Simon Donaldson - Donaldson's Work

Donaldson's Work

A thread running through Donaldson's work is the application of mathematical analysis (especially the analysis of elliptic partial differential equations) to problems in geometry. The problems mainly concern 4-manifolds, complex differential geometry and symplectic geometry. The following theorems rank among his most striking achievements:

• The diagonalizability theorem (Donaldson 1983a, 1983b): if the intersection form of a smooth, closed, simply connected 4-manifold is positive- or negative-definite then it is diagonalizable over the integers. (The simple connectivity hypothesis has since been shown to be unnecessary using Seiberg-Witten theory.) This result is sometimes called Donaldson's theorem.

• A smooth h-cobordism between 4-manifolds need not be trivial (Donaldson 1987a). This contrasts with the situation in higher dimensions.

• A stable holomorphic vector bundle over a non-singular projective algebraic variety admits a Hermitian–Einstein metric (Donaldson 1987b). (A second proof of this, strongly influenced by Donaldson's work, was given by Karen Uhlenbeck and Shing-Tung Yau (Uhlenbeck & Yau 1986).)

• A non-singular, projective algebraic surface can be diffeomorphic to the connected sum of two oriented 4-manifolds only if one of them has negative-definite intersection form (Donaldson 1990). This was an early application of the Donaldson invariant (or instanton invariants).

• Any compact symplectic manifold admits a symplectic Lefschetz pencil (Donaldson 1999).

Donaldson's recent work centers on a difficult problem in complex differential geometry concerning a conjectural relationship between algebro-geometric "stability" conditions for smooth projective varieties and the existence of "optimal" Kähler metrics, typically those with constant scalar curvature. Definitive results have not yet been obtained, but substantial progress has been made (see for example Donaldson 2001).

See also Donaldson theory.