• Hilbert schemes
• Chow groups
• Beauville-Voisin conjecture
• algebraic cycles
• Virasoro algebra

The Beauville-Voisin conjecture for a hyperkähler variety X states that the subring of the Chow ring of X generated by the divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of X.
In the paper "Lehn's formula in Chow and conjectures of Beauville and Voisin", Maulik and Negut addressed this question when X is the Hilbert scheme of points on a K3 surface.
The aim of this thesis is to present a part of the article by Negut and Maulik: specifically, the goal is to prove that the cycle class map from the Chow ring of the Hilbert scheme to its cohomology is injective on the subring generated by the so-called small tautological classes.
The thesis is initially devoted to the construction of the Hilbert scheme, specializing in the case of the Hilbert scheme of points on a surface and generalizing the latter construction to construct parameter spaces for flags of sheaves of ideals.
Then, we study some properties of the Chow group derived from the geometric constructions carried out earlier and derive formulas connecting Chern classes of bundles of tautological ideals and some endomorphisms of the Chow group.
Finally, in the last chapter, the previous results are reorganized to construct the action of a product of an infinite-dimensional Heisenberg algebra and a Virasoro algebra on the Chow group of the Hilbert scheme of points on a K3 surface, by which we prove our main result.