An Analytic Description of Local Intersection Numbers at Non-Archimedian Places for Products of Semi-Stable Curves

Abstract

We generalise a formula of Shou-Wu Zhang [Zha10, Thm 3.4.2], which describes local arithmetic intersection numbers of three Cartier divisors with support in the special fibre on a a self-product of a semi-stable arithmetic surface using elementary analysis. By an approximation argument, Zhang extends his formula to a formula for local arithmetic intersection numbers of three adelic metrized line bundles on the self-product of a curve with trivial underlying line bundle. Using the results on intersection theory from [Kol] we generalize these results to d-fold self-products for arbitrary d. For the approximations to converge, we have to assume that d satisfies the vanishing condition [Kol, 4.7], which is true at least for d ∈ {2, 3, 4, 5}. Contents 11 3.2. The Intersection Pairing 19 Appendix A. The geometric realization of simplicial sets and their subdivision 28 References 31

The combinatorial Chow ring of products of graphs

“…With these preliminaries, combining our Theorem 1.8 with the results in [17], we get the following generalization of Equation (4). Theorem 1.9 (Kolb [17]). For any collection of functions f 0 , .…”

“…For each w i , denote by α(w i , P) the number of indices 1 ≤ i ≤ k such that there exists j ∈ P i with w j = 1. This property was conjectured by Kolb and is required in [17] in order to get the analytic description of the local degree map, that we briefly describe in the next section. 1.3.…”

“…Consider the Chow ring with support Chow Xs (X) (see [11] for the definition of Chow rings with support). The intersection products between X v , for v ∈ V , in Chow Xs (X) verify three types of equations (see (R1), (R2), (R3) below), given by Kolb in [16,17], two of which are higher dimensional analogous of the relations (A 1) and (A 2). This leads to the definition of a Chow ring for the product of graphs, that we describe next.…”

“…By approximation one may take a sequence of piecewise linear functions converging to each of the piecewise smooth functions f i . This has been carried out in great detail in [17]. The well-definedness of the extension as well as the analytic generalization of the Formula ( 4) is guaranteed if the vanishing condition in Theorem 1.8 holds.…”

“…as the space of functions f : T → R which are smooth on simplices of G [17]. This means, for any cube e ≃ [0, 1] d , the restriction of f to each triangle ∆ of [0, 1] d can be extended to a smooth function in a neighborhood of ∆.…”