Some of us, especially those of us who have been hanging out with Nick Proudfoot and Roman Bezrukavnikov, are familiar with the notion of an algebraic symplectic manifold. Let me review this concept.
Recall that a symplectic manifold is a manifold along with a non-degenerate closed 2-form. An algebraic symplectic manifold is smooth algebraic variety over C (or any other field), again with a non-degenerate closed 2-form. In particular algebraic symplectic manifolds are always even complex dimensional and hence their real dimension is a multiple of 4. They are closely related to the differential geometry notion of hyperKahler manifold. There are many nice exampls of algebraic symplectic manifolds. The simplest is the cotangent bundle of a smooth variety.
The other day I was wondering whether anyone has ever considered “algebraic Riemannian manifolds”. By this I mean a smooth algebraic variety with a non-degenerate symmetric bilinear form on each tangent space (note that we drop positive definite, since this makes no sense over C).
I don’t really have much to say about such things, I just wanted to throw it out here to see if anyone can tell me whether such a concept exists or if not, why it is a bad idea to consider such a notion. So any answers?