Stable bundles with torsion Chern classes on non-Kählerian elliptic surfaces

If π:X→B is a non-Kählerian elliptic surface with generic fibreF, the moduli space of stable holomorphic vector bundles with torsion Chern classes onX has an induced fibred structure with base Pic^o(F) and the moduli space of stable parabolic bundles onB ^orb as fibre. This is specific to the non-Kähler case.     

Stable bundles on hypercomplex surfaces

A hypercomplex manifold is a manifold equipped with three complex structures I, J, K satisfying the quaternionic relations. Let M be a 4-dimensional compact smooth manifold equipped with a hypercomplex structure, and E be a vector bundle on M. We show that the moduli space of anti-self-dual connections on E is also hypercomplex, and admits a strong HKT metric. We also study manifolds with (4,4)-supersymmetry, that is, Riemannian manifolds equipped with a pair of strong HKT-structures that have opposite torsion. In the language of Hitchin’s and Gualtieri’s generalized complex geometry, (4,4)-manifolds are called “generalized hyperkähler manifolds”. We show that the moduli space of anti-self-dual connections on M is a (4,4)-manifold if M is equipped with a (4,4)-structure.     

Stable pairs on elliptic K3 surfaces

We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman.     

Connections on differentiable fiber bundles

We present a theory of external connections on differentiable fiber bundles and on the basis of this theory we give a survey of papers on the theory of connections of bundles of a special form that have additional structures.Translated from Itogi Nauki i Tekhniki, Seriya Problemy Geometrii, Vol. 15, pp. 61–93, 1983.     

Rank two vector bundles over regular elliptic surfaces

Research partially supported by NSF grants DMS-85-03743 and DMS-87-03569 and the Alfred P. Sloan Foundation     

Stein and Weinstein structures on disk cotangent bundles of surfaces

We prove that the three-dimensional periodic Burgers’ equation has a unique global in time solution in a critical Gevrey–Sobolev space. Comparatively to Navier–Stokes equations, the main difficulty is the lack of an incompressibility condition. In our proof of existence, we overcome the bootstrapping argument, which was a technical step in a precedent proof in Sololev spaces. This makes our proof shorter and gives sense of considering the Gevrey class for a mathematical study to Burgers’ equation. To prove that the unique solution is global in time, we use the maximum principle. Energy methods, Sobolev product laws, compactness methods, and Fourier analysis are the main tools.     

Positivity of certain functions associated with analysis on elliptic surfaces

In this paper, we study functions of one variable that are called boundary terms of two-dimensional zeta integrals established in recent works of Ivan Fesenko?s two-dimensional adelic analysis attached to arithmetic elliptic surfaces. It is known that the positivity of the fourth log derivatives of boundary terms around the origin is a sufficient condition for the Riemann hypothesis of Hasse-Weil L-functions of elliptic curves. We show that such positivity is also a necessary condition under some reasonable technical assumptions.     

2-Descent on elliptic curves and rational points on certain Kummer surfaces

The first part of this paper further refines the methodology for 2-descents on elliptic curves with rational 2-division points which was introduced in [J.-L. Colliot-Thélène, A.N. Skorobogatov, Peter Swinnerton-Dyer, Hasse principle for pencils of curves of genus one whose Jacobians have rational 2-division points, Invent. Math. 134 (1998) 579-650]. To describe the rest, let E⁽¹⁾ and E⁽²⁾ be elliptic curves, D⁽¹⁾ and D⁽²⁾ their respective 2-coverings, and X be the Kummer surface attached to D⁽¹⁾×D⁽²⁾. In the appendix we study the Brauer-Manin obstruction to the existence of rational points on X. In the second part of the paper, in which we further assume that the two elliptic curves have all their 2-division points rational, we obtain sufficient conditions for X to contain rational points; and we consider how these conditions are related to Brauer-Manin obstructions. This second part depends on the hypothesis that the relevent Tate-Shafarevich group is finite, but it does not require Schinzel's Hypothesis.     

Correction to: Stein and Weinstein structures on disk cotangent bundles of surfaces

Archiv der Mathematik - After our article entitled “Stein and Weinstein structures on disk cotangent bundles of surfaces” was accepted and appeared online, it was pointed out by Sylvain...     

Differential geometric structures on differentiable manifolds

In this paper we present a study of differential geometric structures arising on manifolds imbedded in almost complex spaces and the differential geometry of such manifolds.Translated from Itogi Nauki i Tekhniki. Problemy Geometrii, Vol. 8, pp. 89–111, 1977.     

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.     

Tensor structures on a differentiable manifold

Summary A tensor structure is a class of equivalent G-structures and it is defined by a special tensor field. Such fields are characterized by the existence of a linear connection relative to which they have covariant derivative zero. Two tensor structures may admit a common subordinate structure. Exaples of such subordinate stuctures are given and some cases, when one stucture is a Riemannian metric, are considered. To Enrico Bompiani on his scientific Jubilee     

Ulrich bundles on ruled surfaces

In this short note, we study the existence problem for Ulrich bundles on polarized ruled surfaces, focusing our attention on the smallest possible rank. We show that existence of Ulrich line bundles occurs if and only if the coefficient α of the minimal section in the numerical class of the polarization equals one. For other polarizations, we prove the existence of rank two Ulrich bundles.