The Geometric Unitary Kudla Conjecture

TL;DR

This paper proves the modularity of certain Kudla generating series over CM fields, confirming the geometric unitary Kudla Conjecture and removing previous modularity assumptions in related formulas.

Contribution

It establishes the modularity of Chow-valued Kudla generating series for unitary Shimura varieties over arbitrary CM fields, confirming the geometric Kudla Conjecture in all codimensions.

Findings

Proves convergence and equality of symmetric formal Fourier-Jacobi series to genuine Hermitian Hilbert modular forms.

Shows the Kudla generating series is modular of weight p+1 for a Weil representation.

Removes the modularity hypothesis from the arithmetic inner product formula by Li-Liu.

Abstract

We prove that, over an arbitrary CM field, every symmetric formal Fourier-Jacobi series converges and equals the Fourier-Jacobi expansion of a genuine Hermitian Hilbert modular form. As an application, we show that the Chow-valued Kudla generating series of special cycles on unitary Shimura varieties for Hermitian lattices over CM fields of signature $(p,1)$ at one infinite place and $(p+1,0)$ at all others is modular of weight $p+1$ for a Weil representation, establishing the geometric unitary Kudla Conjecture in arbitrary codimension. This removes the modularity hypothesis from the arithmetic inner product formula by Li-Liu.