Arithmetic intersection theory

Chern classes of vector bundles on arithmetic varieties

“…Namely X is a regular integral scheme and / is flat, projective of pure relative dimension d with the smooth generic fiber XQ. For an integer p > 0, we denote by CH (X) the arithmetic Chow group of codimension p. In [GS1] (see also [SABK]), H. Gillet and C. Soule showed that there exist an intersection product …”

Green currents for modular cycles in arithmetic quotients of complex hyperballs

“…In the arithmetic intersection theory developed by Gillet and Soulé, the role played by the algebraic cycles in the conventional intersection theory is replaced with the arithmetic cycles. In a heuristic sense, Green currents are regarded as the "archimedean" ingredient of such arithmetic cycles [Gillet and Soulé 1990]. Consider the case when X is the quotient of a Hermitian symmetric domain G/K by an arithmetic lattice in the semisimple Lie group G, and Y is a modular cycle stemming from a modular imbedding H/H ∩ K → G/K , where H is a reductive subgroup of G such that H ∩ K is maximally compact in H .…”

Arakelov theory of even orthogonal Grassmannians

“…Let G = (OG, ω OG ) denote the corresponding Arakelov variety, in the sense of [GS1]. The Chow ring CH(OG) and the ring Harm(OG R ) of real ω OG -harmonic differential forms on OG(C) are related by natural isomorphisms…”

“…Elements in the Arakelov Chow group CH p (OG) are represented by arithmetic cycles (Z, g Z ), where Z is a codimension p cycle on OG and g Z is a current of type (p − 1, p − 1) such that the current dd c g Z + δ Z (C) is represented by a differential form in Harm p,p (OG R ). Since the homogeneous space OG admits a natural cellular decomposition, it follows that for each p, the exact sequence of [GS1,§3.3.5] is of the form…”