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Arakelov intersection indices of linear cycles and the geometry of buildings and symmetric spaces. (English) Zbl 1075.14022

This paper generalizes work of Yu. Manin [Invent. Math. 104, 223–243 (1991; Zbl 0754.14014)] for curves of genus \(0\), to Arakelov intersections of certain linear subspaces in projective space \(\mathbb P^{n-1}\).
First, this is done at non-archimedean places, by considering the Bruhat-Tits building \(X\) for the group \(G:=\text{PGL}(V)\), where \(V\) is an \(n\)-dimensional vector space over a non-Archimedean local field \(K\) of characteristic zero. The boundary \(X_\infty\) of \(X\) in the Borel-Serre compactification is identified with the Tits building of \(G\), which is the flag complex in \(V\). Fix an \(R\)-lattice \(M\) in \(V\); this corresponds to choosing a model for \(\mathbb P(V)=\mathbb P^{n-1}_K\) over the ring of integers \(R\) of \(K\) (here \(\mathbb P(V)\) means the projective space of lines in \(V\) passing through the origin). Let \(A\) and \(B\) be linear subspaces of \(V\) of dimension \(p\), and let \(C\) and \(D\) be linear subspaces of dimension \(n-p\). Assume that \(A\cup B\) and \(C\cup D\) are disjoint except at the origin. Under certain additional conditions, an oriented geodesic is explicitly defined such that its oriented length times \(p\) equals the intersection number \(\langle\mathbb P(A)-\mathbb P(B),\mathbb P(C)-\mathbb P(D)\rangle\).
An archimedean analogue of this picture is then constructed, as follows. In place of \(X\), let \(Z\) be the symmetric space \(\text{SL}_n(\mathbb C)\). It is compactified by adding the set of geodesic rays emanating from a fixed point [see W. Ballmann, M. Gromov and V. Schroeder, “Manifolds of nonpositive curvature”, Prog. Math. 61 (1985; Zbl 0591.53001)]. Again, under certain conditions, an oriented geodesic is explicitly defined whose oriented distance times \(\sqrt p/\sqrt q\) is the Arakelov intersection number as above, thus giving a close parallel to the non-archimedean case.

MSC:

14G40 Arithmetic varieties and schemes; Arakelov theory; heights
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
20E42 Groups with a \(BN\)-pair; buildings
51E24 Buildings and the geometry of diagrams
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References:

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