Maria Rosaria Pati (Genova) : Special cycles on Kuga varieties over a Siegel threefold

We construct special cycles on the $k$-fold Kuga variety $\mathcal{A}^{(k)}$ over a Siegel threefold $S_2(n)$, where $k$ and $n$ are positive integers with $n\geq 3$. These are cycles fibered over the points of Humbert surfaces in $S_2(n)$. We prove that the generating series of the cohomology classes of these cycles is a modular form of weight $k+\frac{5}{2}$ and level $\Gamma\subset\tilde{\mathrm{SL}}_2(\mathbb{Z})$ depending on $n$, for the metaplectic cover $\tilde{\mathrm{SL}}_2$ of $\mathrm{SL}_2$. Our result generalizes to the higher weight setting previous works of Kudla and Millson on special cycles on Shimura varieties of orthogonal type.