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The computation of Stiefel-Whitney classes

Pierre Guillot (2010)

Annales de l’institut Fourier

The cohomology ring of a finite group, with coefficients in a finite field, can be computed by a machine, as Carlson has showed. Here “compute” means to find a presentation in terms of generators and relations, and involves only the underlying (graded) ring. We propose a method to determine some of the extra structure: namely, Stiefel-Whitney classes and Steenrod operations. The calculations are explicitly carried out for about one hundred groups (the results can be consulted on the Internet).Next,...

The Euler number of the normalization of an algebraic threefold with ordinary singularities

Shoji Tsuboi (2004)

Banach Center Publications

By a classical formula due to Enriques, the Euler number χ(X) of the non-singular normalization X of an algebraic surface S with ordinary singularities in P³(ℂ) is given by χ(X) = n(n²-4n+6) - (3n-8)m + 3t - 2γ, where n is the degree of S, m the degree of the double curve (singular locus) D S of S, t is the cardinal number of the triple points of S, and γ the cardinal number of the cuspidal points of S. In this article we shall give a similar formula for an algebraic threefold with ordinary singularities...

The generalized Hodge and Bloch conjectures are equivalent for general complete intersections

Claire Voisin (2013)

Annales scientifiques de l'École Normale Supérieure

We prove that Bloch’s conjecture is true for surfaces with p g = 0 obtained as 0 -sets X σ of a section σ of a very ample vector bundle on a variety X with “trivial” Chow groups. We get a similar result in presence of a finite group action, showing that if a projector of the group acts as 0 on holomorphic 2 -forms of  X σ , then it acts as 0 on  0 -cycles of degree 0 of  X σ . In higher dimension, we also prove a similar but conditional result showing that the generalized Hodge conjecture for general X σ implies the...

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