On the Integral Cohomology of Fano Varieties of Linear Subspaces

For each $n$, each dimension $r$, and each subscheme $X\subset {\mathbb{P}}^n$ defined as the common zero-locus of $s$ hypersurfaces, of degrees $\mathbf{d}=(d_1,\dots ,d_s)$ say, the Fano variety $F_r(X)$ of projective $r$-spaces contained in $X$ is a subvariety of the Grassmannian $G(r+1,n+1)$. We prove that the inclusion $F_r(X)\subset G(r+1,n+1)$ induces an isomorphism $H^i(G(r+1,n+1);\mathbb{Z})\to H^i(F_r(X);\mathbb{Z})$ on integral cohomology for certain indices $i$ (i.e., depending only on $n$, $r$, $s$ and $\mathbf{d}$). Our result extends to the integral setting a result proved for rational cohomology by Debarre and Manivel (Math. Ann. '98), and answers a question of Benoist and Voisin. Our techniques adapt ones introduced by Tu (Trans. Am. Math. Soc. '89) for a different purpose.