Generalised Cross-Ratio of Projective Linear Spaces and Limits of Local Height Pairings

Abstract

This thesis generalises the classical cross-ratio of four projective points in complex projective 1-space to a pairing of two k-dimensional and two (n-k-1)-dimensional linear subspaces of complex projective n-space. We prove that the absolute values of the generalised cross-ratio determine the local height pairings of the corresponding cycles. For the Archimedean height, the proof avoids integrating Green’s forms over high-dimensional subspaces by relating this height, via an incidence correspondence, to an explicit height pairing of points and a principal divisor on a Grassmannian.

We then study degenerating families of these linear subspaces and describe the asymptotic behaviour of all local height pairings as the cycles intersect. We show that, in any degeneration, the asymptotics are governed by the intersection degree of the corresponding families of cycles.

Furthermore, using a global construction, we show that the generalised cross-ratio itself, rather than only its norms, computes a generalised Hodge-theoretic height. This is done by showing that the relative homology group induced by the two k-planes and two (n-k-1)-planes is naturally an extension of mixed Hodge structures of Z(0) by Z(1), and that the height of this extension equals the generalised cross-ratio.

Finally, using local methods, we study regularised limits of local heights and interpret these limits using only the central degenerate geometry, together with small perturbations. This thesis provides a new class of examples associating geometric shapes to limits of local heights.