Consider an involution of a smooth projective variety over a field of characteristic not two. We look at the relations between the variety and the fixed locus of the involution from the point of view of cobordism. We show in particular that the fixed locus has dimension larger than its codimension when certain Chern numbers of the variety are not divisible by two, or four. Some of those results, but not all, are analogues of theorems in algebraic topology obtained by Conner-Floyd and Boardman in the sixties. We include versions of our results concerning the vanishing loci of idempotent global derivations in characteristic two. Our approach to cobordism, following Merkurjev's [Mer02], is elementary, in the sense that it does not involve resolution of singularities or homotopical methods.
Abstract. We construct a weak version of the homological first Steenrod square, a natural transformation from the modulo two Chow group to the Chow group modulo two and two-torsion. No assumption is made on the characteristic of the base field. As an application, we generalize a theorem of Nikita Karpenko on the parity of the first Witt index of quadratic forms to the case of a base field of characteristic two.
We investigate some relations between the duality and the topological
filtration in algebraic K-theory. As a result, we obtain a construction of the
first Steenrod square for Chow groups modulo two of varieties over a field of
arbitrary characteristic. This improves previously obtained results, in the
sense that it is not anymore needed to mod out the image modulo two of torsion
integral cycles. Along the way we construct a lifting of the first Steenrod
square to algebraic connective K-theory with integral coefficients, and
homological Adams operations in this theory. Finally we provide some
applications to the Chow groups of quadrics.Comment: To appear in Math. Ann. The numbering of the statements has been
modified, in order to be compatible with the published versio
We prove an integrality property of the Chern character with values in Chow
groups. As a consequence we obtain, for a prime number p, a construction of the
p-1 first homological Steenrod operations on Chow groups modulo p and p-primary
torsion, over an arbitrary field. We provide applications to the study of
correspondences between algebraic varieties.Comment: Correct some typos; add an appendix extending the results to schemes
of finite type over a regular bas
Abstract. Using Lipman's results on resolution of two-dimensional singularities, we provide a form of resolution of singularities in codimension two for reduced quasi-excellent schemes. We deduce that operations of degree less than two on algebraic cycles are characterised by their values on classes of regular schemes. We provide several applications of this "detection principle", when the base is an arbitrary regular excellent scheme: integrality of the Chern character in codimension less than three, existence of weak forms of the second and third Steenrod squares, Adem relation for the first Steenrod square, commutativity and Poincaré duality for bivariant Chow groups in small degrees. We also provide an application to the possible values of the Witt indices of non-degenerate quadratic forms in characteristic two.
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