For a smooth projective complex variety X, we study the problem of when there exists a birational morphism \(X\times X\rightarrow Y\) 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.
