Canonical metrics in complex geometry

In this post, I’ll attempt to provide an intorduction to the study of canonical (or extremal) metrics in complex geometry. The emphasis of this post is on the differential geometry side of the picture, and not the algebro-geometric side. In particular, there are interesting links to moduli theory which I won’t cover. Finally, this is supposed to be an overview post, and so I won’t cover the geometric analysis aspects on the subject.

Schemes from varieties

In scheme-theoretic algebraic geometry, we can define a variety as an reduced1 separated scheme which is finite type over a field $k$. In this post, we’ll start off with the notions of affine and projective varieties from classical algebraic geometry, and then generalise these incrementally, to end up at the definition of a scheme. First of all, recall that we have a bijection between affine varieties and finitely generated reduced $k$ algebras, given by sending an affine variety $X$ to it’s ring of regular functions $k[X]$.

Functor of points

One idea which has been coming up a lot in the SMSTC course on Algebraic Geometry, lectured by Clark Barwick, has been the idea of the functor of points. Here, we’ll try and motivate that definition with some examples. Throughout, we work over an algebraically field $k$, and so schemes and morphisms are relative to $k$. Let $X$ be a scheme. Associated to this, we have a functor $h_X : \mathrm{Sch}/k^{\mathrm{op}} \to \mathrm{Sets}$, given by $$h_X(Y) = \mathrm{Mor}(Y, X)$$ and this is called the functor of points.