Narrow Results

For any n>1, we construct examples of branched Galois coverings M→ℙⁿ where M is one of (ℙ¹)ⁿ, and , where is the 1-ball. In terms of orbifolds, this amounts to giving examples of orbifolds over ℙⁿ uniformized by M. We also discuss the related "orbifold braid groups".

We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton's equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, Vol. 18-I (Academic Press, Boston, MA, 1990), pp. 327–337]. Our examples can be viewed a generalization of previous examples by Cao [Existense of gradient Kähler–Ricci solitons, in Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), pp. 1–16], Chave and Valent [On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B 478 (1996) 758–778], Pedersen, Tønnesen-Friedman, and Valent [Quasi-Einstein Kähler metrics, Lett. Math. Phys. 50(3) (1999) 229–241], and Feldman, Ilmanen and Knopf [Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169–209]. We also prove a uniformization result on complete steady gradient Kähler–Ricci solitons with non-negative Ricci curvature under additional assumptions.

We give a brief survey of the so-called Fenchel's problem for the projective plane, that is the problem of existence of finite Galois coverings of the complex projective plane branched along a given divisor and prove the following result: Let p, q be two integers greater than 1 and C be an irreducible plane curve. If there is a surjection of the fundamental group of the complement of C into a free product of cyclic groups of orders p and q, then there is a finite Galois covering of the projective plane branched along C with any given branching index.

Given a compact connected Riemann surface $\mathrm{\Sigma }$ of genus $g_{\mathrm{\Sigma }}\ge 2$, and an effective divisor $D={\sum }_i{n}_i{x}_i$ on $\mathrm{\Sigma }$ with $\text{degree}(D)<2(g_{\mathrm{\Sigma }}-1)$, there is a unique cone metric on $\mathrm{\Sigma }$ of constant negative curvature $-4$ such that the cone angle at each point $x_i$ is $2\pi n_i$ [R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc.103 (1988) 222–224; M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991) 793–821]. We describe the Higgs bundle on $\mathrm{\Sigma }$ corresponding to the uniformization associated to this conical metric. We also give a family of Higgs bundles on $\mathrm{\Sigma }$ parametrized by a nonempty open subset of $H^0(\mathrm{\Sigma },K_{\mathrm{\Sigma }}^{\otimes 2}\otimes {{𝒪}}_{\mathrm{\Sigma }}(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchin’s results in [N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987) 59–126] for the case $D=0$.

Assume $ZF+{AD}^{+}+V=L(\mathcal{𝒫}(ℝ))$. Let $E$ be a ${\mathrm{\Sigma }}_1^1$ equivalence relation coded in $\mathrm{\text{HOD}}$. $E$ has an ordinal definable equivalence class without any ordinal definable elements if and only if $\mathrm{\text{HOD}}\vDash E$ is unpinned.

$ZF+{AD}^{+}+V=L(\mathcal{𝒫}(ℝ))$ proves $E$-class section uniformization when $E$ is a ${\mathrm{\Sigma }}_1^1$ equivalence relation on $ℝ$ which is pinned in every transitive model of $ZFC$ containing the real which codes $E$: Suppose $R$ is a relation on $ℝ$ such that each section $R_x=\{y:(x,y)\in R\}$ is an $E$-class, then there is a function $f:ℝ\to ℝ$ such that for all $x\in ℝ$, $R(x,f(x))$.

$ZF+AD$ proves that $ℝ\times \kappa$ is Jónsson whenever $\kappa$ is an ordinal: For every function $f:{[ℝ\times \kappa ]}_{=}^{<\omega }\to ℝ\times \kappa$, there is an $A\subseteq ℝ\times \kappa$ with $A$ in bijection with $ℝ\times \kappa$ and $f[{[A]}_{=}^{<\omega }]\ne ℝ\times \kappa$.

Anti-elementarity is a strong way of ensuring that a class of structures, in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form ${ℒ}_{\infty \lambda }$. We prove that many naturally defined classes are anti-elementary, including the following:

• the class of all lattices of finitely generated convex $\ell$-subgroups of members of any class of $\ell$-groups containing all Archimedean $\ell$-groups;
• the class of all semilattices of finitely generated $\ell$-ideals of members of any nontrivial quasivariety of $\ell$-groups;
• the class of all Stone duals of spectra of MV-algebras — this yields a negative solution to the MV-spectrum Problem;
• the class of all semilattices of finitely generated two-sided ideals of rings;
• the class of all semilattices of finitely generated submodules of modules;
• the class of all monoids encoding the nonstable K₀-theory of von Neumann regular rings, respectively, C*-algebras of real rank zero;
• (assuming arbitrarily large Erdős cardinals) the class of all coordinatizable sectionally complemented modular lattices with a large $4$-frame.

The main underlying principle is that under quite general conditions, for a functor $\mathrm{\Phi }:{𝒜}\to {ℬ}$, if there exists a noncommutative diagram $\overrightarrow{D}$ of ${𝒜}$, indexed by a common sort of poset called an almost join-semilattice, such that

• $\mathrm{\Phi }{\overrightarrow{D}}^I$ is a commutative diagram for every set $I$,
• $\mathrm{\Phi }\overrightarrow{D}\ncong \mathrm{\Phi }\overrightarrow{X}$ for any commutative diagram $\overrightarrow{X}$ in ${𝒜}$,

then the range of $\mathrm{\Phi }$ is anti-elementary.

Motivated by Felix Klein's notion that geometry is governed by its group of symmetry transformations, Charles Ehresmann initiated the study of geometric structures on topological spaces locally modeled on a homogeneous space of a Lie group. These locally homogeneous spaces later formed the context of Thurston's 3-dimensional geometrization program. The basic problem is for a given topology Σ and a geometry X = G/H, to classify all the possible ways of introducing the local geometry of X into Σ. For example, a sphere admits no local Euclidean geometry: there is no metrically accurate Euclidean atlas of the earth. One develops a space whose points are equivalence classes of geometric structures on Σ, which itself exhibits a rich geometry and symmetries arising from the topological symmetries of Σ.

We survey several examples of the classification of locally homogeneous geometric structures on manifolds in low dimension, and how it leads to a general study of surface group representations. In particular geometric structures are a useful tool in understanding local and global properties of deformation spaces of representations of fundamental groups.