Scalar curvature and projective embeddings, II

The paper uses the technique of finite-dimensional approximation to show that a constant scalr curvature Kahler metric (on a polarised algebraic variety without holomorphic vector fields) minimises the Mabuchi functional

b-Stability and blow-ups

We extend an argument of Stoppa to make some prgress towards a proof that K\"ahler-Einstein manifolds are "b-stable". We point out some algebro-geometric questions, involving finite generation, that arise

Nahm's equations and free-boundary problems

This paper is a discussion of relations between some free-boundary problems and infinite dimensional Lie groups; particularly a version of Nahm's equations for the group of Hamiltonian diffeomorphisms in two dimensions

Constant scalar curvature metrics on toric surfaces

This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature

A generalised Joyce construction for a family of nonlinear partial differential equations

We explain a simple construction of solutions to a family of PDE's in two dimensions which includes that defining zero scalar curvature Kahler metrics, with two Killing fields, and the affine maximal equation

Kahler geometry on toric manifolds, and some other manifolds with large symmetry

This is an expository article. Among other topics, we discuss the existence of Kahler-Ricci soliton metrics on toric Fano manifolds, and Kahler-Einstein metrics on deformations of the Mukai-Umemura 3-foldComment: Section 3.3 has been changed to correct a mistake in the original versio

Two-forms on four-manifolds and elliptic equations

We define a general class of elliptic equations for 2-forms on 4-manifolds, of which the complex Monge-Ampere equation is a prototype. We obtain some regularity results and discuss various connections (some speculative) with modern symplectic 4-manifold theory

Extremal metrics on toric surfaces, I

The paper develops a continiuty method for solutions of the Abreu equation, which include extremal metrics on toric surfaces. Results are obtained, assuming a hypothesis (the "M-condition") on the solutions

Lie algebra theory without algebra

This is an expository paper in which we explain how basic, standard, results about simple Lie algebras can be obtained by geometric arguments, following ideas of Cartan, Richardson and others

Topological field theories and formulae of Casson and Meng-Taubes

The goal of this paper is to give a new proof of a theorem of Meng and Taubes that identifies the Seiberg-Witten invariants of 3-manifolds with Milnor torsion. The point of view here will be that of topological quantum field theory. In particular, we relate the Seiberg-Witten equations on a 3-manifold with the Abelian vortex equations on a Riemann surface. These techniques also give a new proof of the surgery formula for the Casson invariant, interpreted as an invariant of a homology S^2 x S^1.Comment: 16 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTMon2/paper4.abs.htm