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LEHN’S FORMULA IN CHOW AND CONJECTURES OF BEAUVILLE AND VOISIN
Published online by Cambridge University Press: 30 July 2020
Abstract
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
$S$.
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- © The Author(s) 2020. Published by Cambridge University Press
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