We prove the regularity of the free boundary in the Monge–Ampère obstacle problem $det{D}^{2}v=f(y){\mathit{\chi }}_{\{v>0\}}$. By duality, the regularity of the free boundary is equivalent to that of the asymptotic cone of the solution to the singular Monge–Ampère equation $det{D}^{2}u=1∕f(Du)+{\mathit{\delta }}_0$ at the origin. We first establish an asymptotic estimate for the solution u near the singular point, then use a partial Legendre transform to change the Monge–Ampère equation to a singular, fully nonlinear elliptic equation, and establish the regularity of solutions to the singular elliptic equation.

We show that the ring of Siegel–Jacobi forms of fixed degree and of fixed or bounded ratio between weight and index is not finitely generated. Our main tool is the theory of toroidal b-divisors and their relation to convex geometry. As a byproduct of our methods, we prove a conjecture of Kramer about the representation of all Siegel–Jacobi forms as sections of certain line bundles and we recover a formula due to Tai for the asymptotic dimension of the space of Siegel–Jacobi forms of given ratio between weight and index.

Let G be a connected reductive algebraic group over an algebraically closed field $\mathbb{k}$ of characteristic $p\ge 0$, and let $\mathcal{N}$ be its nilpotent cone. Under mild hypotheses, we construct for each nilpotent G-orbit C and each indecomposable tilting vector bundle $\mathcal{T}$ on C a certain complex $\mathcal{S}(C,\mathcal{T})\in D^{\mathrm{b}}{\mathsf{Coh}}^{G\times {\mathbb{G}}_{\mathrm{m}}}(\mathcal{N})$. We prove that these objects are (up to shift) precisely the indecomposable objects in the coheart of a certain co-t-structure.

We then show that if p is larger than the Coxeter number, then the hypercohomology $H^{\bullet }(\mathcal{N},\mathcal{S}(C,\mathcal{T}))$ is identified with the cohomology of a tilting module for G. This confirms a conjecture of Humphreys on the support of the cohomology of tilting modules.

In this paper, we establish the theory of local newforms for irreducible tempered generic representations of unramified odd unitary groups over a non-Archimedean local field of characteristic 0. For the proof, we prove an analogue of the fundamental lemma for our compact groups.