Hyperelliptic surface

Abstract: We explicitly construct the fully back-reacted half-BPS solutions in Type IIB supergravity which are dual to Wilson loops with 16 supersymmetries in $\mathcal{N}=4$ super Yang-Mills. In a first part, we use the methods of a companion paper to derive the exact general solution of the half-BPS equations on the space $AdS_2 \times S^2 \times S^4 \times \Sigma$, with isometry group $SO(2,1)\times SO(3) \times SO(5)$ in terms of two locally harmonic functions on a Riemann surface $\Sigma$ with boundary. These solutions, generally, have varying dilaton and axion, and non-vanishing 3-form fluxes. In a second part, we impose regularity and topology conditions. These non-singular solutions may be parametrized by a genus $g \geq 0$ hyperelliptic surface $\Sigma$, all of whose branch points lie on the real line. Each genus $g$ solution has only a single asymptotic $AdS_5 \times S^5$ region, but exhibits $g$ homology 3-spheres, and an extra $g$ homology 5-spheres, carrying respectively RR 3-form and RR 5-form charges. For genus 0, we recover $AdS_5 \times S^5$ with 3 free parameters, while for genus $g \geq 1$, the solution has $2g+5$ free parameters. The genus 1 case is studied in detail. Numerical analysis is used to show that the solutions are regular throughout the $g=1$ parameter space. Collapse of a branch cut on $\Sigma$ subtending either a homology 3-sphere or a homology 5-sphere is non-singular and yields the genus $g-1$ solution. This behavior is precisely expected of a proper dual to a Wilson loop in gauge theory.

Abstract: This paper concerns a family of pseudo-Anosov braids with dilatations arbitrarily close to one. The associated graph maps and train tracks have stable "star-like" shapes, and the characteristic polynomials of their transition matrices form Salem-Boyd sequences. These examples show that the logarithms of least dilatations of pseudo-Anosov braids on $2g+1$ strands are bounded above by $\log(2 + \sqrt{3})/g$. It follows that the asymptotic behavior of least dilatations of pseudo-Anosov, hyperelliptic surface homeomorphisms is identical to that found by Penner for general surface homeomorphisms. The family includes pseudo-Anosov braids with minimum dilatation for 3,4, and 5 strands; the latter according to a recent anouncement of J.-Y. Ham and W.-T. Song [math.GT/0506295].

Abstract: Introduction. Let X/k be a smooth, projective, geometrically connected variety over a number field k. Assume that we are given a proper flat morphism p : X → Pk with (smooth) geometrically integral generic fibre. In a series of recent papers ([R], [CT/SwD], [SwD], [Sk], [CT/Sk/SwD.1], [CT/Sk/SwD.2]), under specific assumptions on the pencil, various authors have investigated the Zariski density of the set X(k) of k-rational points and its topological density with respect to some embedding X(k) ⊂ X(kv), where kv denotes a completion of k. In these various papers, the assumptions made imply that no fibre of p is multiple. In the present paper, we go the other way. We look at pencils which possess multiple fibres, and we investigate the consequences for the rational points of the total space X. In the interest of simplicity, we restrict attention to fibres which are double. A fibre XP = p−1(P ) over a closed point P ∈ Pk is called double if as a divisor on X it is a double. Here are the main results:

Abstract: Gordon and Benson have proved that if a compact nilmanifold admits a Kahler structure then it is a torus [5] more precisely they proved that the condition (iv) fails for any symplectic structure on a non-toral nilmanifold M. This result was independently proved by Hasegawa [12] by showing that (v) fails for M. For a compact solvmanifold M of dimension 4 it is known that M has a Kahler structure if and only if it is a complex torus or a hyperelliptic surface. In fact, Auslander and Szczarba in [4] proved that if the first Betti number bι(M) of M is 2, M is a fiber bundle over T with fiber T. Then by Ue [19] M has a complex structure only if it is a hyperelliptic surface or a primary Kodaira surface which is a compact nilmanifold. Thus, if M is a Kahler manifold, it must be a hyperelliptic surface. Since !<&ι(M)<4, M can be a Kahler manifold only if it is a complex torus or a hyperelliptic surface. The fact that a hyperelliptic surface is a solvmanifold follows from Auslander [3]. The above result may be generalized as the following conjecture : A compact solvmanifold has a Kahler structure if and only if it is a finite quotient of a complex torus. In contrast to the case of compact nilmanifolds there are compact symplectic