Algebraic cycles and topology of real algebraic varieties (CWI Tract, 129) |  | Author: J. Van Hamel
Publisher: CWI Tract
Category: Book
Buy New: $14.99
Media: Paperback
ISBN: 9061964938
Publication Date: 2000
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Product Description
This work studies the question which part of the homology of a real algebraic variety is represented by real algebraic subvarieties. In other words, it concerns the image of the real algebraic cycle map. The approach taken here is to consider the associated complex variety on which the Galois group acts via complex conjugation. The set of fixed points under this Galois action is then the original real algebraic variety. I develop a Galois equivariant homology theory, which admits maps into the ordinary homology of the total space (in this case the complex variety) and into the homology of the fixed point set (in this case the real variety). The real algebraic cycle map factors via this equivariant cohomology. This construction gives non-trivial restrictions on the image of the cycle map, purely depending on the equivariant topology of the complexification. In codimension 1 there is an equivariant Lefschetz (1,1)-theorem, allowing to compute in principle the image of the real algebraic cycle map for divisors on nonsingular projective varieties. This is applied to several concrete cases, in particular to real Enriques surfaces and to complex projective varieties endowed with their underlying real algebraic structure. Thomas Stieltjes Award. This book was awarded by the Dutch Research School `Thomas Stieltjes Institute for Mathematics' with a prize for the best 1997 thesis written in this Research School.
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