We prove that every smooth Fano complete intersection of index $1$ and codimension $r$ in ${\mathbb{P}}^{n+r}$ is birationally superrigid and K-stable if $n\ge 10r$. We also propose a generalization of Tian’s criterion of K-stability and, as an application, prove the K-stability of the complete intersection of a quadric and a cubic in ${\mathbb{P}}^5$. In the appendix (written jointly with C. Stibitz), we prove the conditional birational superrigidity of Fano complete intersections of higher index in large dimension.

For any right-angled Coxeter group $\Gamma$ on $k$ generators, we construct proper actions of $\Gamma$ on $\mathrm{O}(p,q+1)$ by right-and-left multiplication and on the Lie algebra $\mathfrak{o}(p,q+1)$ by affine transformations, for some $p,q\in \mathbb{N}$ with $p+q+1=k$. As a consequence, any virtually special group admits proper affine actions on some ${\mathbb{R}}^n$: this includes in particular surface groups, hyperbolic $3$-manifold groups, and examples of word hyperbolic groups of arbitrarily large virtual cohomological dimension. We also study some examples in cohomological dimensions two and four, for which the dimension of the affine space may be substantially reduced.

We develop a Kobayashi–Hitchin correspondence for the extended Bogomolny equations, that is, the dimensionally reduced Kapustin–Witten equations, on the product of a compact Riemann surface $\Sigma$ with ${\mathbb{R}}_y^{+}$, with generalized Nahm pole boundary conditions at $y=0$. The correspondence is between solutions of these equations satisfying these singular boundary conditions and also limiting to flat connections as $y\to \infty$ and certain holomorphic data consisting of triplets $(\mathcal{E},\phi ,L)$, where $(\mathcal{E},\phi )$ is a stable $SL(n+1,\mathbb{C})$ Higgs pair and $L\subset \mathcal{E}$ is a holomorphic line bundle. This corroborates a prediction of Gaiotto and Witten and serves as an extension of our earlier article which treats only the $SL(2,\mathbb{R})$ case.

Fix a semisimple Lie algebra $\mathfrak{g}$. Gaudin algebras are commutative algebras acting on tensor product multiplicity spaces for $\mathfrak{g}$-representations. These algebras depend on a parameter in the Deligne–Mumford moduli space of marked stable genus $0$ curves. When the parameter is real, then the Gaudin algebra acts with simple spectrum on the tensor product multiplicity space and gives us a basis of eigenvectors. We study the monodromy of these eigenvectors as the parameter varies within the real locus, giving an action of the fundamental group of this moduli space called the cactus group. We prove that the monodromy of eigenvectors for Gaudin algebras agrees with the action of the cactus group on tensor products of $\mathfrak{g}$-crystals (as conjectured by Etingof), and we prove that the coboundary category of normal $\mathfrak{g}$-crystals can be reconstructed using the coverings of the moduli spaces. To prove the conjecture, we construct a crystal structure on the set of eigenvectors for the shift of argument algebras, another family of commutative algebras acting on any irreducible $\mathfrak{g}$-representation. We also prove that the monodromy of such eigenvectors is given by the internal cactus group action on $\mathfrak{g}$-crystals.