A generalised cross-ratio and limits of local heights

Abstract

We generalise the standard cross-ratio of four points on a projective line to a cross-ratio of a configuration of four planes in projective $n$-space, the first pair $A_1,\,A_2$ being $k$-dimensional and the second pair $B_1,\,B_2$ being $(n-k-1)$-dimensional, with $A_i \cap B_j = \emptyset$. Over the complex numbers, we show that this cross-ratio equals the augmented height pairing of the corresponding cycles $A_1-A_2, \, B_1-B_2$. Over a discretely valued field, we show that the valuation of the cross-ratio equals the intersection degree of the cycles once they are spread out over the valuation ring. Putting the two together, we conclude that the asymptotics of the Archimedean height pairing of a holomorphic family of configurations are governed by this intersection degree. We also define a degenerate cross-ratio for when $A_i \cap B_j \neq \emptyset$ and interpret the "limit height" of a degenerating holomorphic family of planes as the degenerate cross-ratio of the central plane configuration.