The Quarterly Journal of Mathematics - recent issues

MULTIPLICATIVE CHARACTER SUMS WITH TWICE-DIFFERENTIABLE FUNCTIONS

Banks, W. D., Shparlinski, I. E. — Mon, 09 Nov 2009 07:23:04 PST

For a nontrivial multiplicative character modulo p, we bound character sums taken on the integer parts of a real-valued, twice-differentiable function f whose second derivative decays at an appropriate rate. For the special case that f(x) = x with some positive real number , our bounds extend recent results of several authors.

SUMS OF OPERATOR LOGARITHMS

Clark, S. — Mon, 09 Nov 2009 07:23:04 PST

Let A and B be a pair of resolvent commuting invertible sectorial operators. We shall show that, under Kalton–Weis-type conditions, the operator log A + log B is closed and equal to log(AB).

CONTACT 5-MANIFOLDS WITH SU(2)-STRUCTURE

de Andres, L. C., Fernandez, M., Fino, A., Ugarte, L. — Mon, 09 Nov 2009 07:23:04 PST

We consider 5-manifolds with a contact form arising from a hypo structure, which we call hypo-contact. We provide existence conditions for such a structure on an oriented hypersurface of a 6-manifold with a half-flat SU(3)-structure. For half-flat manifolds with a Killing vector field X preserving the SU(3)-structure we study the geometry of the orbits space. Moreover, we describe the solvable Lie algebras admitting a hypo-contact structure. This allows us to exhibit examples of Sasakian -Einstein manifolds, as well as to prove that such structures give rise to new metrics with holonomy SU(3) and G₂.

EXAMPLES OF FREE ACTIONS ON PRODUCTS OF SPHERES

Hambleton, I., Unlu, O. — Mon, 09 Nov 2009 07:23:04 PST

We construct a non-abelian extension of S¹ by Z/3 x Z/3, and prove that acts freely and smoothly on S⁵ x S⁵. This gives new actions on S⁵ x S⁵ for an infinite family P of finite 3-groups. We also show that any finite odd-order subgroup of the exceptional Lie group G₂ admits a free smooth action on S¹¹ x S¹¹. This gives new actions on S¹¹ x S¹¹ for an infinite family E of finite groups. We explain the significance of these families P, E for the general existence problem, and correct some mistakes in the literature.

HYPERBOLIC SECTIONS IN SEIFERT-FIBERED SURFACE BUNDLES

Ichihara, K., Motegi, K. — Mon, 09 Nov 2009 07:23:04 PST

Let M be a small Seifert fiber space which has also a structure of surface bundle F x [0, 1]/{(x, 0) = (f(x), 1)} over the circle, where f: F -> F is a monodromy map with non-empty fixed point set. A typical example of such a manifold appears as the result of 0-surgery on a torus knot. For each section in M, we have a ‘projection’ in F in a natural way. We give a condition assuring that the given section in M is hyperbolic in terms of the ‘projection’ in the fiber surface. By translating the result, we give a condition to obtain pseudo-Anosov automorphisms of once punctured surfaces from a periodic automorphism.

FLOWS OF G2-STRUCTURES, I

Karigiannis, S. — Mon, 09 Nov 2009 07:23:04 PST

This is a foundational paper on flows of G₂-structures. We use local coordinates to describe the four torsion forms of a G₂-structure and derive the evolution equations for a general flow of a G₂-structure on a 7-manifold M. Specifically, we compute the evolution of the metric g, the dual 4-form and the four independent torsion forms. In the process we obtain a simple new proof of a theorem of Fernández–Gray.

As an application of our evolution equations, we derive an analogue of the second Bianchi identity in G₂-geometry which appears to be new, at least in this form. We use this result to derive explicit formulas for the Ricci tensor and part of the Riemann curvature tensor in terms of the torsion. These in turn lead to new proofs of several known results in G₂-geometry.

THE GAP BETWEEN LOCAL MULTIPLIER ALGEBRAS OF C*-ALGEBRAS

Argerami, M., Farenick, D., Massey, P. — Fri, 14 Aug 2009 08:13:01 PDT

The local multiplier algebra M_loc(A) of a C*-algebra A has the property that M_loc (A) M_loc(M_loc(A)). In this paper we show that there is a separable liminal C*-algebra A such that the inclusion is proper.

A QUADRIC WITH ARITHMETIC PAUCITY

Blomer, V., Brudern, J. — Fri, 14 Aug 2009 08:13:01 PDT

The quadric given by the equations x₁² + x₂² + x₃² = y₁² + y₂² + y₃², x₁ + x₂ + x₃ = y₁ + y₂ + y₃ has almost all its solutions with prime coordinates on the diagonals. This is shown in quantitative form. A similar statement holds for integral solutions whose coordinates can be written as the sum of two squares.

THE EULER OBSTRUCTION AND BRUCE-ROBERTS' MILNOR NUMBER

De Goes Grulha, N. — Fri, 14 Aug 2009 08:13:01 PDT

In this work we determine relations between the local Euler obstruction of an analytic function f and the Milnor number of f defined by Bruce and Roberts for functions on singular spaces.

ETA-INVARIANTS FROM MOLIEN SERIES

Degeratu, A. — Fri, 14 Aug 2009 08:13:01 PDT

We look at the orbifold Cⁿ/ with a finite subgroup of U(n) from two perspectives: from a differential point of view it is a non-compact orbifold with boundary at infinity S^2n–1/, while from an algebraic point of view it is a scheme with coordinate ring the -invariant polynomials in n variables. The main result is a relation between the -invariant of the boundary (an analytical object) and the Molien series of the singularity (an algebraic object).

FREIHEITSSATZE FOR ONE-RELATOR QUOTIENTS OF SURFACE GROUPS AND OF LIMIT GROUPS

Howie, J., Saeed, M. S. — Fri, 14 Aug 2009 08:13:01 PDT

Three versions of the Freiheitssatz are proved in the context of one-relator quotients of limit groups, where the latter are equipped with 1-acylindrical splittings over cyclic subgroups. These are natural extensions of previously published corresponding statements for one-relator quotients of orientable surface groups. Two of the proofs are new even in that restricted context.

SOME Z/2-GRADED REPRESENTATION THEORY

Parshall, B. J., Scott, L. L. — Fri, 14 Aug 2009 08:13:01 PDT

In representation theory, the existence of a Z⁺-grading on a related finite dimensional algebra often plays an important role. For example, such a grading arises from the Koszul structure of the finite dimensional algebra representing the principal block of the BGG category O associated to a complex semisimple Lie algebra. But Koszul gradings in positive characteristic have proved elusive. For example, except for small values of a positive integer n, it is not known if the Schur algebra S(n, n) has such a Koszul grading, assuming the characteristic p of the base field satisfies p ≥ n, though this grading would suffice to establish Lusztig's character formula for these algebras. (And even though the character formula is known for p sufficiently large [H. Andersen, J. Jantzen and W. Soergel, Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p, Astérique, Vol. 220, 1994], it is not known if the Schur algebra is Koszul for p sufficiently large.) This paper introduces Z/2-gradings on quasi-hereditary algebras, and shows that these gradings are almost as useful as a full Z⁺-grading, while being possibly much easier to find. We define the notion of a Z/2-based Kazhdan–Lusztig theory, which appears to be more flexible than, and generalizes, the notion of a Kazhdan–Lusztig theory (as first defined in [E. Cline, B. Parshall and L. Scott, Abstract Kazhdan–Lusztig theories, Tôhoku Math. J. 45 (1993), 511–534]). However, its existence suffices, as was the case with the original notion, to establish character formulas in the standard settings, determine Extⁿ-groups, and show that homological duals behave well. Finally, we present some suggestive symmetric group examples involving Schur algebras which were an outgrowth of this work.

LARGE INDECOMPOSABLE MINIMAL GROUPS

Shelah, S., Strungmann, L. — Fri, 14 Aug 2009 08:13:01 PDT

Assuming V = L we prove that there exist indecomposable almost-free minimal groups of size for every regular cardinal below the first weakly compact cardinal. This is to say that there are indecomposable almost-free torsion-free abelian groups of cardinality which are isomorphic to all of their finite index subgroups.

TWISTOR SPACES, PLURIHARMONIC MAPS AND HARMONIC MORPHISMS

Simoes, B. A., Svensson, M. — Fri, 14 Aug 2009 08:13:01 PDT

The application of twistor methods to construct harmonic morphisms has proved to be a fruitful approach in the 4-dimensional case, where a variety of examples and, in some cases, even a complete classification of harmonic morphisms have been found. In this paper, we generalize this construction to obtain higher-dimensional analogues of these maps. We also prove several results on twistor lifts of pluriharmonic and (1, 1)-geodesic maps.

ON MORITA THEORY FOR SELF-DUAL MODULES

Willems, W., Zimmermann, A. — Fri, 14 Aug 2009 08:13:01 PDT

Let G be a finite group and k be a field of characteristic p. It is known that a kG-module V carries a non-degenerate G-invariant bilinear form b if and only if V is self-dual. We show that whenever a Morita bimodule M that induces an equivalence between two blocks such as B(kG) and B(kH) of group algebras kG and kH is self-dual, then the correspondence preserves self-duality. Even more, if the bilinear form on M is symmetric, then, for p odd, the correspondence preserves the geometric type of simple modules. In characteristic 2, this holds also true for projective modules.

THE ADAPTED COMPLEXIFICATION OF THE TWO-SPHERE WITH A LIOUVILLE METRIC

Aguilar, R. M. — Tue, 12 May 2009 09:12:27 PDT

We show that the two-sphere with a Riemannian metric that is Liouville with finite isometry group does not admit an unbounded adapted complexification in the sense of Lempert and Szoke and of Guillemin and Stenzel; that is, its Grauert tube cannot have infinite radius. We prove this by first extending a classical theorem valid for umbilical geodesics in a triaxial ellipsoid to general Liouville metrics. Furthermore, we derive an isometric rigidity result for the Monge–Ampère foliation of a two-dimensional Grauert tube with infinite radius.

DIVISIBILITY OF EXPONENTIAL SUMS AND SOLVABILITY OF CERTAIN EQUATIONS OVER FINITE FIELDS

Castro, F. N., Rubio, I., Vega, J. M. — Tue, 12 May 2009 09:12:27 PDT

Carlitz [Solvability of certain equations in a finite field, Quart. J. Math. (Oxford) 7 (1956), 3–4] determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend some of Carlitz's results by computing the exact p-divisibility of certain exponential sums. As a by-product we obtain an upper bound for the Waring number for polynomials over extensions of finite fields.

MODULI SPACES OF PARABOLIC U(p, q)-HIGGS BUNDLES

Garcia-Prada, O., Logares, M., Munoz, V. — Tue, 12 May 2009 09:12:27 PDT

Using the L²-norm of the Higgs field as a Morse function, we study the moduli space of parabolic U(p, q)-Higgs bundles over a Riemann surface with a finite number of marked points, under certain genericity conditions on the parabolic structure. When the parabolic degree is zero this space is homeomorphic to the moduli space of representations of the fundamental group of the punctured surface in U(p, q), with fixed compact holonomy classes around the marked points. By means of this homeomorphism we count the number of connected components of this moduli space of representations. Finally, we apply our results to the study of representations of the fundamental group of elliptic surfaces of general type.

HODGE POLYNOMIALS OF THE MODULI SPACES OF TRIPLES OF RANK (2, 2)

Munoz, V., Ortega, D., Vazquez-Gallo, M.-J. — Tue, 12 May 2009 09:12:27 PDT

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple (E₁, E₂, ) on X consists of two holomorphic vector bundles E₁ and E₂ over X and a holomorphic map : E₂ -> E₁. There is a concept of stability for triples which depends on a real parameter . In this paper, we determine the Hodge polynomials of the moduli spaces of -stable triples with rk(E₁) = rk(E₂) = 2, using the theory of mixed Hodge structures (in the cases that these moduli spaces are smooth and compact). This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we also give the Hodge polynomial of the moduli space of even degree rank 2 stable vector bundles.