Proalgebraic toric completion of toric varieties J. M. Burgos, A. Verjovsky January 23, 2020 Abstract 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 ∗ n invariant divisor under the action of (C ) . This is the proalgebraic toric- completion XQ of X. The ramification over the invariant divisor and the singular invariant divisors of X impose topological constraints on the automorphisms of XQ. 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. Keywords: Toric variety, Proalgebraic completion, Adelic, Lamination. AMS Subclass: 14M25, 13B35, 11R56, 57R30. To Misha Lyubich on occasion of his 60th birthday 1 Introduction A proalgebraic variety is a projective system of complex algebraic varieties fij : Vj ! Vi i; j 2 I; i ≤ j where (I; ≤) is a directed poset. Each Vi is a complex algebraic variety and the bonding maps fij; i ≤ j are algebraic maps. Proalgebraic varieties play an important role in several branches of mathematics in particular in the theory of arXiv:2001.07997v1 [math.AG] 22 Jan 2020 dynamical systems. For instance, in [Gr1] M. Gromov studies the surjunctivity of algebraic maps f : V ! V (i.e., regular algebraic maps which are injective are necessarily also surjective) and in [Gr2], using the inverse system f f V − V − V ··· Gromov initiated what he calls symbolic algebraic geometry because one can study the properties of f by means of the associated shift map in the inverse system i.e., through symbolic dynamics. Examples of these profinite structures 1 appear constantly in dynamical systems, for instance, the dyadic solenoid of Van-Danzig Vietoris is an invariant set of a diffeomorphism f : S3 ! S3 which is the Smale{Willams expansive attractor of an Axiom A diffeomorphism of the 3-sphere. Riemann surface laminations which are transversally like the Cantor set were introduced into conformal dynamics by Sullivan [Su1]. These Riemann surface laminations play a role in the dynamics similar to the role played by Riemann surfaces in the action of Kleinian groups ([Su2], [Su3]). The type of laminations we consider in this paper are called solenoidal manifolds ([Su4, Ve]). These are Hausdorff topological spaces X equipped with a covering fUigi2I by open sets and coordinate charts φi : Ui ! Di × Ti, where Di is homeomorphic to a n domain in R and Ti is homeomorphic to the Cantor set. The transition maps −1 φi ◦ φj = φij : φj(Ui \ Uj) ! φi(Ui \ Uj) required to be homeomorphisms that take plaques to plaques , i.e., they are of the form: φij(x; t) = (fij(x; t); γij(t)): For a fixed t = t0 one may impose some regularity on the maps φij(· ; t0) = (fij(· ; t); γij(t0)) in their domain of definition which can be identified with a map between two open sets in Rn. These are objects which locally look like a product of open n-disk with a Cantor set. When n = 2 and we identify R2 with C and require that the maps between plaques is holomorphic we obtain a Riemann surface lamination. Sullivan associated a Riemann surface lamination to any C2-smooth expanding map of the circle. M. Lyubich and Yair Minsky [LM] use laminations in dimension 3 called hyperbolic orbifold 3-laminations (a 3-dimensional solenoidal manifold) to study holomorphic dynamics. Both Dennis Sullivan and Misha Lyubich-Yair Minsky obtained remarkable results in conformal dynamics using these solenoidal manifolds. If XF is a normal complex toric projective variety of complex dimension n determined by the fan F then, by definition, there is a holomorphic action of the algebraic torus (C∗)n = C∗ ×· · ·×C∗ (see Definition 2 below) with a Zariski dense principal orbit U ⊂ XF . The complement D := XF −U is a divisor which is invariant under the action. The profinite completion XF of X is the inverse Q F limit of the tower of finite coverings branched over D. Normality implies that the action of (C∗)n lifts to any such finite branched covering and therefore these coverings are also normal toric complex varieties [AP]. The actions on the covers determine an action of the proalgebraic (or adelic) torus (see Section 3) ( ∗ )n = ∗ × · · · × ∗ on XF , where ∗ is the profinite CQ CQ CQ Q CQ completion of C∗ (see Section 2). Thus the proalgebraic toric varieties are equivariant compactifications of the \solenoidal" abelian group ( ∗ )n. CQ In this paper we study these toric proalgebraic varieties from several per- spectives, in particular we show that that the vector bundle categories of these proalgebraic varieties are the direct limit of their respective categories of their finite toric varieties coverings defining their completions. We also study their automorphism groups. 2 2 Adelic solenoid For every n; m 2 Z+ such that n divides m, then there exists a covering map 1 1 m=n pn;m : S ! S such that pn;m(z) = z . This determines a projective system 1 of covering spaces fS ; pn;mgn;m≥1;njm whose projective limit is the universal one{dimensional solenoid or adelic solenoid: S1 := lim S1: Q − pn 1 Thus, as a set, S consists of sequences (zn) 1 such that pn;m(zm) = zn Q n2N; z2S if n divides m. The canonical projections of the inverse limit are the functions S1 !πn S1 Q defined by πn (zj) = zn and they define the solenoid topology as the initial j2N topology of the family. The solenoid is an abelian topological group and each πn is an epimorphism. In particular each πn is a character which determines a locally trivial Z^{bundle structure where the group ^ := lim =m Z − Z Z pn is the profinite completion of Z, which is a compact, perfect and totally discon- nected abelian topological group homeomorphic to the Cantor set. Being Z^ the profinite completion of Z, it admits a canonical inclusion of Z ⊂ Z^ whose image φ is dense. We have an inclusion ^ ! S1 and a short exact sequence: Z Q φ 0 ! ^ ! S1 !π1 S1 ! 1 Z Q The solenoid S1 can also be realized as the orbit space of the {bundle Q Q structure Q ,! A ! A=Q, where A is the ad`elegroup of the rational numbers which is a locally compact Abelian group, Q is a discrete subgroup of A and = =∼ S1 is a compact Abelian group (see [RV]). From this perspective, = A Q Q A Q can be seen as a projective limit whose n{th component corresponds to the unique covering of degree n ≥ 1 of S1 . Q By considering the properly discontinuously diagonal free action of Z on Z^ × R given by n · (x; t) = (x + n; t − n); (n 2 Z; x 2 Z^; t 2 R) the solenoid S1 is identified with the orbit space ^ × . Here, is acting on Q Z Z R Z R by covering transformations and on Z^ by translations. The path{connected component of the identity element 1 2 S1 will be called the baseleaf [Od] and Q it is a densely immersed copy of R. 3 Summarizing the above discussion we have the commutative diagram: pm;n S1 = lim S1 / :::S1 / S1 / :::S1 (1) Q − O O O O φ l 7!e2πil=n l 7!e2πil=m 0 7!1 ? ? pm;n ? ? ^ = lim =n ::: =n =m ::: f0g Z − Z Z / Z Z / Z Z / where Z^ is the adelic profinite completion of the integers and the image of the group monomorphism φ :(^; +) ! (S1 ; ·) is the principal fiber. We notice that Z Q −1 −1 ^ πn(x) = πn(y) implies πn(y x) = 1 and therefore y x = φ(a) where a 2 nZ for some n 2 Z ⊂ Z^. We define the baseleaf as the image of the monomorphism ν : ! S1 R Q defined as follows: pm;n S1 = lim S1 / :::S1 / S1 / :::S1 (2) Q − O O O O ν t 7!eit=n t 7!eit=m t 7!eit ? = = = R; / ::: R / R / ::: R In particular, the immersion ν is a group morphism and comparing the diagrams (1) and (2), we have ν(2πn) = φ(n) for every integer n. Define: exp : ^ × ! S1 Z R Q such that exp(a; θ) = φ(a).ν(θ). The previous construction can be extended to the inverse limit of the multi- plicative group C∗ as well as to the multiplicative semigroup C. In other words, n if zn : C ! C denotes the map z 7! z we can take the inverse limits under the partial order of divisibility of the integers of the finite coverings of C∗ and the branched coverings of C: def def Definition 1 lim ∗ = ∗ and lim = , respectively. − C CQ − C CQ In the first case we obtain the topological abelian group: the algebraic solenoidal group ∗ . This group is laminated by densely immersed copies of CQ C. In the second case, we obtain the semigroup CQ which is a ramified covering of C at zero and topologically it is homeomorphic to the open cone over the adelic solenoid. The singularity of a cone over a solenoidal torus S1 × :::S1 Q Q will be called a solenoidal cusp or simply a cusp. The following is the complex version of the diagram 1: pm;n ∗ = lim ∗ / ::: ∗ / ∗ / ::: ∗ (3) CQ − C C C C O O O O φ l 7!e2πil=n l 7!e2πil=m 0 7!1 ? ? pm;n ? ? ^ = lim =n ::: =n =m ::: f0g Z − Z Z / Z Z / Z Z / 4 3 Algebraic abelian pro-Lie groups and proalge- braic normal toric varieties.