Cones of special cycles of codimension 2 on orthogonal Shimura varieties

TL;DR

This paper studies the structure of the cone of special codimension 2 cycles on orthogonal Shimura varieties, proving its polyhedrality and developing an algorithm to verify this property.

Contribution

It demonstrates the polyhedrality of the accumulation cone of special cycles and provides an algorithm for certification in specific cases.

Findings

The accumulation cone is polyhedral.

Accumulation rays are generated by Heegner divisors intersected with the Hodge class.

An implemented SageMath algorithm certifies polyhedrality in certain cases.

Abstract

Let $X$ be an orthogonal Shimura variety associated to a unimodular lattice. We investigate the polyhedrality of the cone $\mathcal{C}_X$ of special cycles of codimension 2 on $X$. We show that the rays generated by such cycles accumulate towards infinitely many rays, the latter generating a non-trivial cone. We also prove that such an accumulation cone is polyhedral. The proof relies on analogous properties satisfied by the cones of Fourier coefficients of Siegel modular forms. We show that the accumulation rays of $\mathcal{C}_X$ are generated by combinations of Heegner divisors intersected with the Hodge class of $X$. As a result of the classification of the accumulation rays, we implement an algorithm in SageMath to certify the polyhedrality of $\mathcal{C}_X$ in some cases.