[Paper Review] Algebraic cycles on the relative symmetric powers and on the relative Jacobian of a family of curves. II

B.J. Moonen, Alexander Polishchuk|ArXiv.org|May 23, 2008

Advanced Differential Equations and Dynamical Systems21 references3 citations

TL;DR

This paper establishes a canonical isomorphism between the Chow homology of the infinite relative symmetric powers of a family of curves and a PD-polynomial algebra over the Chow homology of the relative Jacobian, using a Pontryagin product structure. The key result is that $\operatorname{CH}_{*}(C^{[\bullet]}/S) \cong \operatorname{CH}_{*}(J/S)[t]\langle u\rangle$, with two distinct isomorphisms differing by a class $\psi$, and it introduces a new grading on $\operatorname{CH}_{*}(J/S)$ compatible with the action of multiplication-by-$N$ maps and distinct from Beauville's decomposition.

ABSTRACT

Let C be a curve over a non-singular base variety S. We study algebraic cycles on the symmetric powers C^[n] and on the Jacobian J. The Chow homology of C^[*], the sum of all C^[n], is a ring using the Pontryagin product. We prove that this ring is isomorphic to CH(J)[t], the PD-polynomial algebra (variable: u) over the usual polynomial ring (variable: t) over the Chow ring CH(J). We give two such isomorphisms that over a general base are different. Further we give some precise results on how CH(J) sits embedded in CH(C^[*]) and we give an explicit geometric description of how the derivations with regard to t and u act. Our results give rise to a new grading on the Chow ring of the Jacobian. After tensoring with Q the associated descending filtration coincides with the one coming from Beauville's decomposition. The grading we obtain is in general different from Beauville's. Finally we give a version of our main result for tautological classes, and we show how our methods give a very simple and geometric proof of some relations obtained by Herbaut and van der Geer-Kouvidakis.

Motivation & Objective

• To understand the structure of algebraic cycles on the relative symmetric powers $C^{[n]}$ and the relative Jacobian $J$ of a family of curves over a base $S$.
• To establish a ring isomorphism between the Chow homology of the infinite symmetric power $C^{[\bullet]}$ and a PD-polynomial algebra over the Chow homology of the Jacobian.
• To introduce a new grading on $\operatorname{CH}_{*}(J/S)$ that is stable under the action of multiplication-by-$N$ maps and distinct from Beauville’s decomposition.
• To provide a geometric proof of tautological relations previously obtained via cohomological or Fourier-duality methods.

Proposed method

• Use of the Pontryagin product to endow $\operatorname{CH}_{*}(C^{[\bullet]}/S)$ with a bigraded ring structure.
• Construction of two distinct isomorphisms $\tilde{\beta}, \tilde{\gamma}: \operatorname{CH}_{*}(J/S)[t]\langle u\rangle \xrightarrow{\sim} \operatorname{CH}_{*}(C^{[\bullet]}/S)$, differing by the class $\psi = p_0^*K$, where $K$ is the relative canonical class.
• Explicit geometric description of the action of the operators $\partial_t^{[m]}$ and $\partial_u$ on the Chow homology via geometrically defined operators $P_{i,j}(a)$ from Part I.
• Application of the Manin principle to interpret the isomorphism motivically in the ind-category of Chow motives over $S$, yielding $R_*(J/S)[\mathbf{1}]\langle\mathbf{1}(1)\rangle \cong R_*(C^{[\bullet]}/S)$.
• Proof of compatibility of the isomorphism with PD-structures on the relevant ideals, using the fact that $\psi$-torsion is controlled and the PD-structures are natural in the context of smooth quasi-projective bases.
• Use of symmetric functions and Newton identities to derive vanishing results in $\operatorname{CH}_0(C^{[N]})_{\mathbb{Q}}$ and $\operatorname{CH}_0(J)_{\mathbb{Q}}$ under conditions on linear systems.

Experimental results

Research questions

• RQ1 How is the Chow homology of the infinite relative symmetric power $C^{[\bullet]}$ structured as a ring under the Pontryagin product?
• RQ2 What is the precise relationship between the Chow homology of the relative Jacobian $J$ and that of the symmetric powers $C^{[n]}$?
• RQ3 Can a new grading on $\operatorname{CH}_{*}(J/S)$ be defined that is stable under the action of multiplication-by-$N$ maps and distinct from Beauville’s decomposition?
• RQ4 How do the operators $\partial_t^{[m]}$ and $\partial_u$ act on the Chow homology, and what is their geometric meaning?
• RQ5 Can the tautological relations of Herbaut and van der Geer–Kouvidakis be recovered via a geometric, motivic argument?

Key findings

• The Chow homology $\operatorname{CH}_{*}(C^{[\bullet]}/S)$ is isomorphic to $\operatorname{CH}_{*}(J/S)[t]\langle u\rangle$, the PD-polynomial algebra in $u$ over the polynomial algebra in $t$ over $\operatorname{CH}_{*}(J/S)$, with integral coefficients and compatible with PD-structures.
• There exist two distinct isomorphisms $\tilde{\beta}, \tilde{\gamma}$ from $\operatorname{CH}_{*}(J/S)[t]\langle u\rangle$ to $\operatorname{CH}_{*}(C^{[\bullet]}/S)$, differing by the class $\psi = p_0^*K$, which is non-zero in general.
• The embedding of $\operatorname{CH}_{*}(J/S)$ into $\operatorname{CH}_{*}(C^{[\bullet]}/S)$ is not canonical and depends on the choice of isomorphism, reflecting a geometric ambiguity tied to the relative canonical class.
• A new grading on $\operatorname{CH}_{*}(J/S)$ is defined such that the associated descending filtration is stable under all $[N]_*$ maps, and $[N]_*$ acts on the $m$-th graded piece as multiplication by $N^m$, after tensoring with $\mathbb{Q}$.
• The new grading is distinct from Beauville’s decomposition, and the two decompositions are related via Fourier duality.
• The paper gives a geometric proof of the tautological relations of Herbaut and van der Geer–Kouvidakis, showing that $\sum_{n_1+\cdots+n_r=N} (n_1-1)!\cdots(n_r-1)! \cdot \mathrm{cw}_{n_1}(C)*\cdots*\mathrm{cw}_{n_r}(C) = 0$ in $A_*(J)_{\mathbb{Q}}$ when $C$ admits a $g^r_d$ over $k$ and $N > d$.