Are two nice equidimensional varieties with the same Euler characteristic fibers of a flat morphism? I'm very amused by the ability of Euler characteristic to withstand any deformation.
I would be very impressed if one had a geometric analogue of homotopy (much rougher) that captured all pairs with the same Euler characteristic.
Recently I saw an interesting theorem that might offer such a solution at least in the context of algebraic geometry; under good conditions (everything projective etc) if $X \to Y$ is flat, then the fibers have the same Euler characteristic (of the structure sheaf).
My question is thus if under good conditions, if $X,Y$ are equidimensional with the same euler characteristic there is a flat map connecting them; a 'very rough deformation' if you will.

No, for example, a $K3$ surface and an elliptic surface have the same Eular characteristic $\chi(\mathcal{O})$ and dimension, but they can't fit in a flat family. (For smooth varieties to fit in a flat family, they are deformation equivalent, so they have to be diffeomorphic.)
For a given projective family $X\to T$, it is flat if and only if the Hilbert polynomials on the fibers are constant. So a refined question to ask is probably
Question: If two polarized projective varieties $X,Y$ have the Hilbert polynomial, is there a flat family connecting $X$ and $Y$.
