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LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 3 / May 2022
- Published online by Cambridge University Press:
- 30 July 2020, pp. 933-971
- Print publication:
- May 2022
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- Article
- Export citation
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The Beauville–Voisin conjecture for a hyperkähler manifold
$X$ states that the subring of the Chow ring 
$A^{\ast }(X)$ generated by divisor classes and Chern characters of the tangent bundle injects into the cohomology ring of 
$X$. We prove a weak version of this conjecture when 
$X$ is the Hilbert scheme of points on a K3 surface for the subring generated by divisor classes and tautological classes. This in particular implies the weak splitting conjecture of Beauville for these geometries. In the process, we extend Lehn’s formula and the Li–Qin–Wang 
$W_{1+\infty }$ algebra action from cohomology to Chow groups for the Hilbert scheme of an arbitrary smooth projective surface 
$S$.
