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Motives of some acyclic varieties. (English) Zbl 1244.14015
A smooth connected complex variety \(X\) is said to be \({\mathbb{Z}}\)-acyclic (resp. \({\mathbb{Q}}\)-acyclic) if \(X({\mathbb{C}})\), viewed as a a complex manifold, has trivial reduced integral (resp. rational) singular homology. Let \(DM_{gm}({\mathbb{C}})_{{\mathbb{Q}}}\) be Voevodsky’s triangulated category of motives with \({\mathbb{Q}}\)-coefficients. Then, assuming some “standard conjectures”’ about motives, the Hodge conjecture predicts that the Hodge realization functor from \(DM_{gm}({\mathbb{C}})_{{\mathbb{Q}}}\) to a derived category of Hodge structures is conservative.
Therefore a \({\mathbb{Q}}\)-acyclic smooth complex variety should conjecturally have a trivial rational motive.
In the case \(\dim X= 2\), due to results by Fujita, Gurjar, Pradeep and Shastry, a \({\mathbb{Z}}\)-acyclic (resp. \({\mathbb{Q}}\)-acyclic) smooth complex surface \(X\) is rational and affine and there exists an open immersion \(X\to\widetilde X\), with \(\widetilde X\) a smooth projective surface such that the boundary \(\widetilde X- X\) is a simple normal crossing divisor and each irreducible component of it is a rational curve. Using this result, the author in this paper proves the following theorem, which gives some evidence to the conjecture above, in the case of surfaces.
Theorem 1. If \(X\) is a \({\mathbb{Z}}\)-acyclic (resp. \({\mathbb{Q}}\)-acyclic) smooth complex variety of dimension 2, then the canonical morphism \(M(X)\to{\mathbb{Z}}\) (resp. \(M(X)\to{\mathbb{Q}}\)) is an isomorphism in \(DM_{gm}({\mathbb{C}})\) (resp.\(DM_{gm}({\mathbb{C}})_{{\mathbb{Q}}}\)).

14F42 Motivic cohomology; motivic homotopy theory
14R05 Classification of affine varieties
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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