An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with p-adic applications
Abstract
Let X be a nonsingular Hermitian variety of even dimension over a finite field, and let Λ be the lattice spanned by the classes of its maximal totally isotropic linear subspaces defined over the base field. By work of Dummigan, Dummigan--Tiep and Shimada, these classes span the middle cohomology, and the intersection form on the primitive part is definite, as predicted by the standard conjecture of Hodge type. We give an elementary proof, which avoids the representation theory of finite unitary groups and uses only the dual polar graph attached to the Hermitian form. It shows that the intersection form on the primitive part is an explicit positive multiple of a Euclidean inner product. The advantage of the new proof is that it also controls Λ p-adically. If X is defined over Fq2 and has dimension 2m, every p-adic elementary divisor of the intersection matrix of these subspaces divides qm. Consequently qm times every algebraic cycle of codimension m is numerically equivalent to an integral combination of them. As a by-product, the standard conjecture of Hodge type holds for varieties finitely covered by Hermitian varieties, and the p-adic bound descends along finite morphisms of degree prime to p.
Related
- Explicit lower bounds on the conductors of elliptic curves and abelian varieties over number fields
- Explicit equations of Galois subfields of Hermitian function fields with respect to decomposition groups
- Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
- Suslin's cancellation conjecture on smooth real affine varieties with few real points
- Arithmetic jet spaces and the unramified Zilber--Pink conjecture for subvarieties of abelian varieties
BibTeX
@misc{harashita2026explicit,
title = {{An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with $p$-adic applications}},
author = {Shushi Harashita},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01731},
eprint = {2610.01731},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01731},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An explicit form of the standard conjecture of Hodge type for Hermitian varieties, with p-adic applications
AU - Shushi Harashita
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01731
UR - https://arxiv.org/abs/2610.01731
AB - Let X be a nonsingular Hermitian variety of even dimension over a finite field, and let Λ be the lattice spanned by the classes of its maximal totally isotropic linear subspaces defined over the base field. By work of Dummigan, Dummigan--Tiep and Shimada, these classes span the middle cohomology, and the intersection form on the primitive part is definite, as predicted by the standard conjecture of Hodge type. We give an elementary proof, which avoids the representation theory of finite unitary groups and uses only the dual polar graph attached to the Hermitian form. It shows that the intersection form on the primitive part is an explicit positive multiple of a Euclidean inner product. The advantage of the new proof is that it also controls Λ p-adically. If X is defined over F_q^2 and has dimension 2m, every p-adic elementary divisor of the intersection matrix of these subspaces divides q^m. Consequently q^m times every algebraic cycle of codimension m is numerically equivalent to an integral combination of them. As a by-product, the standard conjecture of Hodge type holds for varieties finitely covered by Hermitian varieties, and the p-adic bound descends along finite morphisms of degree prime to p.
ER -