On the Integral Cohomology of Fano Varieties of Linear Subspaces

TL;DR

This paper investigates the integral cohomology of Fano varieties of linear subspaces within hypersurfaces, extending known rational cohomology results to the integral setting and answering longstanding questions.

Contribution

It proves that the inclusion of Fano varieties into Grassmannians induces isomorphisms on integral cohomology in certain degrees, extending previous rational cohomology results.

Findings

Integral cohomology of Fano varieties matches that of Grassmannians in specific degrees.

Extends rational cohomology results to the integral setting.

Answers a question posed by Benoist and Voisin.

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 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 $H^i(G(r + 1, n + 1); \mathbb{Z}) \rightarrow H^i(F_r(X); \mathbb{Z})$ 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.