Complex orientations of real algebraic surfaces
Abstract
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.
Cite
@article{arxiv.math/0611396,
title = {Complex orientations of real algebraic surfaces},
author = {Oleg Viro},
journal= {arXiv preprint arXiv:math/0611396},
year = {2007}
}