The Quarterly Journal of Mathematics - recent issues

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

Aguilar, R. M. — 2009-05-12

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. — 2009-05-12

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. — 2009-05-12

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. — 2009-05-12

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.

DESINGULARIZATIONS OF CALABI-YAU 3-FOLDS WITH CONICAL SINGULARITIES. II. THE OBSTRUCTED CASE

Chan, Y.-M. — 2009-02-17

This is the second of two papers studying Calabi–Yau 3-folds with conical singularities and their desingularizations. In our first paper [Y.-M. Chan, Quart. J. Math. 57 (2006), 151–181] we constructed the desingularization of the conically singular manifold M₀ by gluing an asymptotically conical (AC) Calabi–Yau 3-fold Y into M₀ at the singular point, thus obtaining a 1-parameter family of compact, non-singular Calabi–Yau 3-folds M_t for small t > 0. During the gluing process one may encounter a kind of cohomological obstruction to defining a 3-form _t on M_t which interpolates between the 3-form ₀ on M₀ and the scaled 3-form t³ _Y on Y if the rate at which the AC Calabi–Yau 3-fold Y converges to the Calabi–Yau cone is equal to – 3. The first paper [3] studied the simpler case < –3 where there is no obstruction. This paper extends the result in the first one by considering a more complicated situtation when = –3. Assuming the existence of singular Calabi–Yau metrics on compact complex 3-folds with ordinary double points, our result in this paper can be applied to repairing such kinds of singularities, which is an analytic version of Friedman's result giving necessary and sufficient conditions for smoothing ordinary double points.

NON-COMMUTATIVE VITALI-HAHN-SAKS THEOREM HOLDS PRECISELY FOR FINITE W*-ALGEBRAS

Chetcuti, E., Hamhalter, J. — 2009-02-17

It is shown that the bona fide generalization of the Vitali–Hahn–Saks theorem to von Neumann algebras is possible if, and only if, the algebra is finite. This settles the problem on the non-commutative Vitali–Hahn–Saks theorem completely and provides new means of characterizing finite von Neumann algebras.

A NOTE ON THE LEAST TOTIENT OF A RESIDUE CLASS

Garaev, M. Z. — 2009-02-17

Let q be a large prime number, a be any integer and be a fixed small positive quantity. Friedlander and Shparlinksi (Least totient in a residue class, Bull. London Math. Soc. 39 (2007), 425–432) have shown that there exists a positive integer n << q^5/2+ such that (n) falls into the residue class a (mod q. Here, (n) denotes Euler's function. In the present paper we improve this bound to n << q^{2 +}.

THE IDENTITIES OF A LIE ALGEBRA VIEWED AS A LIE RING

Krasilnikov, A. — 2009-02-17

Let G be a Lie algebra over the field Q of rationals. Then G can be viewed as a Lie ring as well. If the identities of the Lie ring G have a finite basis then so have the identities of the Lie algebra G. We give an example which shows that the converse is not always true.

MIXED WEAK TYPE INEQUALITIES FOR ONE-SIDED OPERATORS

Martin-Reyes, F. J., Ombrosi, S. J. — 2009-02-17

We discuss mixed weak type inequalities in weighted spaces for one-sided operators. In particular, we prove that if T_c f(x) = (x – c)^–1 _c^x f(y) dy, x > c, is the Hardy averaging operator, u A₁^– (one-sided Muckenhoupt A₁ class), and v A₁⁺ (another one-sided Muckenhoupt A₁ class), then there exists a constant C such that sup_cR _{{x:|T_cf(x)| > v(x)}}uv ≤ C _R|f|u.

HOMOLOGICAL SYSTEMS IN MODULE CATEGORIES OVER PRE-ORDERED SETS

Mendoza, O., Saenz, C., Xi, C. — 2009-02-17

We introduce the so-called homological systems in a module category over a pre-ordered set, which generalize the notion of a stratifying system over a linearly ordered set, and study both the corresponding modules filtrated by the systems and algebras stratified by the systems. In particular, we recover the tilting theory for pre-standardly stratified algebras, and get a general formula for computing the Cartan determinants of pre-standardly stratified algebras in terms of standard modules and simple modules. Also, the finitistic dimension of a given algebra, and the relative homological dimensions of full subcategories of the modules related to a homological system, are discussed. As an application, we get a new bound for the finitistic dimension of a pre-standardly stratified algebra.

CLASSIFICATION OF A CLASS OF CONTINUOUS MAPS ON THE UNIT INTERVAL

Sawyer, D. J., Truss, J. K. — 2009-02-17

We give a classification of certain continuous maps from the unit interval (or any non-trivial closed bounded interval of the real line) to itself up to conjugacy. The maps studied are those for which there is a partition of the interval into finitely many subintervals whose endpoints form a cycle of the map, and on each of which the map is monotonic.

POSITIVE LAWS IN DERIVED SUBGROUPS OF FIXED POINTS

Shumyatsky, P. — 2009-02-17

Let A be an elementary abelian group of order at least q⁴ acting on a finite q'-group G in such a manner that the derived subgroup of C_G(a) satisfies a positive law of degree n for any a A^#. It is proved that G' satisfies a positive law of degree bounded by a function of q and n only.

WHEN JORDAN SUBMODULES ARE BIMODULES

Bresar, M., Kissin, E., Shulman, V. S. — 2008-11-18

Let A be an algebra and let X be an A-bimodule. We call a linear subspace Y of X a Jordan A-submodule of X if Ay + yA Y for all A A and y Y (if X = A, then this coincides with the classical concept of a Jordan ideal). When is a Jordan A-submodule a submodule? We give a thorough analysis of this question in both algebraic and analytic context. In the first part of the paper, we consider general algebras and general Banach algebras. In the second part, we treat some more specific topics, such as symmetrically normed Jordan A-submodules. Some of our results are of interest also in the classical situation; in particular, we show that there exist C*-algebras having Jordan ideals that are not ideals.

MULTIPLICATIVE STRUCTURES FOR KOSZUL ALGEBRAS

Buchweitz, R.-O., Green, E. L., Snashall, N., Solberg, O. — 2008-11-18

Let = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over is given in [E. L. Green and Ø. Solberg, An algorithmic approach to resolutions, J. Symbolic Comput., 42 (2007), 1012–1033]. This resolution is shown to have a ‘comultiplicative’ structure in [E. L. Green, G. Hartman, E. N. Marcos and Ø. Solberg, Resolutions over Koszul algebras, Arch. Math. 85 (2005), 118–127.], and this is used to find a minimal projective resolution P of over the enveloping algebra ^e. Using these results, we show that the multiplication in the Hochschild cohomology ring of relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of is shown to be surjective onto the graded centre of the Koszul dual.

POLYNOMIAL NUMERICAL INDEX FOR SOME COMPLEX VECTOR-VALUED FUNCTION SPACES

Choi, Y. S., Garcia, D., Maestre, M., Martin, M. — 2008-11-18

We study the relation between the polynomial numerical indices of a complex vector-valued function space and the ones of its range space. It is proved that the spaces C(K, X) and L(µ, X) have the same polynomial numerical index as the complex Banach space X for every compact Hausdorff space K and every -finite measure µ, which does not hold any more in the real case. We give an example of a complex Banach space X such that, for every k ≥ 2, the polynomial numerical index of order k of X is the greatest possible, namely 1, while the one of X** is the least possible, namely k^k/(1–k). We also give new examples of Banach spaces with the polynomial Daugavet property, namely L(µ, X) when µ is atomless, and C_w(K, X), C_w*(K, X*) when K is perfect.

ON THE SUM OF THE FIRST n PRIMES

Cilleruelo, J., Luca, F. — 2008-11-18

In this note, we show that the set of n such that the arithmetic mean of the first n primes is an integer is of asymptotic density zero. We use the same method to show that the set of n such that the sum of the first n primes is a square is also of asymptotic density zero. We also prove that both the arithmetic mean of the first n primes as well as the square root of the sum of the first n primes are well distributed modulo 1.

ON THE ERROR TERMS AND EXCEPTIONAL SETS IN CONJECTURAL SECOND MAIN THEOREMS

Levin, A., McKinnon, D., Winkelmann, J. — 2008-11-18

We study the error terms and exceptional sets in conjectural Second Main Theorems in Nevanlinna theory and Diophantine approximation (Vojta's conjecture). In particular, we give a geometric description of the exceptional set in the case of surfaces and the trivial divisor. Examples are given which show that, in general, the exceptional sets in conjectural Second Main Theorems must depend on the parameter in these conjectures. As a consequence, we obtain counterexamples to a conjecture of S. Lang on the forms of the error terms in conjectural Second Main Theorems.

ON DAVENPORT-STOTHERS INEQUALITIES AND ELLIPTIC SURFACES IN POSITIVE CHARACTERISTIC

Schutt, M., Schweizer, A. — 2008-11-18

We show that the Davenport–Stothers inequality from characteristic 0 fails in any characteristic p > 3. The proof uses elliptic surfaces over P¹ and inseparable base change. We then present adjusted inequalities. These follow from results of Pesenti and Szpiro. For characteristics 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over P¹. Our ideas are also related to supersingular surfaces (in Shioda's sense).

ASYMPTOTICALLY LINEAR ELLIPTIC PROBLEM ON RN

Zhou, H.-S., Zhu, H. — 2008-11-18

In this paper, we consider the following semilinear elliptic problem:

where f(x, t) tends to p(x) and q(x) L(R^N), respectively, as t -> 0 and t -> +. We prove that there exist two numbers l and L with L < l such that problem (P) has at least one positive solution for (– l, –L) and has no positive solution for all [–l,–L]. The existence and non-existence of positive solutions for problem (P) at = –l and = –L are also discussed.