The topology of toric varieties

We construct an Ahlfors-Bers model for the Teichm\"uller space of the universal hyperbolic lamination (also known as Sullivan's Teichm\"uuller space) and the renormalized Weil-Petersson metric on it as an extension of the usual one In this setting, we prove that Sullivan's Teichm\"uller space is K\"ahler isometric biholomophic to the space of continuous functions from the profinte completion of the fundamental group of a compact Riemann surface of genus greater than or equal to two to the Teichm\"uller space of this surface; ie We find natural K\"ahler coordinates for the Sullivan's Teichm\"uller space This is the main result As a corollary we show the expected fact that the Nag-Verjovsky embedding is transversal to the Sullivan's Teichm\"uller space contained in the universal one Finally, we heuristically comment about some progress in the Hong-Rajeev non-perturbative bosonic string theory action

Teichm\"uller theory of the universal hyperbolic lamination

In this paper we survey n-dimensional solenoidal manifolds for n = 1, 2 and 3, and present new results about them. Solenoidal manifolds of dimension n are metric spaces locally modeled on the product of a Cantor set and an open n-dimensional disk. Therefore, they can be “laminated” (or “foliated”) by n-dimensional leaves. By a theorem of A. Clark and S. Hurder, topologically homogeneous, compact solenoidal manifolds are McCord solenoids i.e., are obtained as the inverse limit of an increasing tower of finite, regular covering spaces of a compact manifold with an infinite and residually finite fundamental group. In this case, their structure is very rich since they are principal Cantor-group bundles over a compact manifold and behave like “laminated” versions of compact manifolds, thus they share many of their properties. These objects codify the commensurability properties of manifolds.

Low-dimensional solenoidal manifolds

We study the proalgebraic space which is the inverse limit of all finite branched covers over a normal toric variety $X$ with branching set the invariant divisor under the action of $(\mathbb{C}^*)^n$. This is the proalgebraic toric-completion $X_{\mathbb{Q}}$ of $X$. The ramification over the invariant divisor and the singular invariant divisors of $X$ impose topological constraints on the automorphisms of $X_{\mathbb{Q}}$. Considering this proalgebraic space as the toric functor on the adelic complex plane multiplicative semigroup, we calculate its automorphic group. Moreover we show that its vector bundle category is the direct limit of the respective categories of the finite toric varieties coverings defining the proalgebraic toric-completion.

Proalgebraic toric completion of toric varieties

Over the past two decades the theory of the Weil-Petersson metric has been extended to general Teichmüller spaces of infinite type, including for example the universal Teichmüller space. In this paper we give a survey of the main results in the Weil-Petersson geometry of infinite-dimensional Teichmüller spaces. This includes the rigorous definition of complex Hilbert manifold structures, Kähler geometry and global analysis, and generalizations of the period mapping. We also discuss the motivations of the theory in representation theory and physics beginning in the 1980s. Some examples of the appearance of Weil-Petersson Teichmüller space in other fields such as fluid mechanics and two-dimensional conformal field theory are also provided.

Weil-petersson teichmüller theory of surfaces of infinite conformal type eric schippers and wolfgang staubach

The universal arithmetic one dimensional solenoid $S^1_{\mathbb{Q}}$ is the Pontryagin dual of the additive rationals $\mathbb{Q}$ and it is isomorphic to the ad\`ele class group ${\mathbb{A}}_{\mathbb{Q}}/{\mathbb{Q}}$. It is also isomorphic to the algebraic universal covering on the unit circle $S^1$ obtained by the inverse limit of the tower of its finite coverings. It is the boundary of the surface lamination with boundary obtained as the algebraic universal covering of the punctured closed disk $\overline{\Delta}-\{0\}\subset{\mathbb{C}}$. The interior of this lamination is the inverse limit of the tower of finite coverings of the open punctured disk $\Delta-\{0\}$. The latter is a Riemann surface lamination denoted $\mathbb{H}_\mathbb Q$ and it is foliated by densely embedded copies of the hyperbolic plane $\mathbb H$. The boundary of the leaves are densely embedded copies of $\mathbb{R}$ in $S^1_{\mathbb{Q}}$. In this framework the pair $(S^1_\mathbb{Q},{\mathbb{H}}_{\mathbb{Q}})$ is the adelic version of the pair $(\mathbb{R},\mathbb{H})$. The stage is set to develop the adelic theory of Beltrami differentials, Ahlfors-Bers theory, quasi-symmetric homeomorphisms of $S^1_\mathbb{Q}$ and Teichm\"uller theory. This paper is a first step towards this goal in parallel with the work by Dennis Sullivan on the universal Teichm\"uller spaces of Riemann surface laminations. \end{abstract}

Adelic Ahlfors-Bers theory

We generalize the Ahlfors-Bers theory to the adelic Riemann sphere. In particular, after defining the appropriate notion of a Beltrami differential in the solenoidal context, we give a sufficient condition on it such that the corresponding Beltrami equation has a quasiconformal homeomorphism solution; i.e. The Ahlfors-Bers Theorem in the solenoidal case. This additional condition on the solenoidal Beltrami differentials can be written as a Banach norm in a subspace of solenoidal differentials. Moreover, this subspace is the completion under this norm of those solenoidal differentials locally constant at the fiber. As a toy example, we show how this technique works on a linear problem: We generalize the diophantine equation complex analytic extension problem to the respective solenoidal space.

Adelic solenoid II: Ahlfors-Bers theory.

Topologically the adelic Riemann sphere is the suspension of the adelic solenoid and because of this relation, here we study the adelic solenoid by studying the adelic Riemann sphere topology. The main result is the Birkhoff-Grothendieck Theorem: A holomorphic vector bundle splits as a sum of holomorphic line bundles whose Chern character is now a rational number. As a consequence, the Picard group is isomorphic to the additive group of rational numbers and the $K$--ring has new elements that factor the tautological class.

Juan Manuel Burgos

Papers

Adelic Ahlfors-Bers theory

Adelic solenoid II: Ahlfors-Bers theory.

Adelic solenoid I: Structure and topology

Teichm\"uller theory of the universal hyperbolic lamination

Proalgebraic toric completion of toric varieties