Gillet–Soulé's arithmetic analogues of the standard conjectures

Let $X$ be an arithmetic variety, meaning a regular scheme projective and flat over $\mathbb Z$ whose generic fibre $X_{\mathbb Q}$ is smooth over $\mathbb Q$. Let $\widehat{CH}^p(X)_{\mathbb R}$ denote its arithmetic Chow groups, and let $H$ be an ample line bundle on $X$. For an $F_{\infty}$-invariant hermitian metric $\|\ \|$ on $H_{\mathbb C}$, write

$$L_{H,\|\ \|}:\widehat{CH}^p(X)_{\mathbb R}\to\widehat{CH}^{p+1}(X)_{\mathbb R}$$

for intersection with the arithmetic first Chern class. Gillet–Soulé's arithmetic standard-conjecture analogue. If $n$ is the relative dimension of $X$ over $\mathbb Z$, there exists an $F_{\infty}$-invariant hermitian metric $\|\ \|$ on $H_{\mathbb C}$ such that, for $2p\leq n+1$, $L_{H,\|\ \|}^{n+1-2p}:\widehat{CH}^p(X)_{\mathbb R}\to\widehat{CH}^{n+1-p}(X)_{\mathbb R}$ is an isomorphism and, for every $0\not= x\in\widehat{CH}^p(X)_{\mathbb R}$ with $L_{H,\|\ \|}^{n+2-2p}(x)=0$, $(-1)^p\widehat{\operatorname{deg}}(L_{H,\|\ \|}^{n+1-2p}(x)x)$ is positive. The paper records proofs in several special cases, including projective spaces and regular quadric hypersurfaces, but not the general conjecture.