I am not an expert, but there seems to be an enormous technical difference between algebraic geometry and differential/metric geometry stemming from the fact that there is apparently no such thing as curvature in the former context while curvature is everywhere in the latter (indeed, it is hard to produce nontrivial results in Riemannian geometry that DON'T involve curvature). Of course it seems unreasonable to just port definitions of curvature into an algebraic context, but maybe there are constructions that play the same role in algebraic geometry that curvature does in other kinds of geometry. Here are two specific ways the notion of curvature shows up which I can imagine making sense in more general contexts.
Algebraic Chern - Weil Theory? In differential geometry one uses the curvature of a connection on a vector bundle to produce explicit cohomology classes. Does this have an algebraic analogue?
Algebraic Curvature Bounds? One supremely important theme in modern geometry involves proving theorems that depend only on the large scale geometry of a space, and the main strategy is to compare the space to a simpler one with the same large scale properties. This reminds me a little bit of tropical geometry wherein one replaces an algebraic variety with a simple combinatorial proxy, but from what little I know the analogy seems to stop there.
Any thoughts? I hope this question is not too vague, but it seems worthwhile and part of the problem is that I can't formulate a precise question along these lines. Thanks in advance!