On the geometry of the higher dimensional Voisin maps

Voisin constructed self-rational maps of Calabi-Yau manifolds obtained as varieties of $r$-planes in cubic hypersurfaces of adequate dimension. This map has been thoroughly studied in the case $r=1$, which is the Beauville-Donagi case. In this paper, we compute the action of $\mathrm{\Psi }$ on holomorphic forms for any $r$. For $r=2$, we compute the action of $\mathrm{\Psi }$ on the Chow group of $0$-cycles, and confirm that it is as expected from the generalized Bloch conjecture.