Algebraic (semi-) Riemannian geometry ?

Joel Kamnitzer had a very similar question a couple years ago, that prompted a nice discussion at the Secret Blogging Seminar. I'm afraid no one ended up citing any literature, and I have been unable to find anything with a quick Google search, but that doesn't rule out the possibility of existence.

This topic in the affine case is extensively studied in Ernst Kunz unpublished book "Algebraic Differential Calculus". You can get it as a collection of several PS files at his webpage (scroll to the bottom):

Kunz' webpage

If a holomorphic Riemannian metric $g=g_{ij}(z) dz^i dz^j$ on a compact Kaehler manifold $X$ is everywhere nondegenerate, then the metric has a holomorphic Levi-Civita connection, so the Atiyah class of the tangent bundle of $X$ is zero. Therefore a finite etale cover of $X$ is a complex torus, and the metric pulls back to be translation invariant. Hence $X$ is a quotient of such a torus by a finite group acting as affine isometries without fixed points. On the other hand, if you allow degeneracies of the holomorphic Riemannian metric, I suppose anything could happen. If you allow $X$ to be a compact complex manifold, perhaps not Kaehler, then you might look at the papers of Sorin Dumitrescu where you find a low dimensional classification.