{j 2h}

g

h^j

h

^{mult (h)xW}

PROOF. – Since E is the unit for relative Pontryagin product and since D^yrelhiW 4 mult(h)xW as well as ^t[D^yrelh]_iW 4 mult(h)^xW , the two equali- ties follow by a straightforward computation. r

LEMMA 7. – Let W be as in the theorem. Then

[ (D 2E)^yrelj]iW 40, ( j F n 1 d 1 1 ; (7_a)

t[ (D 2E)^yrelj]iW 40, ( j F n 1 p 1 1 . (7_b)

PROOF. – We prove (7_b), the proof of (7_a) is very similar. By lemma (6), we have to show that for j Fn1p11 one has

h 40

!

j

(21)^{j 2h}

g

h^j

h

^{mult (h)xW 40 .}

By linearity of the left-hand-side operator we are free to assume that W be- longs to one of the factors from Beauville decomposition (1), namely we are free to assume that W
[CH_d(X)_Q]_sfor some s
[2d, n2d]. Thus (see 2), we assume that mult (m)^xW 4m^{2 p 2s}W , (m
Z . It follows

h 40

!

j

(21)^{j 2h}

g

h^j

h

^{mult (h)xW 4}_{h 40}

^!

^{j (21)j 2h}

g

h^j

h

^{h2 p 2sW .}

For s in the range above, the range for 2 p 2s is [p, n1p]; in particular, we have 2 p 2sEj. By lemma (3), the coefficient

!

h 40 j

(21)^{j 2h}

g

h^j

h

^{h2 p 2s vanishes.}

Then we are done. r

PROOF (of theorem 4). – We start with formula ( 4_a). Let k 42d1s. Then, we have

W_s4 d2 n 22d2sⁱW 4 1

( 2 d 1s)![ log (D)^{yrel2 d 1s}]iW

4

!

j 4k 2 n

a_{k , j}(D 2E)^yreljiW .

Now observe that by lemma (7), we have (D 2E)^yreljiW 40 for jFn1d11.

Thus, the summation above can be taken up to r , provided that r Fn1d. It follows that

Ws4

!

j 4k r

ak , j(D 2E)^yreljiW , (r F n 1 d .

Looking at the definition of the operators L_k^(r)it is then clear that ( 4_a) follows from the first equality from lemma (6),

(D 2E)^yreljiW 4

!

h 40 j

(21)^{j 2h}

g

h^j

h

^{mult (h)xW .}

The proof of formula ( 4_b) is similar. For r Fn1p we have W_s4^td_{2 d 1s}iW 4 1

( 2 p 2s)!

t[ log (D)^{xrel2 p 2s}]iW

4

!

j 42p2s 2 n

a_{2 p 2s, j}^t[ (D 2E)^xrelj]iW

4

!

j 42p2s r

a_{2 p 2s, j}^t[ (D 2E)^xrelj]iW

4

!

j 42p2s r

a_{2 p 2s, j}

!

h 40 j

(21)^{j 2h}

g

h^j

h

^{mult (h)xW}

4 L2 p 2s^(r) (mult ( 0 )^x, R , mult (r)^x) W

where the 4^th equality follows by lemma (7), the 5^th equality follows by lem- ma (6) and the 6^th equality follows by the definition of the operators L_k^(r). r

REMARK 8. – The equality (7_b) can be improved. We have,

t[ (D 2E)^xrelj]iW 40, (j F n 1 p 1 1 2 d (88)

where d4min ]d, 2(. Infact, since [CHd(X)_Q]_s40 provided that sGmin ]2d11, 21 ( (see [Be]), the actual range for s can be shrinked to min ]2d 1 2 , 0 ( G s Gn2d. Thus in turn, one obtains (88) by the same proof of (7b). As a conse- quence, (4_b) can be refined: the equality there also holds for all r Fn1p2d (where d is as above).

Furthermore, for the same reason, if Beauville’s conjecture (B.C.) men- tioned above holds, then

t[ (D 2E)^xrelj]iW 40, ( j F 2 p 1 1

In particular, if W is a divisor (hence it satisfies B.C.), then ^t[(D2E)^xrel3]i

ALGEBRAIC CYCLES ON ABELIAN VARIETIES ETC. 239

W 40, namely 3W23 mult(2)^xW 1mult(3)^xW 40, which is obvious (in the case of divisors, this kind of computations provide trivial results).

Now fix s , working modulo _{l Fs11}

5

^[CHd(X)Q]l, or rather modulo

l Gs21

5

[CH^p(X)_Q]_l, yields simpler formulas than the ones from theorem (4); fur- thermore, it can be used to provide a reformulation for Beauville’s conjecture (B.C.), see corollary (10) and the example below.

PROPOSITION 9. – Let W and W_s be as in the theorem. Then

W_s4 1

( 2 d 1s)!

!

h 40 2 d 1s

(21)^{2 d 1s2h}

g

^{2 d 1s}h

h

^{mult (h)xW ,}

modulo

5

l Fs11[CH_d(X)_Q]_l (9_a)

Furthermore,

W_s4 1

( 2 p 2s)!

!

h 40 2 p 2s

(21)^{2 p 2s2h}

g

^{2 p 2s}h

h

^{mult (h)xW ,}

modulo

5

l Gs21[CH^p(X)_Q]_l (9_b)

PROOF. – We prove (9_b). Let K 4 ¹

( 2 p 2s)!

!

h 40 2 p 2s

(21)^{2 p 2s2h}

g

^{2 p 2s}h

h

^{mult (h)x.}

It suffices to prove that

KW 4. /´ 0 W

if W
[CH_d(X)_Q]_l, l Fs11 if W
[CH_d(X)_Q]_s.

This is clear by the proof of (7_b); as for the case W
[CH_d(X)_Q]_s, the equality KW 4W follows since, by lemma (3), the coefficient

!

h 40 s

(21)^{s 2h}

g

^sh

h

^{hs equals}

s! (here s 42p2s). The proof of (9^a) is similar. r A straightforward consequence of ( 9_b) is the following.

COROLLARY 10. – Let X be as in the theorem. Then, it satisfies Beauville’s conjecture for d-dimensional cycles if and only if

h 40

!

k

(21)^{k 2h}

g

^kh

h

^{mult (h)x}

acts trivially on CH_d(X)_Q for k F2p11, where p4n2d as usual.

For 5-dimensional abelian varieties the only bad component that might exist is [CH₃(X)_Q]₂₁. Then, by the corollary above it follows that a 5-dimen-

sional abelian variety X satisfies Beauville’s conjecture (B.C.) if and only if 5 W 210 mult(2)^xW 110 mult(3)^xW 25 mult(4)^xW 1mult(5)^xW 40, for all W
CH₃(X)_Q.

R E F E R E N C E S

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[Bl] S. BLOCH, Some Elementary Theorems about Algebraic cycles on Abelian Va- rieties Inventiones, Math., 37 (1976), 215-228.

[DM] C. DENINGER - J. MURRE, Motivic decomposition of abelian schemes and the Fourier transform, J. Reine Angew. Math., 422 (1991), 201-219.

[J] U. JANNSEN, Equivalence relation on algebraic cycles, NATO Sci. Ser. C Math.

Phys. Sci., 548 (2000). 225-260.

[GMV] M. GREEN- J. MURRE- C. VOISIN, Algebraic Cycles and Hodge Theory, Lecture Notes in Mathematics, 1594 (1993).

[Ku] K. KU¨NNEMANN, On the Chow Motive of an Abelian Scheme, Proceedings of Symposia in pure Mathematics, 55 (1994), 189-205.

[M] S. MUKAI, Duality between D(X) and D(X×) with its applications to Picard Sheaves, Nagoya Math. J., 81 (1981), 153-175.

[S] S. SAITO, Motives and filtrations on Chow groups, II, NATO Sci. Ser. C Math.

Phys. Sci., 548 (2000), 321-346.

Dipartimento di Matematica, Università degli studi di Roma II

«Tor Vergata», Via della Ricerca Scientifica, I-00133 Roma (Italy) E-mail: mariniHaxp.mat.uniroma2.it

Pervenuta in Redazione

il 7 dicembre 2002 e in forma rivista il 31 marzo 2003