Smoothing low-dimensional algebraic cycles [after Kollár and Voisin]

2024/07/04 by Olivier Benoist, Benoist, Olivier
Computer Science · Mathematics · #14C25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2407.04000

openalex publication_date 2024/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 János Kollár 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.

Related

• On the smoothability problem with rational coefficients
• Smoothing of 1-cycles over finite fields
• Flat pushforwards of Chern classes and the smoothability of cycles below the middle dimension
• Projective smoothing of varieties with simple normal crossings
• Reconstructing a variety from its topology (after Kollár, building on earlier work of Lieblich and Olsson)