Singularities of ball quotients

We prove a result on the singularities of ball quotients ${\Gamma\backslash\mathbb{C}{H^n}}$ by an arithmetic group. More precisely, we show that a ball quotient has at most canonical singularities under certain restrictions on the dimension n and the underlying lattice. We also extend this result to the toroidal compactification.     

Compactification of symmetric varieties

The symmetric varieties considered in this paper are the quotientsG/H, whereG is an adjoint semi-simple group over a fieldk of characteristic 2, andH is the fixed point group of an involutorial automorphism ofG which is defined overk. In the casek=C, De Concini and Procesi (1983) constructed a wonderful compactification ofG/H. We prove the existence of such a compactification for arbitraryk. We also prove cohomology vanishing results for line bundles on the compactification. Dedicated to the memory of C. Chevalley     

The extension of the operation in the Stone-Čech compactification of semigroups

For a large class of locally compact semitopological semigroups S, the Stone-Čech compactification β S is a semigroup compactification if and only if S is either discrete or countably compact. Furthermore, for this class of semigroups which are neither discrete nor countably compact, the quotient contains a linear isometric copy of ℓ _∞. These results improve theorems by Baker and Butcher and by Dzinotyiweyi.     

A Note on the WAP–Compactification and the LUC–Compactification of a Topological Group

Let G be a topological group. Denote by G^LUC and G^WAP the LUC–compactification and the WAP–compactification of G, respectively. G^WAP can be regarded as a quotient of G^LUC and the quotient map denoted by . In this note we shall show that, when $G$ is a SIN--group, there exists a dense open subset of G^LUC\ G$, consisting of points of unicity for , of cardinality at least 2^{2 (G)}, where (G) denotes the compact covering number of G. We give an example to show that this statement does not hold for IN–groups, although G^LUC\ G does contain at least 2^{2 (G)} points if G is an IN–group. We also give characterisations of the completion \tilde{G} of G as a subspace of the uniform compactification uG. A consequence of the first result is an analogue of Veechs theorem for the WAP–compactification of a SIN-group.     

A Constructive Proof of the Gelfand—Kolmogorov Theorem

A construction of the Stone—ech compactification of a locale L is presented in this paper as a quotient of the frame of radical ideals of the algebra C ^*(L). As a corollary, a constructive, localic version of the Gelfand—Kolmogorov theorem is obtained.     

Real Quotient Singularities and Nonsingular Real Algebraic Curves in the Boundary of the Moduli Space

The quotient of a real analytic manifold by a properly discontinuous group action is, in general, only a semianalytic variety. We study the boundary of such a quotient, i.e., the set of points at which the quotient is not analytic. We apply the results to the moduli space M_g/ of nonsingular real algebraic curves of genus g (g2). This moduli space has a natural structure of a semianalytic variety. We determine the dimension of the boundary of any connected component of M_g/. It turns out that every connected component has a nonempty boundary. In particular, no connected component of M_g/ is real analytic. We conclude that M_g/ is not a real analytic variety.     

The moduli space of complete embedded constant mean curvature surfaces

We examine the space of finite topology surfaces in ³ which are complete, properly embedded and have nonzero constant mean curvature. These surfaces are noncompact provided we exclude the case of the round sphere. We prove that the spaceM _k of all such surfaces withk ends (where surfaces are identified if they differ by an isometry of ³) is locally a real analytic variety. When the linearization of the quasilinear elliptic equation specifying mean curvature equal to one has noL ²-nullspace, we prove thatM _k is locally the quotient of a real analytic manifold of dimension 3k–6 by a finite group (i.e. a real analytic orbifold), fork 3. This finite group is the isotropy subgroup of the surface in the group of Euclidean motions. It is of interest to note that the dimension ofM _k is independent of the genus of the underlying punctured Riemann surface to which is conformally equivalent. These results also apply to hypersurfaces of H ⁿ⁺¹ with nonzero constant mean curvature greater than that of a horosphere and whose ends are cylindrically bounded.Research of the first author supported in part by NSF grant # DMS9404278 and an NSF Postdoctoral Fellowship, of the second auther by NSF Young Investigator Award, a Sloan Foundation Postdoctoral Fellowship and NSF grant # DMS9303236, and of the third author by NSF grant # DMS9022140 and an NSF Postdoctoral Fellowship.     

Countable Direct Sums of Banach Spaces

Let (X_n) be a sequence of infinite-dimensional BANACH spaces. We prove that has a non-locally complete quotient if X₁ is not quasi-reflexive.     

The Commuting Graph of Minimal Nonsolvable Groups

The purpose of this paper is to prove that if G is a finite minimal nonsolvable group (i.e. G is not solvable but every proper quotient of G is solvable), then the commuting graph of G has diameter 3. We give an example showing that this result is the best possible. This result is related to the structure of finite quotients of the multiplicative group of a finite-dimensional division algebra.     

Heegner divisors in the moduli space of genus three curves

S. Kondo used periods of surfaces to prove that the moduli space of genus three curves is birational to an arithmetic quotient of a complex 6-ball. In this paper we study Heegner divisors in the ball quotient, given by arithmetically defined hyperplane sections of the ball. We show that the corresponding loci of genus three curves are given by hyperelliptic curves, singular plane quartics and plane quartics admitting certain rational ``splitting curves'.

     

COTANGENT BUNDLE TO THE GRASSMANN VARIETY

We show that there is an affine Schubert variety in the infinite-dimensional partial Flag variety (associated to the two-step parabolic subgroup of the Kac–Moody group Open image in new window , corresponding to omitting α ₀; α _d ) which is a natural compactification of the cotangent bundle to the Grassmann variety.     

On the Epireflectivity of the Wallman–Shanin-Type Compactification for Approach Spaces

In this paper, we study the functorial behaviour of the Wallman–Shanin-type compactification for weakly symmetric T ₁ approach spaces, as defined in (R. Lowen and M. Sioen [13]). We do this by generalizing the technique used by H. L. Bentley and S. A. Naimpally in their paper [2], yielding a recharacterization of our quantified compactification theory in terms of contiguity clusters of [0, ]-valued functionals and we also define a construct WS (which is the counterpart of SEP in [2]) on a suitable non-full subconstruct of which the Wallman–Shanin-type compactification determines an epireflection.     

The uniform structure of Banach spaces

We explore the existence of uniformly continuous sections for quotient maps. Using this approach we are able to give a number of new examples in the theory of the uniform structure of Banach spaces. We show for example that there are two non-isomorphic separable ${\mathcal L_1}$ -subspaces of ? ₁ which are uniformly homeomorphic. We also prove the existence of two coarsely homeomorphic Banach spaces (i.e. with Lipschitz isomorphic nets) which are not uniformly homeomorphic (answering a question of Johnson, Lindenstrauss and Schechtman). We construct a closed subspace of L ₁ whose unit ball is not an absolute uniform retract (answering a question of the author).     

Differential Forms and Smoothness of Quotients by Reductive Groups

Let :XY be a good quotient of a smooth variety X by a reductive algebraic group G and 1k dim (Y) an integer. We prove that if, locally, any invariant horizontal differential k-form on X (resp. any regular differential k-form on Y) is a Kähler differential form on Y then codim(Y _sing)>k+1. We also prove that the dualizing sheaf on Y is the sheaf of invariant horizontal dim(Y)-forms.     

Non-completeness of the Arakelov-induced metric on moduli space of curves

Let X be a compact Riemann surface of genus g>1. We study two different, naturally defined metric forms on X: The hyperbolic metric form μ_hyp,X, arising from hyperbolic geometry, and the Arakelov metric form μ_Ar,X, arising from arithmetic algebraic geometry. Now consider a sequence X_t of Riemann surfaces approaching the Deligne-Mumford boundary of the moduli space of compact Riemann surfaces of genus g. We prove here that As a corollary of this result, we prove that the Weil-Petersson metric on the moduli space induced from the Arakelov metric is not complete, i.e., certain boundary components of the Deligne-Mumford compactification are at finite distance. The first author acknowledges support from grants from the NSF and PSC-CUNY. The second author thanks the Centre de Recerca Matemàtica (CRM) in Barcelona for its support and hospitality.     

Gibbs Measures on Mutually Interacting Brownian Paths under Singularities

We are interested in the analysis of Gibbs measures defined on two independent Brownian paths in ?^d interacting through a mutual self‐attraction. This is expressed by the Hamiltonian with two probability measures μ and ν representing the occupation measures of two independent Brownian motions. We will be interested in a class of potentials V that are singular , e.g., Dirac‐ or Coulomb‐type interactions in ?³, or the correlation function of the parabolic Anderson problem with white noise potential. The mutual interaction of the Brownian paths inspires a compactification of the quotient space of orbits of product measures, which is structurally different from the self‐interacting case introduced in [27], owing to the lack of shift‐invariant structure in the mutual interaction. We prove a strong large‐deviation principle for the product measures of two Brownian occupation measures in such a compactification and derive asymptotic path behavior under Gibbs measures on Wiener paths arising from mutually attracting singular interactions. For the spatially smoothened parabolic Anderson model with white noise potential, our analysis allows a direct computation of the annealed Lyapunov exponents, and a strict ordering of them implies the intermittency effect present in the smoothened model. © 2017 Wiley Periodicals, Inc.     

Slices in the unit ball of the symmetric tensor product of $\mathcal{C}(K)$ and L1(μ)

We prove that for the cases (K infinite) and X=L ₁(μ) (μ σ-finite and atomless) it holds that every slice of the unit ball of the N-fold symmetric tensor product of X has diameter two. In fact, we prove more general results for weak neighborhoods relative to the unit ball. As a consequence, we deduce that the spaces of N-homogeneous polynomials on those classical Banach spaces have no points of Fréchet differentiability. Dedicated to Angel Rodríguez Palacios on the occasion of his 60th birthday.     

Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups

In this paper we study the holomorphic K-theory of a projective variety. This K-theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory is built out of studying algebraic bundles over a variety up to algebraic equivalence. In this paper we will give calculations of this theory for flag like varieties which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors we will show that there is a rational isomorphism of graded rings between holomorphic K-theory and the appropriate morphic cohomology groups, in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the symmetrized loop group U(n)/ _n where the symmetric group _n acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K-theory by inverting the Bott class, then rationally this is isomorphic to topological K-theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K-theory, and show that these obstructions vanish for generalized flag mani-folds.