I have a question about the second example in Hartshorne's Algebraic Geometry, Appendix B, section 3 (given by Hironaka?). It is an example of a compact complex Moishezon 3-fold $X$ which is not an algebraic variety, obtained by "blowing up the two branches at a node one by one locally". The way we show it is not a variety is to show that an irreducible curve $C$ in $X$ is homologically equivalent to zero, which is impossible if $X$ were a variety (because then there would exist a surface in $X$ that intersects transversely with $C$).
I get confused by this argument. I think the intersection theory for (at least) $C^{\infty}$-submanifolds on a compact orientable real manifold is well-defined and well-behaved. In particular, we have Poincar\'e duality. In the example above, if $C$ is homologically equivalent to zero (I guess for the Betti cohomology), one can always take a $C^{\infty}$-surface (i.e. not necessarily algebraic) in $X$ that intersects $C$ transversely; I don't see any obstruction in doing so. This seems to contradict the Poincar\'e duality. How do we explain this?
Also, is there a good intersection theory for algebraic cycles on a proper smooth algebraic space (over $\mathbb C$ or a general field)? For instance, I think we should have the notion of the fundamental class of an algebraic cycle (in Betti cohomology for $\mathbb C$ and at least \'etale cohomology for a general field), Poincar\'e duality, and some normalization condition (i.e. for transversal intersections, we have ...). If we do have a good theory, then it seems that arguments in Hartshorne loc. cit. would lead to the conclusion that $X$ is not an algebraic space either, but it is.
I heard before that even for (say proper smooth) Deligne-Mumford stacks, a good intersection theory exists (with fractional coefficients). If I am right, can any one provide some references on this? Thanks.
Edit: Maybe the special point $P$ in the example above is not a scheme-like point, so that there is no open affine neighborhood.
