S. K. Donaldson

Polynomial invariants for smooth four-manifolds

THE TRADITIONAL methods of geometric topology have not given a clear picture of the classification of smooth 4-manifolds. This gap has been partially bridged by the introduction into 4-manifold theory of methods using Yang-Mills theory (or Gauge theory). Riemannian 4manifolds carry with them an array of moduli spaces, finite dimensional spaces of connections cut out by the first order Yang-Mills equations. These equations depend on the Riemannian geometry of the 4-manifold but at the level of homology we find properties of the moduli spaces which do not change as the metric is varied continuously. Any two Riemannian metrics can be joined by a path so, by default, these properties depend only upon the underlying smooth 4-manifold, and they furnish a mine of potential new differential topological invariants. This point of view was developed in [7], and the invariant defined there did indeed go beyond the classical ones. In fact Friedman and Morgan [13], and Okonek and Van de Ven [29] showed that this invariant could distinguish mutually distinct differentiable structures on an infinite family of homeomorphic 4-manifolds (Dolgachev surfaces). This paper takes the same ideas further. In [7] a single moduli space was used to define an invariant for manifolds with b

Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

-the rank of a maximal positive subspace for the intersection form-equal to 1. Here we will use infinite families of moduli spaces to define an infinite set of invariants for a simply connected 4-manifold X with b: odd and strictly greater than 1. These invariants are distinguished elements of the ring S*(H*(X)) of polynomials in the cohomology of the underlying 4-manifold. They can be viewed equivalently as symmetric multilinear functions:

The Seiberg-Witten equations and 4-manifold topology

This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.

Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2 and completion of the main proof

Since 1982 the use of gauge theory, in the shape of the Yang-Mills instanton equations, has permeated research in 4-manifold topology. At first this use of differential geometry and differential equations had an unexpected and unorthodox flavour, but over the years the ideas have become more familiar; a body of techniques has built up through the efforts of many mathematicians, producing results which have uncovered some of the mysteries of 4-manifold theory, and leading to substantial internal conundrums within the field itself. In the last three months of 1994 a remarkable thing happened: this research area was turned on its head by the introduction of a new kind of differential-geometric equation by Seiberg and Witten: in the space of a few weeks long-standing problems were solved, new and unexpected results were found, along with simpler new proofs of existing ones, and new vistas for research opened up. This article is a report on some of these developments, which are due to various mathematicians, notably Kronheimer, Mrowka, Morgan , Stern and Taubes, building on the seminal work of Seiberg [S] and Seiberg and Witten [SW]. It is written as an attempt to take stock of the progress stemming from this initial period of intense activity. The time period being comparatively short, it is hard to give complete references for some of the new material, and perhaps also to attribute some of the advances precisely. The author is grateful to a number of mathematicians, but most particularly to Peter Kronheimer, for explaining these new developments as they unfolded.

Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2

This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2\pi. We also put all our technical results together to complete the proof of the main theorem that if a K-stable Fano manifold admits a Kahler-Einstein metric.

Floer homology groups in Yang-Mills theory

This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2\pi. We show that these are in a natrual way projective algebraic varieties. In the case when the limiting variety and the limiting divisor are smooth we show that the limiting metric also has standard cone singularities.

Kähler Metrics with Cone Singularities Along a Divisor

1. Introduction 2. Basic material 3. Linear analysis 4. Gauge theory and tubular ends 5. The Floer homology groups 6. Floer homology and 4-manifold invariants 7. Reducible connections and cup products 8. Further directions.

Connected sums of self-dual manifolds and deformations of singular spaces

We develop some analytical foundations for the study of Kahler metrics with cone singularities in codimension one. The main result is an analogue of the Schauder theory in this setting. In the later parts of the paper we discuss connections with the existence problem for Kahler–Einstein metrics,in the positive case.

Singular Lefschetz pencils

The authors give general conditions under which the connected sum of two self-dual Riemannian 4-manifolds again admits a self-dual structure. Their techniques combine twistor methods with the deformation theory of compact complex spaces. They are related on the one hand to the analytical approach which has been used by Floer (1987) and on the other hand to the algebro-geometric results of Hitchin and Poon (1987). They give specific examples involving the projective plane and K3 surfaces.

Floer’s work on instanton homology, knots and surgery

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4–manifold equipped with a “near-symplectic” structure (ie, a closed 2–form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4–manifold (X, ω) can be decomposed into (a) two symplectic Lefschetz fibrations over discs, and (b) a fibre bundle over S 1 which relates the boundaries of the Lefschetz fibrations to each other via a sequence of fibrewise handle additions taking place in a neighbourhood of the zero set of the 2–form. Conversely, from such a decomposition one can recover a near-symplectic structure.