When is the diagonal contractible?

Abstract

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 $\hat{I}{”}_X\subset X\times X$ 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.

Missing condition in Thm 1.4 and further examples added. New Theorem 1.9 about contractions of other diagonals in higher products added. Final version, to appear in Math. Zeit