The Quarterly Journal of Mathematics - current issue

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.