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 $\F_{q^2}$ 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 $p$.