Complex orientations of real algebraic surfaces

Details Complex orientations of real algebraic surfaces Complex orientations of real algebraic surfaces

2006-11-13

arxiv.org/abs/math/0611396v1

Advances in Soviet Math., vol. 18, 1994, 261-284

Mathematics

Algebraic Geometry

Scientific paper

We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of real points is equipped with a pair of opposite orientations and a Spin structure. If the set of real points realizes the same homology class as the complexification of a real curve on the surface, then the complement of the curve in set of real points is equipped a pair of opposite orientations, which do not extend across the curve, and the whole set of real points is equipped with a Pin^- structure. These constructions are similar to the complex orientations of real algebraic curves dividing their complexifications and generalize to high dimensions.

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