On the Integral Cohomology of Fano Varieties of Linear Subspaces

Abstract

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 $\mathbf{d} = (d_1, \ldots , d_s)$ say, the Fano scheme $F_r(X)$ of projective $r$-spaces contained in $X$ is a subscheme 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 $H^i(G(r + 1, n + 1); \mathbb{Z}) \rightarrow H^i(F_r(X); \mathbb{Z})$ 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.