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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.$$DM_{gm}({\mathbb{C}})_{{\mathbb{Q}}}$$).

MSC:
 14F42 Motivic cohomology; motivic homotopy theory 14R05 Classification of affine varieties 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects)
Keywords:
acylic; $$\mathbb{A}^1$$-homotopy; motive
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