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

TL;DR

This paper surveys how Kollár and Voisin addressed the longstanding question of whether low-dimensional algebraic cycles on smooth projective varieties can be approximated by smooth cycles, providing positive results in certain cases.

Contribution

It explains the methods used by Kollár and Voisin to show that subvarieties of dimension less than half the variety can be smoothed and approximated by smooth algebraic cycles.

Findings

Positive answer for subvarieties of dimension less than half of the ambient variety.

Techniques for smoothing algebraic cycles in low dimensions.

Advances in understanding algebraic cycle homology and smoothing.

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 $X$.