Volume 2019, Issue 751

The following dichotomy for affine ℤk\mathbb{Z}^{k} actions on the torus 𝕋d{\mathbb{T}}^{d}, k,d∈ℕk,d\in{\mathbb{N}}, is shown to hold: (i) The linear part of the action has no rank-one factors, and then the affine action is locally rigid. (ii) The linear part of the action has a rank-one factor, and then the affine action is locally rigid in a probabilistic sense if and only if the rank-one factors are trivial. Local rigidity in a probabilistic sense means that rigidity holds for a set of full measure of translation vectors in the rank-one factors.

We prove the existence and uniqueness of Kähler–Einstein metrics on ℚ{{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

Soit G un groupe réductif p -adique de centre connexe et de groupe dérivé simplement connexe. Nous montrons que certaines “chaînes ” de séries principales de G n’existent pas et nous établissons plusieurs propriétés de la construction Π⁢(ρ)ord\Pi(\rho)^{\mathrm{ord}} de Breuil–Herzig. En particulier, nous obtenons une caractérisation naturelle de cette dernière et nous démontrons une conjecture de Breuil–Herzig. Pour cela, nous calculons le δ-foncteur H∙⁢OrdP\mathrm{H^{\bullet}Ord}_{P} des parties ordinaires dérivées d’Emerton relatif à un sous-groupe parabolique P de G sur une série principale. Nous énonçons une nouvelle conjecture sur les extensions entre représentations lisses modulo p de G obtenues par induction parabolique à partir de représentations supersingulières de sous-groupes de Levi de G et nous la démontrons pour les extensions par une série principale. Let G be a split p -adic reductive group with connected centre and simply connected derived subgroup. We show that certain “chains” of principal series of G do not exist and we establish several properties of the Breuil–Herzig construction Π⁢(ρ)ord\Pi(\rho)^{\mathrm{ord}}. In particular, we obtain a natural characterization of the latter and we prove a conjecture of Breuil–Herzig. In order to do so, we partially compute Emerton’s δ-functor H∙⁢OrdP\operatorname{H^{\bullet}Ord}_{P} of derived ordinary parts with respect to a parabolic subgroup on a principal series. We formulate a new conjecture on the extensions between smooth mod p representations of G parabolically induced from supersingular representations of Levi subgroups of G and we prove it in the case of extensions by a principal series.

In this paper, we introduce the notion of maximal actions of compact tori on smooth manifolds and study compact connected complex manifolds equipped with maximal actions of compact tori. We give a complete classification of such manifolds, in terms of combinatorial objects, which are triples (Δ,𝔥,G){(\Delta,\mathfrak{h},G)} of nonsingular complete fan Δ in 𝔤{\mathfrak{g}}, complex vector subspace 𝔥{\mathfrak{h}} of 𝔤ℂ{\mathfrak{g}^{\mathbb{C}}} and compact torus G satisfying certain conditions. We also give an equivalence of categories with suitable definitions of morphisms in these families, like toric geometry. We obtain several results as applications of our equivalence of categories; complex structures on moment-angle manifolds, classification of holomorphic nondegenerate ℂn{\mathbb{C}^{n}}-actions on compact connected complex manifolds of complex dimension n , and construction of concrete examples of non-Kähler manifolds.

On a complex manifold, the Riemann–Hilbert correspondence embeds the triangulated category of (not necessarily regular) holonomic 𝒟{\mathcal{D}}-modules into the triangulated category of ℝ{\mathbb{R}}-constructible enhanced ind-sheaves. The source category has a standard t-structure. Here, we provide the target category with a middle perversity t-structure, and prove that the embedding is exact. In the paper, we also discuss general perversities in the framework of ℝ{\mathbb{R}}-constructible enhanced ind-sheaves on bordered subanalytic spaces.

We study an equation proposed by Fu and Yau as a natural n -dimensional generalization of a Strominger system that they solved in dimension 2. It is a complex Hessian equation with right-hand side depending on gradients. Building on the methods of Fu and Yau, we obtain C0{C^{0}}, C2{C^{2}}, and C2,α{C^{2,\alpha}} a priori estimates. We also identify difficulties in extending the Fu–Yau arguments for non-degeneracy from dimension 2 to higher dimensions.

We prove that very general non-rational Fano threefolds which are not birational to cubic threefolds are not stably rational.

Given a closed complex hypersurface Z⊂ℂN+1{Z\subset\mathbb{C}^{N+1}} (N∈ℕ{N\in\mathbb{N}}) and a compact subset K⊂Z{K\subset Z}, we prove the existence of a pseudoconvex Runge domain D in Z such that K⊂D{K\subset D} and there is a complete proper holomorphic embedding from D into the unit ball of ℂN+1{\mathbb{C}^{N+1}}. For N=1{N=1}, we derive the existence of complete properly embedded complex curves in the unit ball of ℂ2{\mathbb{C}^{2}}, with arbitrarily prescribed finite topology. In particular, there exist complete proper holomorphic embeddings of the unit disc 𝔻⊂ℂ{\mathbb{D}\subset\mathbb{C}} into the unit ball of ℂ2{\mathbb{C}^{2}}. These are the first known examples of complete bounded embedded complex hypersurfaces in ℂN+1{\mathbb{C}^{N+1}} with any control on the topology.

We give a simple necessary and sufficient condition for uniform K-stability of ℚ\mathbb{Q}-Fano varieties.