School of Mathematics & Statistics

Complex Geometry

 

Complex geometry is the intersection of algebraic and differential geometry. Many of the motivating questions ask differential-geometric questions about algebraic varieties, or use analytic methods to prove results in algebraic geometry. A particularly rich direction, which can be viewed as the higher dimensional analogue of the uniformisation theorem for Riemann surfaces, asks for the existence of Riemannian metrics with special curvature properties on a given variety. These special metrics do not always exist, and their existence is conjectured by Yau-Tian-Donaldson to be characterised by a condition in algebraic geometry called K-stability of a variety. This conjecture constitutes one of the deepest links between algebraic geometry, differential geometry and geometric analysis. Miraculously, this same condition of K-stability is expected to be the right one to construct moduli spaces of varieties in higher dimensions; these moduli spaces have points which are naturally in bijection with K-stable varieties. A very active area of complex geometry involves proving such a correspondence in certain cases, understanding the geometric meaning of K-stability, and constructing and studying resulting moduli spaces.

Image courtesy of John McCarthy.