Smoothing low-dimensional algebraic cycles [after Koll\'ar and Voisin]

Olivier Benoist

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arXiv · 2407.04000

Smoothing low-dimensional algebraic cycles [after Koll\'ar and Voisin]

Olivier Benoist

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Abstract

Let $X$ be a smooth projective complex algebraic variety. An old question of Borel and Haefliger asks whether any (possibly singular) algebraic subvariety of $X$ is homologically equivalent to a linear combination with integral coefficients of smooth algebraic subvarieties of $X$. In general, this question is too optimistic, and counterexamples have been known for a long time. The aim of this survey is to explain how J\'anos Koll\'ar and Claire Voisin have provided a positive answer to Borel and Haefliger's question, for subvarieties of dimension less than half the dimension of $X$.

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Olivier Benoist. 2024-07-04. Smoothing low-dimensional algebraic cycles [after Koll\'ar and Voisin]. https://arxiv.org/abs/2407.04000

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