Real loci of non-real varieties - Oleg Viro

Mathematics Department Colloquium - April 18, 2024
Stony Brook University
Oleg Viro, Stony Brook University
Title: Real loci of non-real varieties
Abstract: A smooth submanifold of a manifold is called cooriented, if its normal bundle is oriented. While oriented submanifolds realize integer homology classes, cooriented submanifolds realize integer cohomology classes. Cooriented and oriented submanifolds are very much like cocycles and cycles with integer coefficients. In particular, they may have integer intersection or linking numbers even in a non-oriented ambient manifold.
Coorientations naturally appear in real algebraic geometry. This happens if X is a real variety with complexification X_C and A is a transverse intersection of X with a complex subvariety V of X_C. Then A is a real subvariety of X of a very special kind. For example, if X is real projective space and A has codimension 2, then A is the base of a pencil of real hypersurfaces. The coorientation of A is related to the geometry of the pencil. There is an explicit upper bound for the linking number of A with a real curve B bounding in its complexification.