We study the injectivity of the Kudla–Millson lift of genus-2 Siegel cusp forms, vector-valued with respect to the Weil representation associated to an even lattice L. We prove that if L splits off two hyperbolic planes and is of sufficiently large rank, then the lift is injective. As an application, we deduce that the image of the lift in the degree-4 cohomology of the associated orthogonal Shimura variety has the same dimension as the lifted space of cusp forms. Our results also cover the case of moduli spaces of quasi-polarized K3 surfaces. To prove the injectivity, we introduce vector-valued indefinite Siegel theta functions of genus 2 and of Jacobi type attached to L. We describe their behavior with respect to the split of a hyperbolic plane in L. This generalizes the results of Borcherds to genus higher than 1.

We introduce and explore the Uniform Izumi–Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if R is a normal domain essentially of finite type over a field, there exists a constant C such that for all prime ideals $\mathfrak{p}\subseteq \mathfrak{q}\in\mathrm{Spec}(R)$, if $\mathfrak{p}\subseteq \mathfrak{q}^{(t)}$, then for all $n\in\mathbb{N}$, there is a containment of symbolic powers $\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$.

We prove that all projective crepant resolutions of Nakajima quiver varieties satisfying natural conditions are also Nakajima quiver varieties. More generally, we classify the small birational models of many geometric invariant theory (GIT) quotients by introducing a sufficient condition for the GIT quotient of an affine variety V by the action of a reductive group G to be a relative Mori dream space. Two surprising examples illustrate that our new condition is optimal. When the condition holds, we show that the linearisation map identifies a region of the GIT fan with the Mori chamber decomposition of the relative movable cone of $V{{/\!\!/\!}}_\theta G$. If $V{{/\!\!/\!}}_\theta G$ is a crepant resolution of $Y\!\!:= V{{/\!\!/\!}}_0 G$, then every projective crepant resolution of Y is obtained by varying $\theta$. Under suitable conditions, we show that this is the case for quiver varieties and hypertoric varieties. Similarly, for any finite subgroup $\Gamma\subset \operatorname{SL}(3,{{\mathbb{C}}})$ whose non-trivial conjugacy classes are all junior, we obtain a simple geometric proof of the fact that every projective crepant resolution of $\mathbb{C}^3/\Gamma$ is a fine moduli space of $\theta$-stable $\Gamma$-constellations.

We provide a complete classification of when the homeomorphism group of a stable surface, $\Sigma$, has the automatic continuity property: Any homomorphism from $\mathrm{Homeo}({\Sigma})$ to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of $\Sigma$ has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable, second-countable Stone space has the automatic continuity property. Under the presence of stability, this answers two questions of Mann.

Let A be an abelian variety over a number field $\mathsf{K},$ with algebraic closure $\bar{\mathsf{K}}$. Assuming the Mumford–Tate conjecture for A, we show that the isogeny class of A over $\bar{\mathsf{K}}$ contains only finitely many isomorphism classes of bounded Faltings height. As the Mumford–Tate conjecture is known for many abelian varieties, our theorem is unconditional in those cases.