[Paper Review] On the kernel of the push-forward homomorphism between Chow groups

Kalyan Banerjee, Jaya Nn Iyer|arXiv (Cornell University)|Apr 29, 2015

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper proves that the push-forward homomorphism on rational Chow groups, induced by the closed embedding of a special ample divisor $H_C$ (linearly equivalent to $n\Theta$) into the Jacobian variety $J(C)$ of a smooth projective curve $C$, has a trivial kernel for all $k \geq 0$. The proof relies on localization sequences for higher Chow groups and an extension of Collino's theorem on symmetric powers of curves.

ABSTRACT

In this note we prove that the kernel of the push-forward homomorphism on $d$-cycles modulo rational equivalence, induced by the closed embedding of an ample divisor linearly equivalent to some multiple of the theta divisor inside the Jacobian variety $J(C)$ is trivial. Here $C$ is a smooth projective curve of genus $g$.

Motivation & Objective

• To investigate the kernel of the push-forward homomorphism on rational Chow groups induced by a closed embedding of an ample divisor into a smooth projective variety.
• To verify a dual version of Nori's and Paranjape's conjectures on Chow groups for the case of theta divisors in Jacobian varieties.
• To establish injectivity of the push-forward map on $k$-cycles for divisors linearly equivalent to multiples of the theta divisor in $J(C)$.
• To extend Collino's theorem on symmetric powers of curves to higher Chow groups and open subsets, enabling the main proof.

Proposed method

• Utilizes the localization exact sequence for higher Chow groups to analyze cycle classes on $G$-fixed subvarieties of the Jacobian of a Galois cover $\tilde{C} \to C$.
• Applies a theorem of Collino to prove injectivity of push-forward maps on $k$-cycles for inclusions of symmetric powers $\mathrm{Sym}^m C \hookrightarrow \mathrm{Sym}^n C$ with $m \leq n$.
• Extends Collino's result to higher Chow groups of symmetric powers and their open subsets, establishing well-definedness of pullback maps $\rho^{s*}$ at the level of higher Chow groups.
• Constructs a commutative diagram involving higher Chow groups of symmetric powers and their complements, using boundary maps $\partial_i^*$ and pullbacks $\rho^{s*}$ to analyze cycle classes.
• Employs induction on the symmetric power degree to prove injectivity of the push-forward map $i_{m,n*}^s$ on higher Chow groups $\mathrm{CH}^*(\mathrm{Sym}^m C, s) \to \mathrm{CH}^*(\mathrm{Sym}^n C, s)$.
• Uses the injectivity of the push-forward on $\mathrm{Sym}^{m-1}C$ to deduce triviality of cycles in $\mathrm{Sym}^m C$ under the push-forward, leading to the main result on $H_C \subset J(C)$.

Experimental results

Research questions

• RQ1 Is the push-forward homomorphism on rational Chow groups injective for the inclusion of a special ample divisor $H_C$ into the Jacobian $J(C)$, where $H_C$ is the restriction of a theta divisor from a Galois cover of $C$?
• RQ2 Does the injectivity of the push-forward on $k$-cycles hold for divisors linearly equivalent to $n\Theta$ in $J(C)$, particularly when $H_C$ arises as the pullback of a theta divisor from $J(\tilde{C})$?
• RQ3 Can Collino's theorem on symmetric powers of curves be extended to higher Chow groups and open subsets of symmetric powers?
• RQ4 What is the structure of the kernel of the push-forward map $j_{C*}: \mathrm{CH}_k(H_C) \otimes \mathbb{Q} \to \mathrm{CH}_k(J(C)) \otimes \mathbb{Q}$ for such divisors $H_C$?
• RQ5 How does the injectivity of the push-forward on higher Chow groups relate to the geometry of the Jacobian and its symmetric powers?

Key findings

• The kernel of the push-forward homomorphism $j_{C*}: \mathrm{CH}_k(H_C) \otimes \mathbb{Q} \to \mathrm{CH}_k(J(C)) \otimes \mathbb{Q}$ is trivial for all $k \geq 0$, where $H_C$ is a smooth, irreducible, ample divisor in $|n\Theta|$ arising as the restriction of a theta divisor from $J(\tilde{C})$.
• The push-forward map on higher Chow groups $i_{m,n*}^s: \mathrm{CH}^*(\mathrm{Sym}^m C, s) \to \mathrm{CH}^*(\mathrm{Sym}^n C, s)$ is injective for all $m \leq n$ and all $s \geq 0$.
• The pullback map $\rho^{s*}$ on higher Chow groups is well-defined for open subsets of symmetric powers, specifically from $\mathrm{CH}^*(\mathrm{Sym}^m C \setminus W, s)$ to $\mathrm{CH}^*(W_0(m), s)$, where $W_0(m)$ is a complement in the symmetric power.
• The injectivity of the push-forward on $\mathrm{CH}^*(\mathrm{Sym}^{m-1}C, s)$ implies injectivity on $\mathrm{CH}^*(\mathrm{Sym}^m C, s)$ via the localization exact sequence, enabling inductive proof.
• The extension of Collino's theorem to higher Chow groups and open subsets provides a crucial technical tool for proving the main result on the Jacobian.
• For a very general divisor in $|n\Theta|$, injectivity on $CH_0$ cannot be expected, as noted by C. Voisin, but the result holds for the special $H_C$ constructed via Galois covers.