Integrality of the Chern character in small codimension
Let \(X\) be an algebraic variety, \(K_{0}^{\prime}(X)\) the Grothendieck group of coherent sheaves, \(\mathrm{CH}_{i}(X)_{ Q}\) the Chow group of \(X\) of algebraic cycles of dimension \(i\) and \(\mathrm{ch}_{i}: K_{0}^{\prime}(X) \rightarrow \mathrm{CH}_{i}(X)_{Q}\) the homological Chern character. One can state the following conjecture:NEWLINENEWLINE Conjecture. Let \(X\) be a variety, and \(n\) a positive integer. Then NEWLINE\[NEWLINEp^{[n/(p-1)]} \cdot \mathrm{ch}_{\dim X-n} [{\mathcal O}_{X}]\in \mathrm{im} (\mathrm{CH}_{\dim X-n}(X)_{Z_{(p)}}\rightarrow \mathrm{CH}_{\dim X-n}(X)_{Q_{(p)}}NEWLINE\]NEWLINE The topological analog of this was formulated and proved by \textit{J. F. Adams} [Proc. Camb. Philos. Soc. 57, 189-199 (1961; Zbl 0103.16001)]. For another easier proof of this but with more restrictive assumptions [\textit{M. F. Atiyah}, Q. J. Math., Oxf. II. Ser. 17, 165--193 (1966; Zbl 0144.44901)]. The author gives a proof of this conjecture for any \(p\) and \(n<p(p-1).\) Using Gabber's result on existence of regular alterations the author can prove the conjecture for all \(n\) in the case when the characteristic of the base field is not equal to \(p.\) The consequences of this conjecture especially to study of correspondences between algebraic varieties are also discussed. Assuming that the conjecture holds true for \(n\leq m(p-1),\) given any (possibly singular) variety \(X\) the author constructs for any \(0<i \leq m ,\) a morphism of graded abelian groups: NEWLINE\[NEWLINET_{i}: \mathrm{CH}_{\bullet}(X)\otimes {\mathbb Z}/{p} \rightarrow {\widetilde{\mathrm{CH}}}_{{\bullet}-i(p-1)}(X)\otimes {\mathbb Z}/{p}NEWLINE\]NEWLINE \({\widetilde{\mathrm{CH}}}_{{\bullet}}\) denotes here the Chow group modulo torsion. The operations \(T_{i}\) for \(1<i \leq (p-1) \) are constructed unconditionally and for all \(i,\) when the characteristics of the base field is different from \(p.\) The operations \(T_{i}\) for \(i\leq p\) correspond up to sign to the \(i\)-th Steenrod operations modulo \(p\) for \(i\leq p.\) The author proves also some formulas concerning these operations.
scientific article; zbMATH DE number 6084085
scientific article