b-Stability and blow-ups
We extend an argument of Stoppa to make some prgress towards a proof that Kähler-Einstein manifolds are "b-stable". We point out some algebro-geometric questions, involving finite generation, that arise.
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arXiv subjects
S. K. Donaldson
Publications and source records attributed to S. K. Donaldson.
Volume estimates for Kähler-Einstein metrics: the three dimensional case
We obtain an estimate for the volume of neighbourhoods of sets of large curvature in three-dimensional Kähler-Einstein manifolds.
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Volume estimates for Kähler-Einstein metrics and rigidity of complex structures
This paper extends our earlier results to higher dimensions using a different approach, based on the rigidity of complex structures on certain domains.
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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.
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Constant scalar curvature metrics on toric surfaces
This paper completes a programme to determine which toric surfaces admit Kahler metrics of constant scalar curvature/
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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-fold
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Numerical approximations to extremal metrics on toric surfaces
We give some detailed numerical information about extremal metrics on four different toric surfaces. These are sample of many other cases which can be treated using a computer programme outlined in the paper.
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A note on the α-invariant of the Mukai-Umemura 3-fold
We give an elementary argument to compute the $α$-invariant of this Fano 3-fold, which implies the existence of a Kahler-Einstein metric.
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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.
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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.
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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.
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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.
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Some numerical results in complex differential geometry
The first part of this paper discusses general procedures for finding numerical approximations to distinguished Kahler metrics, such as Calabi-Yau metrics, on complex projective manifolds. These procedures are closely related to ideas from Geometric Invariant Theory, and to the asymptotics of high powers of positive line bundles. In the core of the paper these ideas are illustrated by detailed numerical results for a particular K3 surface.
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Lower bounds on the Calabi functional
The main result of this paper shows that "test configurations" give new lower bounds on the $L^{2}$ norm of the scalar curvature on a Kahler manifold. This is closely analogous to the analysis of the Yang-Mills functional over Riemann surfaces by Atiyah and Bott. The proof uses asymptotic approximation by finite-dimensional problems: the essential ingredient being the Tian-Zelditch-Lu expansion of the "density of states" function.
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Interior estimates for solutions of Abreu's equation
The paper develops various estimates for solutions of a fourth order nonlinear PDE, which corresponds to prescribing the scalar curvature of a toric Kahler metric.
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Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves
We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.
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Fundamental groups of complements of plane curves and symplectic invariants
Introducing the notion of stabilized fundamental group for the complement of a branch curve in $CP^2$, we define effectively computable invariants of symplectic 4-manifolds that generalize those previously introduced by Moishezon and Teicher for complex projective surfaces. Moreover, we study the structure of these invariants and formulate conjectures supported by calculations on new examples.
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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.
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