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We show that if X is a smooth complex projective surface with torsion-free cohomology, then the Hilbert scheme 
$X^{[n]}$ has torsion-free cohomology for every natural number n. This extends earlier work by Markman on the case of Poisson surfaces. The proof uses Gholampour-Thomas’s reduced obstruction theory for nested Hilbert schemes of surfaces.
We prove the integral Hodge conjecture for all 3-folds 
$X$ of Kodaira dimension zero with 
$H^{0}(X,K_{X})$ not zero. This generalizes earlier results of Voisin and Grabowski. The assumption is sharp, in view of counterexamples by Benoist and Ottem. We also prove similar results on the integral Tate conjecture. For example, the integral Tate conjecture holds for abelian 3-folds in any characteristic.
The Hilbert scheme 
$X^{[a]}$ of points on a complex manifold 
$X$ is a compactification of the configuration space of 
$a$ -element subsets of 
$X$ . The integral cohomology of 
$X^{[a]}$ is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of 
$X^{[2]}$ for any complex manifold 
$X$ , and the integral cohomology of 
$X^{[2]}$ when 
$X$ has torsion-free cohomology.
We show that the subcategory of mixed Tate motives in Voevodsky’s derived category of motives is not closed under infinite products. In fact, the infinite product 
$\prod _{n=1}^{\infty }\mathbf{Q}(0)$ is not mixed Tate. More generally, the inclusions of several subcategories of motives do not have left or right adjoints. The proofs use the failure of finite generation for Chow groups in various contexts. In the positive direction, we show that for any scheme of finite type over a field whose motive is mixed Tate, the Chow groups are finitely generated.
