Green forms and the arithmetic Siegel-Weil formula

TL;DR

This paper constructs Green forms for special cycles on Shimura varieties and links local archimedean height pairings to derivatives of Siegel Eisenstein series, confirming part of Kudla's conjecture.

Contribution

It develops Green forms for special cycles in all codimensions and proves their relation to derivatives of Eisenstein series for certain Shimura varieties, settling part of Kudla's conjecture.

Findings

Green forms constructed for all codimensions.

Local archimedean height pairings related to derivatives of Eisenstein series.

Partial confirmation of Kudla's conjecture on arithmetic intersections.

Abstract

We construct natural Green forms for special cycles in orthogonal and unitary Shimura varieties, in all codimensions, and, for compact Shimura varieties of type O(p,2) and U(p,1), we show that the resulting local archimedean height pairings are related to special values of derivatives of Siegel Eisentein series. A conjecture put forward by Kudla relates these derivatives to arithmetic intersections of special cycles, and our results settle the part of his conjecture involving local archimedean heights.