Universal Algebraic Equivalences between Tautological Cycles on Jacobians of Curves

We present a collection of algebraic equivalences between tautological cycles on the Jacobian J of a curve, i.e., cycles in the subring of the Chow ring of J generated by the classes of certain standard subvarieties of J . These equivalences are universal in the sense that they hold for all curves of given genus. We show also that they are compatible with the action of the Fourier transform on tautological cycles and compute this action explicitly. Introduction Let J be the Jacobian of a smooth projective complex curve C of genus g ≥ 2. For every d, 0 ≤ d ≤ g, consider the morphism σd : Sym d C → J : D 7→ OC(D − dp), where p is a fixed point on C (for d = 0 this is an embedding of the neutral point into J). It is well-known that σd is birational onto its image. Let us denote by wd = [σg−d(Sym )] ∈ CH(J), where 0 ≤ d ≤ g, the corresponding classes in the Chow ring CH(J) of J . Following Beauville who studied the subring in CH(J) generated by these classes (see [3]) we call an element of this subring a tautological cycle on J . Note that w0 = 1, while w1 is the class of the theta divisor on J . The Poincaré formula states that wd is homologically equivalent to w 1 d! . However, it is known that in general this formula fails to hold modulo algebraic equivalence (although it does hold for hyperelliptic curves, see [5]). More precisely, Ceresa in [4] has shown that if C is generic of genus g ≥ 3 then wd is not algebraically equivalent to [−1]wd for 1 < d ≤ g− 1, where [−1] ∗ is the involution of the Chow ring induced by the inversion morphism [−1] : J → J . This raises the problem of finding universal polynomial relations between the classes wd that hold modulo algebraic equivalence. In this paper we derive a number of such relations. We do not know whether our set of relations is complete for a generic curve. However, we show that these relations are in some sense consistent with the action of the Fourier transform on CH(J). Let us set pk = −N (w) ∈ CH(J) for k ≥ 1, where N(w) are the Newton polynomials on the classes w1, . . . , wg: N(w) = 1 k! g ∑ i=1 λki , where λ1, . . . , λg are roots of the equation λ g −w1λ g−1 + . . .+(−1)wg = 0. For example, p1 = −w1, p2 = w2 − w 2 1/2, p3 = w2w1/2 + w3/2 − w 3 1/6. From the Poincaré formula it is easy to see that the classes pn for n > 1 are homologically trivial. Let us denote by CH(J)Q/(alg) the quotient of the Chow ring of J with rational coefficients modulo the ideal of classes algebraically equivalent to 0. It is also convenient to have a notation Supported in part by NSF grant. 1 for the divided powers of p1: p [d] 1 := p d 1/d!. Our main result is the following collection of relations in CH(J)Q/(alg). Theorem 0.1. (i) Let us define the differential operator D acting on polynomials in infinitely many variables x1, x2, . . . : D = −g∂1 + 1 2 ∑ m,n≥1 ( m+ n m ) xm+n−1∂m∂n, where ∂i = d/dxi. Then for every polynomial F of the form F (x1, x2, . . . ) = D (x1 1 . . . x mk k ), where m1 + 2m2 + . . .+ kmk = g, m1 < g and d ≥ 0, one has F (p1, p2, . . . ) = 0. in CH(J)Q/(alg). (ii) Here is another description of the same collection of relations. For every k ≥ 1, every n1, . . . , nk such that ni > 1, and every d such that 0 ≤ d ≤ k − 1, one has ∑ [1,k]=I1⊔I2⊔...⊔Im ( m− 1 d+m− k ) b(I1) . . . b(Im)p [g−d−m+k− ∑k i=1 ni] 1 pd(I1) . . . pd(Im) = 0 (0.1) in CH(J)Q/(alg), where the summation is over all partitions of the set [1, k] = {1, . . . , k} into the disjoint union of nonempty subsets I1, . . . , Im such that −d + k ≤ m ≤ g − d + k− ∑k i=1 ni (two partitions differing only by the ordering of the parts are considered to be the same); for a subset I = {i1, . . . , is} ⊂ [1, k] we denote b(I) = (ni1 + . . .+ nis)! ni1 ! . . . nis! , d(I) = ni1 + . . .+ nis − s+ 1. Let us point out some corollaries of these relations. Corollary 0.2. The class pn is algebraically equivalent to 0 for n ≥ g/2 + 1. In [6] Colombo and Van Geemen proved that pn is algebraically equivalent to 0 for n ≥ d, where d is the minimal degree of a nonconstant morphism from C to P (see [3], Proposition 4.1). For generic curve this is equivalent to the statement of the above corollary. Corollary 0.3. Every tautological cycle in CH(J)Q is algebraically equivalent to a linear combination of the classes p g−d− ∑k i=1 ni 1 pn1 . . . pnk , where 0 ≤ k ≤ d and ni > 1 for all i = 1, . . . , k. More precisely, for arbitrary n1, . . . , nk such that ni > 1 and for d such that 0 ≤ d ≤ k − 1 one has p [g−d− ∑k i=1 ni] 1 pn1 . . . pnk = ∑d j=1(−1) k+d ( k−1−j d−j )∑ [1,k]=I1⊔I2⊔...⊔Ij b(I1) . . . b(Ij)p [g−d−j+k− ∑k i=1 ni] 1 pd(I1) . . . pd(Ij). (0.2) in CH(J)Q/(alg). 2 For example, in the case d = 1 the above corollary states that for n1, . . . , nk such that ni > 1 and ∑k i=1 ni ≤ g − 1 one has p [g−1− ∑k i=1 ni] 1 pn1 . . . pnk = (−1) k−1 ( ∑k i=1 ni)! n1! . . . nk! p [g+k−2− ∑k i=1 ni] 1 p1−k+ ∑k i=1 ni (0.3) in CH(J)Q/(alg). The main tool in the proof of Theorem 0.1 is the Fourier transform S : CH(J) → CH(J) introduced and studied by Beauville in [1], [2] and [3]. Theorem 0.1 is closely related to the following explicit formula for the induced action of S on tautological cycles in CH(J)Q/(alg). Theorem 0.4. For every n1, . . . , nk such that ni > 1 and every n ≥ 0 one has (−1)S(p [n] 1 pn1 . . . pnk) = ∑ [1,k]=I1⊔I2⊔...⊔Im b(I1) . . . b(Im)p [g−n−m− ∑k i=1 ni] 1 pd(I1) . . . pd(Im) (0.4) in CH(J)Q/(alg), where the summation is similar to the one in equation (0.1). Note that the expression in the identity (0.1) is equal to the RHS of equation (0.4) in the case n = −1, d = k − 1. More interesting observation is the formal consistency between Theorems 0.1 and 0.4 proved in part (iii) of the next theorem. Theorem 0.5. For a fixed g ≥ 2 let us denote by R g the quotient-space of Q[x1, x2, . . . ] by the linear span of all the polynomials appearing in Theorem 0.1 together with all polynomials of degree > g where deg(xi) = i. In other words, R g = Q[x1, x2, . . . ]/Ig where the subspace Ig is spanned by all polynomials F such that degF > g and by polynomials of the form Dd(x1 1 . . . x mk k ), where d ≥ 0, m1 + 2m2 + . . . + kmk = g, m1 < g. Let us denote by pi the image of xi in R Jac g . Then (i) Ig = ∩n≥1 im(D ), where im(D) denotes the image of the operator D acting on Q[x1, x2, . . . ]; (ii) Ig is an ideal in Q[x1, . . . , xg], so R g has a commutative ring structure; (iii) the formula (0.4) gives a well-defined operator S on R g such that S(p1pn1 . . . pnk) = (−1) k+ ∑k i=1 npn1pn1 . . . pnk , where ni > 1; (iv) the operators e(F ) = x1 · F, f = −D, h = −g + ∑ n≥1 (n+ 1)xn∂n (0.5) on Q[x1, x2, . . . ] define a representation of the Lie algebra sl2. Furthermore, they preserve the ideal Ig and therefore define a representation of sl2 on R Jac g . The operators S, e, f and h on R g satisfy the standard compatibilities: SeS = −f, SfS = −e, ShS = −h.