TITLE: Algebraic (semi-) Riemannian geometry ?
QUESTION [28 upvotes]: I hope these are not to vague questions for MO. 
Is there an analog of the concept of a Riemannian metric, in algebraic geometry? 
Of course, transporting things literally from the differential geometric context, we have to forget about the notion of positive definiteness, cause a bare field has no ordering. So perhaps we're looking to an algebro geometric analog of semi- Riemannian geometry.
Suppose to consider a pair $(X,g)$, where $X$ is a (perhaps smooth) variety and $g$ is a nondegenerate section  of the second symmetric power of the tangent bundle (or sheaf) of $X$.
What can be said about this structure? Can some results of DG be reproduced in this context? Is there a literature about this things?

REPLY [4 votes]: 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.