Projective resolutions of representations of GL (n).
Euler characteristics for -adic Lie groups
The total Chern class is not a map of multiplicative cohomology theories.
Birational geometry of quadrics
We construct new birational maps between quadrics over a field. The maps apply to several types of quadratic forms, including Pfister neighbors, neighbors of multiples of a Pfister form, and half-neighbors. One application is to determine which quadrics over a field are ruled (that is, birational to the projective line times some variety) in a larger range of dimensions. We describe ruledness completely for quadratic forms of odd dimension at most 17, even dimension at most 10, or dimension 14....
Pseudo-abelian varieties
Chevalley’s theorem states that every smooth connected algebraic group over a perfect field is an extension of an abelian variety by a smooth connected affine group. That fails when the base field is not perfect. We define a pseudo-abelian variety over an arbitrary field to be a smooth connected -group in which every smooth connected affine normal -subgroup is trivial. This gives a new point of view on the classification of algebraic groups: every smooth connected group over a field is an extension...
Line bundles with partially vanishing cohomology
Define a line bundle on a projective variety to be -ample, for a natural number , if tensoring with high powers of kills coherent sheaf cohomology above dimension . Thus 0-ampleness is the usual notion of ampleness. We show that -ampleness of a line bundle on a projective variety in characteristic zero is equivalent to the vanishing of an explicit finite list of cohomology groups. It follows that -ampleness is a Zariski open condition, which is not clear from the definition.
Complexifications of nonnegatively curved manifolds
Chern numbers for singular varieties and elliptic homology.
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