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arXiv · 2507.21889

When is the diagonal contractible?

Xi Chen·Frank Gounelas
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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 $\Delta_X\subset X\times X$ 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.

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Xi Chen, Frank Gounelas. 2025-07-29. When is the diagonal contractible?. https://arxiv.org/abs/2507.21889

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