Volume 2019, Issue 747

In this paper we consider the limit set in Thurston’s compactification 𝒫⁢ℳ⁢ℱ{\mathcal{P}\kern-2.27622pt\mathcal{M}\kern-0.284528pt\mathcal{F}} of Teichmüller space of some Teichmüller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that (a) there are quadratic differentials so that the limit set of the geodesic is a unique point, (b) there are quadratic differentials so that the limit set is a line segment, (c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and (d) there are quadratic differentials so that the vertical foliation is ergodic and there is a unique point as its limit set. These give examples of divergent Teichmüller geodesics whose limit sets overlap and Teichmüller geodesics that stay a bounded distance apart but whose limit sets are not equal. A byproduct of our methods is a construction of a Teichmüller geodesic and a simple closed curve γ so that the hyperbolic length of the geodesic in the homotopy class of γ varies between increasing and decreasing on an unbounded sequence of time intervals along the geodesic.

Let k{{k}} be an algebraically closed field of characteristic 0, and let f:X→Y{f:X\to Y} be a morphism of smooth projective varieties over the ring k⁢((t)){k((t))} of formal Laurent series. We prove that if a general geometric fiber of f is rationally connected, then the induced map fan:Xan→Yan{f^{\mathrm{an}}:X^{\mathrm{an}}\to Y^{\mathrm{an}}} between the Berkovich analytifications of X and Y is a homotopy equivalence. Two important consequences of this result are that the Berkovich analytification of any ℙn{\mathbb{P}^{n}}-bundle over a smooth projective k⁢((t)){k((t))}-variety is homotopy equivalent to the Berkovich analytification of the base, and that the Berkovich analytification of a rationally connected smooth projective variety over k⁢((t)){k((t))} is contractible.

Let k be an algebraic closure of a finite field of odd characteristic. We prove that for any rank two graded Higgs bundle with maximal Higgs field over a generic hyperbolic curve X1{X_{1}} defined over k , there exists a lifting X of the curve to the ring W⁢(k){W(k)} of Witt vectors as well as a lifting of the Higgs bundle to a periodic Higgs bundle over X/W⁢(k){X/W(k)}. These liftings give rise to a two-dimensional absolutely irreducible representation of the arithmetic fundamental group π1⁢(XK){\pi_{1}(X_{K})} of the generic fiber of X . This curve X and its associated representation is in close relation to the canonical curve and its associated canonical crystalline representation in the p -adic Teichmüller theory for curves due to S. Mochizuki. Our result may be viewed as an analogue of the Hitchin–Simpson’s uniformization theory of hyperbolic Riemann surfaces via Higgs bundles.

We investigate to what extent the theory of Néron models of jacobians and of abel–jacobi maps extends to relative curves over base schemes of dimension greater than 1. We give a necessary and sufficient criterion for the existence of a Néron model. We use this to show that, in general, Néron models do not exist even after making a modification or even alteration of the base. On the other hand, we show that Néron models do exist outside some codimension-2 locus.

Let E⊆ℙ2{E\subseteq\mathbb{P}^{2}} be a complex rational cuspidal curve and let (X,D)→(ℙ2,E){(X,D)\to(\mathbb{P}^{2},E)} be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E . This is done by analyzing the Minimal Model Program run for the pair (X,12⁢D){(X,\frac{1}{2}D)}. Namely, we show that ℙ2∖E{\mathbb{P}^{2}\setminus E} is ℂ**{\mathbb{C}^{**}}-fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

In this paper we show that the moduli stacks of Bridgeland semistable objects on smooth projective 3-folds are quasi-proper algebraic stacks of finite type if they satisfy the Bogomolov–Gieseker (BG for short) inequality conjecture proposed by Bayer, Macrì and the second author. The key ingredients are the equivalent form of the BG inequality conjecture and its generalization to arbitrary very weak stability conditions. This result is applied to define Donaldson–Thomas invariants counting Bridgeland semistable objects on smooth projective Calabi–Yau 3-folds satisfying the BG inequality conjecture, for example on étale quotients of abelian 3-folds.

We introduce a sheaf of infinite order differential operators 𝒟⌢{\overset{\frown}{\mathcal{D}}} on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle. We show that the sections of this sheaf over sufficiently small affinoid varieties are Fréchet–Stein algebras, and use this to define co-admissible sheaves of 𝒟⌢{\overset{\frown}{\mathcal{D}}}-modules. We prove analogues of Cartan’s Theorems A and B for co-admissible 𝒟⌢{\overset{\frown}{\mathcal{D}}}-modules.

We prove that for any infinite countable amenable group G , any ε>0{\varepsilon>0} and any finite subset K⊂G{K\subset G}, there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are (K,ε){(K,\varepsilon)}-invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

Let 𝔤{{\mathfrak{g}}} be a symmetrizable Kac–Moody algebra and let Uq⁢(𝔤){{U_{q}(\mathfrak{g})}} denote the corresponding quantized enveloping algebra. In the present paper we show that quantum symmetric pair coideal subalgebras B𝐜,𝐬{{B_{\mathbf{c},\mathbf{s}}}} of Uq⁢(𝔤){{U_{q}(\mathfrak{g})}} have a universal K-matrix if 𝔤{{\mathfrak{g}}} is of finite type. By a universal K-matrix for B𝐜,𝐬{{B_{\mathbf{c},\mathbf{s}}}} we mean an element in a completion of Uq⁢(𝔤){{U_{q}(\mathfrak{g})}} which commutes with B𝐜,𝐬{{B_{\mathbf{c},\mathbf{s}}}} and provides solutions of the reflection equation in all integrable Uq⁢(𝔤){{U_{q}(\mathfrak{g})}}-modules in category 𝒪{{\mathcal{O}}}. The construction of the universal K-matrix for B𝐜,𝐬{{B_{\mathbf{c},\mathbf{s}}}} bears significant resemblance to the construction of the universal R-matrix for Uq⁢(𝔤){{U_{q}(\mathfrak{g})}}. Most steps in the construction of the universal K-matrix are performed in the general Kac–Moody setting. In the late nineties T. tom Dieck and R. Häring-Oldenburg developed a program of representations of categories of ribbons in a cylinder. Our results show that quantum symmetric pairs provide a large class of examples for this program.