# The Raymond and Belerly Sackler Distinguished Lecture

## Singular Points of Complex Surfaces - Geometry and Smoothing

Abstract

 Suppose a complex polynomial $f(x,y,z)$ has a zero and an isolated critical point at the origin. Then the two-dimensional complex hypersurface $V=\{f=0\}$ has a singular point at $0$. Topologically, it is the cone over its neighborhood boundary $\Sigma$, the compact $3$-manifold which is the intersection of $V$ with a small sphere in $\C^3$. $\Sigma$ may be understood explicitly by resolution of singularities, and its intrinsic topology gives much (but not enough!) information about geometry of the singular point. Smoothing" of the singularity takes place by considering $\{f=\delta\}$, and the change in topology as $\delta$ goes to $0$ is described by the Milnor fibre" $M$ (the intersection of $\{f=\delta\}$ with a small ball); it is a compact $4$-manifold with boundary $\Sigma$. Milnor's classical work proves that $M$ is simply-connected, and describes its homotopy type. Our goal is consider the more general (and richer) situation of a smoothing of a normal surface singularity $(V,0)$". The local topology of $V$ is again determined by its neighborhood boundary $\Sigma$, and a smoothing (if one exists) gives a $4$-manifold $M$ with boundary $\Sigma$. The questions and techniques which arise cover diverse areas in geometry, including: Fintushel-Stern' s rational blow-down" in symplectic geometry; construction of new algebraic surfaces of general type with $p_g=0$; and log-terminal singularities, introduced by Koll\'ar and Mori. Of special interest in geometry will be the short list of examples of singular points possessing smoothings for which the Milnor fibre is particularly simple", i.e. has the same rational homology as a disk. These lecture will be as self-contained as possible; we will assume essentially no prior knowledge of the subject.