Hyper-Kähler quotients of solvable Lie groups

Orbifold K{ä}hler Groups related to arithmetic complex hyperbolic lattices

We study fundamental groups of toroidal compactifications of non compact ball quotients and show that the Shafarevich conjecture on holomorphic convexity for these complex projective manifolds is satisfied in dimension 2 provided the corresponding lattice is arithmetic and small enough. The method is to show that the Albanese mapping on an {é}tale covering space generates jets on the interior, if the lattice is small enough. We also explore some specific examples of Picard-Eisenstein type.

Hyper-K{ä}hler Hierarchies and their twistor theory

A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyper-Kähler equations. It is shown that R acts on the twistor data by multiplication with a rational function. The structures are illustrated by the example of the Sparling-Tod (Eguchi-Hansen) solution. An extended space-time ${\cal N}$ is constructed whose extra dimensions corresp…

Twistor theory of hyper-K{ä}hler metrics with hidden symmetries

We briefly review the hierarchy for the hyper-Kähler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kähler metric admits a hidden symmetry if it admits a certain Killing spinor. We show that if the hidden symmetry is tri-holomorphic, then this is equivalent to requiring symmetry along a higher time and the hidden symmetry determines a `twistor group' action as introduced by Bielawski \cite{B00}. This leads to a construction for the …

具有H-可解李代数的超复幂零流形中的复曲线

Complex curves in hypercomplex nilmanifolds with H-solvable Lie algebras

本文研究幂零李代数上的超复结构，提出四元数可解性概念并猜想所有超复结构均满足该性质，进而证明在相应的紧超复幂零流形中一般复结构下不存在复曲线。

平坦超复幂零流形是H-可解的

Flat hypercomplex nilmanifolds are H-solvable

该数学研究定义了$\u210d$-可解超复幂零李代数的概念，证明了带平坦Obata联络的超复幂零流形对应的李代数满足$\u210d$-可解性，丰富了超复几何与幂零李代数的关联理论。

On the non-vanishing of the first Betti number of hyperbolic three manifolds

We show the non-vanishing of cohomology groups of sufficiently small congruence lattices in $SL(1,D)$, where $D$ is a quaternion division algebras defined over a number field $E$ contained inside a solvable extension of a totally real number field. As a corollary, we obtain new examples of compact, arithmetic, hyperbolic three manifolds, with non-torsion first homology group, confirming a conjecture of Thurston. The proof uses the characterisation of the image of solvable base change by the author, and the construction of cusp forms with non-zero cusp cohomology by Labesse and Schwermer.