In this paper we investigate the geometry of the orbit space of the closure of the subscheme parameterizing smooth Kähler–Einstein Fano manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski-open condition, and we establish the uniqueness of the Gromov–Hausdorff limit for a punctured flat family of Kähler–Einstein Fano manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\mathbb{Q}$-Gorenstein smoothable, K-semistable $\mathbb{Q}$-Fano varieties, and we verify various necessary properties to guarantee that it is a good moduli space.

A purity conjecture due to Grothendieck and Auslander–Goldman predicts that the Brauer group of a regular scheme does not change after removing a closed subscheme of codimension $\ge 2$. The combination of several works of Gabber settles the conjecture except for some cases that concern $p$-torsion Brauer classes in mixed characteristic $(0,p)$. We establish the remaining cases by using the tilting equivalence for perfectoid rings. To reduce to perfectoids, we control the change of the Brauer group of the punctured spectrum of a local ring when passing to a finite flat cover.

We study local and global properties of positive solutions of $-\Delta u=u^p|\nabla u{|}^q$ in a domain $\Omega$ of ${\mathbb{R}}^N$, in the range $p+q>1$, $p\ge 0$, $0\le q<2$. We first prove a local Harnack inequality and nonexistence of positive solutions in ${\mathbb{R}}^N$ when $p(N-2)+q(N-1)<N$. Using a direct Bernstein method, we obtain a first range of values of $p$ and $q$ in which $u(x)\le c(dist(x,\partial \Omega ){)}^{\frac{q-2}{p+q-1}}$. This holds in particular if $p+q<1+\frac{4}{N-1}$. Using an integral Bernstein method, we obtain a wider range of values of $p$ and $q$ in which all the global solutions are constants. Our result contains Gidas and Spruck’s nonexistence result as a particular case. We also study solutions under the form $u(x)=r^{\frac{q-2}{p+q-1}}\omega (\sigma )$. We prove existence, nonexistence, and rigidity of the spherical component $\omega$ in some range of values of $N$, $p$, and $q$.

We show that for any bounded operator $T$ acting on an infinite-dimensional Banach space there exists an operator $F$ of rank at most one such that $T+F$ has an invariant subspace of infinite dimension and codimension. We also show that whenever the boundary of the spectrum of $T$ or $T^{\ast }$ does not consist entirely of eigenvalues, we can find such rank-one perturbations that have arbitrarily small norm. When this spectral condition is not satisfied, we can still find suitable finite-rank perturbations of arbitrarily small norm, but not necessarily of rank one.