an:05998029
Zbl 1244.14015
Asok, Aravind
Motives of some acyclic varieties
EN
Homology Homotopy Appl. 13, No. 2, 329-335 (2011).
00286593
2011
j
14F42 14R05 19E15
acylic; \(\mathbb{A}^1\)-homotopy; motive
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.\(DM_{gm}({\mathbb{C}})_{{\mathbb{Q}}}\)).
Claudio Pedrini (Genova)