Volume 22, Issue 1

A Ricci almost soliton whose associated vector field is projective is shown to have vanishing Cotton tensor, divergence-free Bach tensor and Ricci tensor as conformal Killing. For the compact case, a sharp inequality is obtained in terms of scalar curvature.We show that every complete gradient Ricci soliton is isometric to the Riemannian product of a Euclidean space and an Einstein space. A complete K -contact Ricci almost soliton whose associated vector field is projective is compact Einstein and Sasakian.

We give an overview of the research related to the topological characterization of Stein domains in complex two-dimensional space, and an instance of their many important connections to smooth manifold topology in dimension four. One goal is to motivate and explain the following remarkable conjecture of Gompf: no Brieskorn integral homology sphere (other than S 3 ) admits a pseudoconvex embedding in ℂ 2 , with either orientation. We include some new examples and results that consider the conjecture for families of rational homology spheres which are Seifert fibered, and integral homology spheres which are hyperbolic.

The radial map w ( x ) = x ‖ x ‖ −1 is a well-known example of harmonic maps and p -harmonic ones into spheres with a point singularity. In this paper we give two examples of harmonic maps and p -harmonic ones into spheres of higher dimensions with the singularity x i x j ‖ x ‖ −2 or the singularity x i x j x k ‖ x ‖ −3 .

Several recent papers have examined a rational polyhedron P m whose integer points are in bijection with the numerical semigroups (cofinite, additively closed subsets of the non-negative integers) containing m . A combinatorial description of the faces of P m was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of P m containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of P m . In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of P m .

Given a lattice polygon, we study the moduli space of all tropical plane curves with that Newton polygon. We determine a formula for the dimension of this space in terms of combinatorial properties of that polygon. If this polygon is nonhyperelliptic or maximal and hyperelliptic, then this formula matches the dimension of the moduli space of nondegenerate algebraic curves with the given Newton polygon.

We study the automorphisms of a function field of genus g ≥ 2 over an algebraically closed field of characteristic p > 0. More precisely, we show that the order of a nilpotent subgroup G of its automorphism group is bounded by 16 ( g – 1) when G is not a p -group. We show that if | G | = 16( g – 1), then g – 1 is a power of 2. Furthermore, we provide an infinite family of function fields attaining the bound.

In this paper, we classify the compact locally homogeneous non-gradient m -quasi Einstein 3- manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is m -quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial m -quasi Einstein metrics that are a compact quotient of the product of two Einstein metrics. We also show that S 1 is the only compact manifold of any dimension which admits a metric which is nontrivially m -quasi Einstein and Einstein.

This paper presents a direct and simple proof of a result concerning the existence of metrics of positive Ricci curvature on the total space of fiber bundles with compact structure groups. In particular, it generalizes and puts in a unified framework the results of Nash [12] and Poor [15]. With the intention of disseminating this result,we apply it to build new examples of manifolds with positive Ricci curvature, including bundles whose base consists of gradient shrinking Ricci solitons.

Let X be an Enriques surface over the field of complex numbers. We prove that there exists a nontrivial quaternion algebra 𝓐 on X . Then we study the moduli scheme of torsion free 𝓐-modules of rank one. Finally we prove that this moduli scheme is an étale double cover of a Lagrangian subscheme in the corresponding moduli scheme on the associated covering K3 surface.

We give an upper bound for the Matveev complexity of the whole class of closed connected orientable prime graph manifolds; this bound is sharp for all 14502 graph manifolds of the Recogniser catalogue (available at http://matlas.math.csu.ru/?page=search)

Assume that we have a fibration of smooth projective varieties X → S over a surface S such that X is of dimension four and that the geometric generic fiber has finite-dimensional motive and the first étale cohomology of the geometric generic fiber with respect to ℚ l coefficients is zero and the second étale cohomology is spanned by divisors. We prove that then A 3 ( X ), the group of codimension three algebraically trivial cycles modulo rational equivalence, is dominated by finitely many copies of A 0 ( S ); this means that there exist finitely many correspondences Γ i on S × X such that Σ i Γ i is surjective from A 2 ( S ) to A 3 ( X ).

We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results.