Poincaré series of degree-two Chow varieties
Abstract
Let Cp, 2(ℙn) denote the Chow variety parametrizing effective algebraic p-cycles of degree two on ℙn. We show that its rational cohomology is pure Tate and concentrated in even degrees. An anti-invariant Chern class identity yields a presentation of the cohomology ring as a fiber product, which in turn determines the cohomology over ℤ[1/2]. We identify the ordinary Poincaré polynomial with the known virtual formula, determine the optimal ranges over which the coefficients stabilize, and compute the nonrational Poincaré series of the stable space D(2). For the Lawson--Michelsohn map D(2)⟶ D(∞), the first nonzero rational homotopy kernel occurs in degree eight and is one-dimensional, while the integral kernel has rank one. Finally, we compute the relative homology of the pair $(D(2),BU)$ over ℤ[1/2] and show that additional cohomology generators and nonzero rational homotopy kernels arise in unbounded degrees.
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BibTeX
@misc{chen2026poincaré,
title = {{Poincaré series of degree-two Chow varieties}},
author = {Youming Chen and Wenchuan Hu and Deyuan Ma},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39498},
eprint = {2610.39498},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39498},
}
RIS
TY - UNPB
TI - Poincaré series of degree-two Chow varieties
AU - Youming Chen
AU - Wenchuan Hu
AU - Deyuan Ma
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39498
UR - https://arxiv.org/abs/2610.39498
AB - Let C_p, 2(ℙ^n) denote the Chow variety parametrizing effective algebraic p-cycles of degree two on ℙ^n. We show that its rational cohomology is pure Tate and concentrated in even degrees. An anti-invariant Chern class identity yields a presentation of the cohomology ring as a fiber product, which in turn determines the cohomology over ℤ[1/2]. We identify the ordinary Poincaré polynomial with the known virtual formula, determine the optimal ranges over which the coefficients stabilize, and compute the nonrational Poincaré series of the stable space D(2). For the Lawson--Michelsohn map D(2)⟶ D(∞), the first nonzero rational homotopy kernel occurs in degree eight and is one-dimensional, while the integral kernel has rank one. Finally, we compute the relative homology of the pair $(D(2),BU)$ over ℤ[1/2] and show that additional cohomology generators and nonzero rational homotopy kernels arise in unbounded degrees.
ER -