On the Kernel of the Gysin Homomorphism on Chow Groups of Zero Cycles

Paucar, Rina; Schoemann, Claudia

Publications mathématiques de Besançon. Algèbre et théorie des nombres (2024), pp. 59-104

Given a smooth projective connected surface over $ℂ$ embedded into a projective space ${\mathbb{P}}^d$ and a smooth projective curve $C$ embedded into the surface we study the kernel of the Gysin homomorphism between the Chow groups of 0-cycles of degree zero of the curve and the surface induced by the closed embedding. Following the approach of Bannerjee and Guletskii we prove that the kernel of the Gysin homomorphism is a countable union of translates of an abelian subvariety $A$ inside the Jacobian $J$ of the curve $C$. We also prove that there is a $c$-open subset $U_0$ contained in the set $U\subset {\left({\mathbb{P}}^d\right)}^{*}$ parametrizing the smooth projective curves such that $A=0$ or $A=B$ for all curves parametrized by $U_0$, where $B$ is the abelian subvariety of $J$ corresponding to the vanishing cohomology $H^1{(C,\mathbb{Q})}_{\mathrm{van}}$ of $C$.

We give a background of algebraic cycles, Chow groups, Hodge structures, the Abel–Jacobi map, Lefschetz pencils and the irreducibility of the monodromy representation.

Étant donné une surface lisse projective connexe sur $ℂ$ plongée dans un espace projectif ${\mathbb{P}}^d$ et $C$ une courbe lisse projective plongée dans la surface, on étudie le noyau de l’homomorphisme de Gysin entre les groupes de Chow des 0-cycles de degré zéro de la courbe et de la surface induit par l’injection fermée. Suivant l’approche de Bannerjee and Guletskii, on démontre que le noyau de l’homomorphisme de Gysin est une union dénombrable de translatées d’une sous-variété abélienne $A$ dans la Jacobienne $J$ de la courbe $C$. On démontre également qu’il existe un sous-ensemble c-ouvert $U_0$ de l’ensemble $U\subset {\left({\mathbb{P}}^d\right)}^{*}$ paramétrisant les courbes lisses projectives tel que $A=0$ ou $A=B$ pour toute courbe paramétrisée par $U_0$, où $B$ est la sous-variété abélienne de $J$ correspondant à la cohomologie évanescente $H^1{(C,\mathbb{Q})}_{\mathrm{van}}$ de $C$.

On donne une introduction aux cycles algébriques, aux groupes de Chow, aux structures de Hodge, à l’application d’Abel–Jacobi, aux pinceaux de Lefschetz et à l’irréductibilité de la représentation de monodromie.

Keywords: $0$-cycles, rational, algebraic and homological equivalence, Chow groups, Hodge theory, Lefschetz pencils, the monodromy argument, Gysin homomorphism

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