Chow groups of symmetric products of projective spaces
Abstract
Let SPd(ℙn) denote the d-th symmetric product of complex projective space. We prove that its integral cycle class map is split injective and has finite cokernel annihilated by d!. We also compute the ranks of its Chow groups. By constructing compatible integral retractions on Chow groups and singular homology, we prove that addition of a fixed point induces an isomorphism on the Chow group of q-dimensional cycles whenever d≥ q. For n≥ 3 and d≥ 6, we construct a nonalgebraic class of order two in codimension two. Its image generates a direct summand of the cycle class cokernel and remains nonalgebraic after addition of fixed points. Consequently, integral surjectivity fails both at certain finite stages with q≤ d and in the infinite symmetric product. We further identify the codimension two cokernel with third unramified cohomology of the smooth configuration space. Finally, we provide an example where multiplication by an even integer annihilates a nonzero Chow class of order two.
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BibTeX
@misc{chen2026chow,
title = {{Chow groups of symmetric products of projective spaces}},
author = {Youming Chen and Wenchuan Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20239},
eprint = {2610.20239},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20239},
}
RIS
TY - UNPB
TI - Chow groups of symmetric products of projective spaces
AU - Youming Chen
AU - Wenchuan Hu
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.20239
UR - https://arxiv.org/abs/2610.20239
AB - Let SP^d(ℙ^n) denote the d-th symmetric product of complex projective space. We prove that its integral cycle class map is split injective and has finite cokernel annihilated by d!. We also compute the ranks of its Chow groups. By constructing compatible integral retractions on Chow groups and singular homology, we prove that addition of a fixed point induces an isomorphism on the Chow group of q-dimensional cycles whenever d≥ q. For n≥ 3 and d≥ 6, we construct a nonalgebraic class of order two in codimension two. Its image generates a direct summand of the cycle class cokernel and remains nonalgebraic after addition of fixed points. Consequently, integral surjectivity fails both at certain finite stages with q≤ d and in the infinite symmetric product. We further identify the codimension two cokernel with third unramified cohomology of the smooth configuration space. Finally, we provide an example where multiplication by an even integer annihilates a nonzero Chow class of order two.
ER -