On the Integral Cohomology of Fano Varieties of Linear Subspaces

2025/12/08 by Diamond, Benjamin E.
Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Geometry and complex manifolds

paper · doi:10.48550/arxiv.2512.07572

Abstract

For each n, each dimension r, and each subscheme X ⊂ ℙⁿ defined as the common zero-locus of s hypersurfaces, of degrees d = (d₁, … , 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) ⊂ G(r + 1, n + 1) induces an isomorphism Hⁱ(G(r + 1, n + 1); ℤ) → Hⁱ(F_r(X); ℤ) on integral cohomology for certain indices i (i.e., depending only on n, r, s and 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.

Related

• Schémas de Fano
• Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces
• On the degree of Fano schemes of linear subspaces on hypersurfaces
• Fano varieties and linear sections of hypersurfaces
• Cohomology of hypersurfaces of weighted projective space and the intersection form on H²