Real loci

Duistermaat first introduced and studied real loci. A real locus is the fixed point set of an anti-symplectic involution on a symplectic manifold. The most familiar examples are the real points of a complex projective variety. Many results of symplectic geometry apply to these Lagrangian submanifolds. I worked on several projects confirming this idea.

The mod 2 equivariant cohomology of real loci
(with Daniel Biss and Victor Guillemin) Adv. Math. 185 (2004) 370--399.
Preprint math.SG/0107151
This paper gives a combinatorial description of the equivariant cohomology of the real points of the space X
Real loci of symplectic reductions
(with Rebecca Goldin) Trans. AMS posted on the web April 2004, to appear in print.
Preprint math.SG/0302265
We discuss the real loci of symplectic reductions. We prove an analogue of Kirwan surjectivity.

The equivariant cohomology of hypertoric varieties and their real loci
(with Megumi Harada) Commun. Anal. and Geom. to appear.
Preprint math.DG/0405422
We discuss the real loci of hypertoric varieties, and relate their topology to the topology of the ambient space.
Conjugation spaces
(with Jean-Claude Hausmann and Volker Puppe) Algebr. Geom. Topol. 5 (2005) 923--964.
Preprint math.AT/0412057
We study the real loci of conjugation spaces. Our results are phrased in an algebraic context.

I am currently working on two projects in this area. With Reyer Sjamaar, I am working to understand the real loci of symplectic reductions by non-abelian groups. With Jean-Claude Hausmann, I am trying to make a more systematic study of real loci in a context somewhere between the very algebraic approach in our paper with Volker Puppe and the very symplectic approach in other works on real loci.