Canonical Embedding Research Articles

Computing fundamental domains for the Bruhat–Tits tree for , -adic automorphic forms, and the canonical embedding of Shimura curves

AbstractWe describe algorithms that allow the computation of fundamental domains in the Bruhat–Tits tree for the action of discrete groups arising from quaternion algebras. These algorithms are used to compute spaces of rigid modular forms of arbitrary even weight, and we explain how to evaluate such forms to high precision using overconvergent methods. Finally, these algorithms are applied to the calculation of conjectural equations for the canonical embedding of p-adically uniformizable rational Shimura curves. We conclude with an example in the case of a genus 4 Shimura curve.

Growth rate of canonical and minimal group embeddings of spherical latin trades

A spherical latin trade is a partial latin square associated with a face 2-colourable triangulation of the sphere. A latin trade W embeds into an abelian group G if the Cayley table of G contains a partial latin square isotopic to W. A trade W embeds minimally into G if W embeds into G but not into any proper subgroup of G. The canonical groupAW⁎ of a spherical latin trade W is a finite abelian group that W embeds into, and possesses a “universal property”: every abelian group that W embeds into minimally is isomorphic to a quotient group of AW⁎.We study simple graph operations that create larger spherical latin trades from smaller ones. In particular, we analyse the effect of these operations on the canonical groups of the trades. We then apply our results to answer several open problems about canonical and minimal groups recently posed by Cavenagh and Wanless and by Blackburn and McCourt.A family of trades is constructed that exhibits the fastest known asymptotic growth for |AW⁎| with respect to the size |W|. It achieves |AW⁎|=r|W|(1+o(1)), where r≈1.329. Another family is shown to embed minimally into groups of exponentially increasing order; previously exponential growth had only been observed for canonical groups.Questions about the rank (size of a smallest generating set) of canonical and minimal groups are also resolved. The rank of any minimal group is shown to be at most logarithmic in the size of the trade, resulting in an example where the canonical group grows much faster than any minimal group, with regards to both order and rank.

Canonical embeddings of [formula omitted] into orientable n-dimensional closed PL manifolds for [formula omitted

For each embedded oriented circle c in an n-dimensional closed PL manifold M, we define a canonical embedding with respect to a triangulation of M, of S1×Δn−1 into a regular neighborhood of the embedded circle in M. For n>4, we give a necessary and sufficient condition for an embedding of a surface with boundary in M, such that the embedding together with the canonical embeddings on the boundary components of the surface, extends to an embedding of the regular neighborhood of the surface in Rn to a regular neighborhood of the embedded surface in M.

Powerset residuated algebras

We present an algebraic approach to canonical embeddings of arbitrary residuated algebras into powerset residuated algebras. We propose some construction of powerset residuated algebras and prove a representation theorem for symmetric residuated algebras.

The Operator Space Projective Tensor Product: Embedding into the Second Dual and Ideal Structure

AbstractWe prove that, for operator spaces V and W, the operator space V** ⊗hW** can be completely isometrically embedded into (V ⊗hW)**, ⊗h being the Haagerup tensor product. We also show that, for exact operator spaces V and W, a jointly completely bounded bilinear form on V × W can be extended uniquely to a separately w*-continuous jointly completely bounded bilinear form on V× W**. This paves the way to obtaining a canonical embedding of into with a continuous inverse, where is the operator space projective tensor product. Further, for C*-algebras A and B, we study the (closed) ideal structure of which, in particular, determines the lattice of closed ideals of completely.

On the finite presentation of subdirect products and the nature of residually free groups

We establish {\it virtual surjection to pairs} (VSP) as a general criterion for the finite presentability of subdirect products of groups: if $\Gamma_1,\ldots,\Gamma_n$ are finitely presented and $S<\Gamma_1\times\cdots\times\Gamma_n$ projects to a subgroup of finite index in each $\Gamma_i\times\Gamma_j$, then $S$ is finitely presentable, indeed there is an algorithm that will construct a finite presentation for $S$. We use the VSP criterion to characterize the finitely presented residually free groups. We prove that the class of such groups is recursively enumerable. We describe an algorithm that, given a finite presentation of a residually free group, constructs a canonical embedding into a direct product of finitely many limit groups. We solve the (multiple) conjugacy problem and membership problem for finitely presentable subgroups of residually free groups. We also prove that there is an algorithm that, given a finite generating set for such a subgroup, will construct a finite presentation. New families of subdirect products of free groups are constructed, including the first examples of finitely presented subgroups that are neither ${\rm FP}_\infty$ nor of Stallings-Bieri type.

A Curve and its Abstract Jacobian

Let C(K) be the K-points of a smooth projective curve C of genus g > 1 and J(K) its Jacobian. Fixing a point on the curve one has a canonical embedding of C(K) into J(K) with the point identified as 0 of the group. Consider 〈J(K); +, C(K)〉 as an abstract structure of a group with a distiguished subset. Using model theory we prove that, for K algebraically closed, one can recover from this data the field K and the curve C, up to isomorphisms of fields and a bijective isogeny of J. In characteristic 0 the bijective isogeny is just a regular isomorphism, and in positive characteristic such an isogeny can be seen as a “Frobenius twisting”. In such an interpretation our theorem, in particular, proves a conjecture posed by F.Bogomolov, M.Korotaev and Yu.Tschinkel.

Segre embeddings and the canonical image of a curve

We prove that there is no g for which the canonical embedding of a general curve of genus g lies on the Segre embedding of any product of three or more projective spaces.

Measure-theoretic rigidity for Mumford curves

AbstractOne can describe isomorphism of two compact hyperbolic Riemann surfaces of the same genus by a measure-theoretic property: a chosen isomorphism of their fundamental groups corresponds to a homeomorphism on the boundary of the Poincaré disc that is absolutely continuous for Lebesgue measure if and only if the surfaces are isomorphic. In this paper, we find the corresponding statement for Mumford curves, a non-Archimedean analogue of Riemann surfaces. In this case, the mere absolute continuity of the boundary map (for Schottky uniformization and the corresponding Patterson–Sullivan measure) only implies isomorphism of the special fibers of the Mumford curves, and the absolute continuity needs to be enhanced by a finite list of conditions on the harmonic measures on the boundary (certain non-Archimedean distributions constructed by Schneider and Teitelbaum) to guarantee an isomorphism of the Mumford curves. The proof combines a generalization of a rigidity theorem for trees due to Coornaert, the existence of a boundary map by a method of Floyd, with a classical theorem of Babbage, Enriques and Petri on equations for the canonical embedding of a curve.

Nerve complexes and moment-angle spaces of convex polytopes

We introduce spherical nerve complexes that are a far-reaching generalization of simplicial spheres, and consider the differential ring of simplicial complexes. We show that spherical nerve complexes form a subring of this ring, and define a homomorphism from the ring of polytopes to this subring that maps each polytope P to the nerve KP of the cover of the boundary ∂P by facets. We develop a theory of nerve complexes and apply it to the moment-angle spaces ZP of convex polytopes P. In the case of a polytope P with m facets, its moment-angle space ZP is defined by the canonical embedding in the cone ℝ≥m. It is well-known that the space ZP is homeomorphic to the polyhedral product (D2, S1)∂P* if the polytope P is simple. We show that the homotopy equivalence \(\mathcal{Z}_P \simeq (D^2 ,S^1 )^{K_P }\) holds in the general case. On the basis of bigraded Betti numbers of simplicial complexes, we construct a new class of combinatorial invariants of convex polytopes. These invariants take values in the ring of polynomials in two variables and are multiplicative with respect to the direct product or join of polytopes. We describe the relation between these invariants and the well-known f-polynomials of polytopes. We also present examples of convex polytopes whose flag numbers (in particular, f-polynomials) coincide, while the new invariants are different.

More on upper bicompletion-true functorial quasi-uniformities

Let T : QU 0 → Top 0 denote the usual forgetful functor from the category of quasi-uniform T 0 -spaces to that of the topological T 0 -spaces. We regard the bicompletion reflector as a (pointed) endofunctor K : QU 0 → QU 0 . For any section F : Top 0 → QU 0 of T we consider the (pointed) endofunctor R = T K F : Top 0 → Top 0 . The T-section F is called upper bicompletion-true (briefly, upper K-true) if the quasi-uniform space KFX is finer than FRX for every X in Top 0 . An important known characterisation is that F is upper K-true iff the canonical embedding X → R X is an epimorphism in Top 0 for every X in Top 0 . We show that this result admits a simple, purely categorical formulation and proof, independent of the setting of quasi-uniform and topological spaces. We thus mention a few other settings where the result is applicable. Returning then to the setting T : QU 0 → Top 0 , we prove: Any T-section F is upper K-true iff for all X the bitopology of KFX equals that of FRX; and iff the join topology of KFX equals the strong topology (also called the b- or Skula topology) of RX.

The spectrum of the twisted Dirac operator on Kähler submanifolds of the complex projective space

We establish an upper estimate for the small eigenvalues of the twisted Dirac operator on Kahler submanifolds in Kahler manifolds carrying Kahlerian Killing spinors. We then compute the spectrum of the twisted Dirac operator of the canonical embedding $${{\mathbb C}P^d\rightarrow {\mathbb C}P^n}$$ in order to test the sharpness of the upper bounds.

Canonical embeddings of compact metric spaces

We prove that for any infinite compact metric space, two canonical embeddings (the Hausdorff–Kuratowski and Kantorovich–Rubinshtein embeddings) do not coincide. Bibliography: 3 titles.

Green’s conjecture for curves on arbitrary K3 surfaces

AbstractGreen’s conjecture predicts than one can read off special linear series on an algebraic curve, by looking at the syzygies of its canonical embedding. We extend Voisin’s results on syzygies of K3 sections, to the case of K3 surfaces with arbitrary Picard lattice. This, coupled with results of Voisin and Hirschowitz–Ramanan, provides a complete solution to Green’s conjecture for smooth curves on arbitrary K3 surfaces.

A classification of finite rank dimension groups by their representations in ordered real vector spaces

This paper systematically studies finite rank dimension groups, as well as finite-dimensional ordered real vector spaces with Riesz interpolation. We provide an explicit description and classification of finite rank dimension groups, in the following sense. We show that for each n, there are (up to isomorphism) finitely many ordered real vector spaces of dimension n that have Riesz interpolation, and we give an explicit model for each of them in terms of combinatorial data. We show that every finite rank dimension group can be realized as a subgroup of a finite-dimensional ordered real vector space with Riesz interpolation via a canonical embedding. We then characterize which of the subgroups of a finite-dimensional ordered real vector space have Riesz interpolation (and are therefore dimension groups).

The fundamental group of a locally finite graph with ends

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the free groups π 1 ( G ′ ) with G ′ ⊆ G finite.

Generalised Moore spectra in a triangulated category

In this paper we consider a construction in an arbitrary triangulated category $${\fancyscript {T}}$$ which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of $${\fancyscript {T}}$$ satisfying some finite tilting assumptions, we obtain a functor which “approximates” objects from the module category of the endomorphism algebra of C in $${\fancyscript {T}}$$ . This provides a higher analogue of a construction of Jørgensen which appears in (Manuscr Math 110:381–406, 2003) in connection with lifts of certain homological functors of derived categories. We show that this new functor is well-behaved with respect to short exact sequences and distinguished triangles, and as a consequence we obtain a new way of embedding a module category in a triangulated category. As an example of the theory, we recover Keller’s canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category of $${u \, \geqslant \, 2}$$ .

The canonical embedding of an unramified morphism in an étale morphism

We show that every unramified morphism X->Y has a canonical and universal factorization X->E->Y where the first morphism is a closed embedding and the second is etale (but not separated).

The quadrics through the Hurwitz curves of genus 14

A basis for the space of quadratic polynomials in the ideal of the canonical embedding of one of the three Hurwitz curves of genus 14 is found. Our method enables us to recover equations for other curves such as the Macbeath curve of genus 7 and the A-embedding of the Klein quartic in ℙ3.

On holomorphic isometric embeddings of the unit n-ball into products of two unit m-balls

We study holomorphic isometric embeddings of the complex unit n-ball into products of two complex unit m-balls with respect to their Bergman metrics up to normalization constants (the isometric constant). There are two trivial holomorphic isometric embeddings for m ≥ n, given by F1(z) = (0, In;m(z)) with the isometric constant equal to (m + 1)/(n + 1) and F2(z) = (In;m(z), In;m(z)) with the isometric constant equal to 2(m + 1)/(n + 1). Here \({I_{n;m}:\mathbb{C}^n \longrightarrow \mathbb{C}^m}\) is the canonical embedding. We prove that when m < 2n, these are the only holomorphic isometric embeddings up to unitary transformations.