When is the diagonal contractible?

For a smooth projective complex variety $X$, we study the problem of when there exists a birational morphism $X\times X\to Y$ to a projective variety $Y$ contracting the diagonal $\Delta_X\subset X\times X$ to a subvariety of smaller dimension.

We prove this happens if and only if various conditions related to the Albanese morphism of $X$ are satisfied.

We also give necessary and sufficient conditions for the existence of a contraction which is an isomorphism outside the diagonal and initiate the problem of understanding contractions of diagonals in higher products.