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On the coniveau of rationally connected threefolds

### Claire Voisin

Geometry & Topology 26 (2022) 2731–2772
DOI: 10.2140/gt.2022.26.2731
##### Abstract

We prove that the integral cohomology modulo torsion of a rationally connected threefold comes from the integral cohomology of a smooth curve via the cylinder homomorphism associated to a family of $1$–cycles. Equivalently, it is of strong coniveau $1$. More generally, for a rationally connected manifold $X$ of dimension $n$, we show that the strong coniveau ${Ñ}^{n-2}{H}^{2n-3}\left(X,ℤ\right)$ and coniveau ${N}^{n-2}{H}^{2n-3}\left(X,ℤ\right)$ coincide for cohomology modulo torsion.

##### Keywords
Cohomology, coniveau, birational invariants, rationally connected manifolds
##### Mathematical Subject Classification
Primary: 14C25, 14E08, 14F99, 14J30, 14M22