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!
Best Answer
An algebraic analog of Chern-Weil theory (explicitly taking symmetric polynomials of curvature) is given by the Atiyah class. Given a vector bundle $E$ on a smooth variety we can consider the short exact sequence $$ 0\to End(E) \to A(E) \to T_X\to 0$$ where $T_X$ is the tangent sheaf and $A(E)$ is the "Atiyah algebroid" --- differential operators of order at most one acting on sections of $E$, whose symbol is a scalar first order diffop (hence the map to the tangent sheaf). A (holomorphic or algebraic) connection is precisely a splitting of this sequence, and a flat connection is a Lie algebra splitting. Now algebraically such splittings will often not exist (having a holomorphic connection forces your characteristic classes to have type $(p,0)$ rather than the $(p,p)$ you want..) but nonetheless we can define the extension class, which is the Atiyah class $$a_E\in H^1(X, End(E)\otimes \Omega^1_X).$$ This is the analog of the curvature form in the Riemannian world -- we now can take symmetric polynomials in the $End(E)$ factor to get the characteristic classes of $E$ in $H^p(X,\Omega_X^p)$ as desired.
This answer and Mariano's agree of course in the sense that Atiyah classes can be interpreted via Hochschild and cyclic (co)homology and generalized to arbitrary coherent sheaves (or complexes) on varieties (or stacks) (let me stick to characteristic zero to be safe). Namely the Atiyah class of the tangent sheaf can be used to define a Lie algebra structure (or more precisely $L_\infty$) on the shifted tangent sheaf $T_X[-1]$, and Hochschild cohomology is its enveloping algebra. This Lie algebra acts as endomorphisms of any coherent sheaf (which is another way to say Hochschild cohomology is endomorphisms of the identity functor on the derived category), and one can take characters for these modules, recovering the characteristic classes defined concretely above.
(In fact the notion of characters is insanely general... for example an object of any category - with reasonable finiteness - defines a class (or "Chern character") in the Hochschild homology of that category, which is cyclic and so descends to cyclic homology. An example of this is the category of representations of a finite group, whose HH is class functions, recovering usual characters, or coherent sheaves on a variety, recovering usual Chern character. or one can go more general.)