Essential p-dimension and Chern numbers
Abstract
Let X be a smooth, projective, connected variety over a field k containing a root of unity of order p. If X has a Chern number prime to p, we show that every finite p-group acting on X acts through a subgroup of GLn(k), where n=dim X. This allows one to transfer properties of representations of finite p-groups to their actions on X. We deduce a fixed-point theorem which, unlike earlier results of this kind, is sensitive to the arithmetic of the base field, and whose dimensional bound is sharp for each individual group (apart from one low-dimensional case when p=2). We also obtain restrictions on finite p-groups of birational automorphisms of rationally connected varieties; for instance the Cremona group Crn(ℚ) contains no element of order p2 when p ≥ n+2. More generally, we prove variants of these results in the presence of a Chern number not divisible by a given power of p. The method is based on the following observation about the essential p-dimension edp(G) of an affine algebraic group G: over a field of characteristic zero, the number edp(G) + dim G is the least dimension of a smooth projective variety Y with a generically free G-action such that the degree map CHG(Y) → 𝔽p from the equivariant Chow group is nonzero. The Chern numbers modulo p of Y lie in the image of this map; combined with Karpenko and Merkurjev's computation of the essential p-dimension of finite p-groups, this yields our results.
Related
- Essential Dimension and Faithful Rank of Finite p-Gerbes
- The finite-degree profile of essential dimension I: general theory and characteristic prime to |G|
- The finite-degree profile of essential dimension II: cyclic p-groups in characteristic p
- Point-Splitting-Perturbative-Chern-Simons II
- Second Chern class and Fujiki constants of hyperkähler manifolds
BibTeX
@misc{haution2026essential,
title = {{Essential $p$-dimension and Chern numbers}},
author = {Olivier Haution},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.08877},
eprint = {2608.08877},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.08877},
}
RIS
TY - UNPB
TI - Essential p-dimension and Chern numbers
AU - Olivier Haution
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2608.08877
UR - https://arxiv.org/abs/2608.08877
AB - Let X be a smooth, projective, connected variety over a field k containing a root of unity of order p. If X has a Chern number prime to p, we show that every finite p-group acting on X acts through a subgroup of GL_n(k), where n=dim X. This allows one to transfer properties of representations of finite p-groups to their actions on X. We deduce a fixed-point theorem which, unlike earlier results of this kind, is sensitive to the arithmetic of the base field, and whose dimensional bound is sharp for each individual group (apart from one low-dimensional case when p=2). We also obtain restrictions on finite p-groups of birational automorphisms of rationally connected varieties; for instance the Cremona group Cr_n(ℚ) contains no element of order p^2 when p ≥ n+2. More generally, we prove variants of these results in the presence of a Chern number not divisible by a given power of p. The method is based on the following observation about the essential p-dimension ed_p(G) of an affine algebraic group G: over a field of characteristic zero, the number ed_p(G) + dim G is the least dimension of a smooth projective variety Y with a generically free G-action such that the degree map CH_G(Y) → 𝔽_p from the equivariant Chow group is nonzero. The Chern numbers modulo p of Y lie in the image of this map; combined with Karpenko and Merkurjev's computation of the essential p-dimension of finite p-groups, this yields our results.
ER -