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arXiv · 2610.20239

Chow groups of symmetric products of projective space

Youming Chen·Wenchuan Hu
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Abstract

We prove that the integral cycle class map for $\SP^d(\PP^n)$ is split injective, with finite cokernel annihilated by $d!$,and we determine the ranks of Chow groups for $\mathrm{SP}^d(\mathbb{P}^n)$. Compatible integral transfer retractions show that $\CH_q(\SP^d(\PP^n))$ stabilizes for $d\geq q$. An explicit nonalgebraic order two class persists under addition of points, producing a split cokernel summand even in the range $q\leq d$ and in the infinite symmetric product. For the smooth configuration open set, we identify the codimension two cokernel with third unramified cohomology and detect its ramification on a rational resolution. In contrast to addition, even multiplicities annihilate a nonzero Chow class.

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Youming Chen, Wenchuan Hu. 2026-10-14. Chow groups of symmetric products of projective space. https://arxiv.org/abs/2610.20239

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