The cohomological Kudla conjecture for unitary Shimura varieties

TL;DR

This paper proves a cohomological version of Kudla's conjecture for unitary Shimura varieties by constructing generating series of cohomology classes that are holomorphic Hermitian modular forms, and develops related quasi-modular form theory.

Contribution

It constructs natural extensions of Kudla--Millson series for unitary Shimura varieties and proves their holomorphicity, confirming a long-standing conjecture in all codimensions up to the middle.

Findings

Generated series are holomorphic Hermitian modular forms.

Established the theory of Hermitian quasi-modular forms.

Proved generating series of special cycles are Hermitian quasi-modular forms.

Abstract

We construct natural extensions of the Kudla--Millson generating series of cohomology classes of special cycles in compactified unitary Shimura varieties of signature $(n+1,1)$ and prove that they are holomorphic Hermitian modular forms. This proves the cohomological version of a conjecture of Kudla and Bruinier--Rosu--Zemel, in all codimensions up to the middle. We also develop the theory of Hermitian quasi-modular forms, with a particular focus on polynomial weighted theta functions, and prove that the generating series of Zariski closures of special cycles is a Hermitian quasi-modular form.