MANIFOLDS WITH MANY COMPLEX STRUCTURES

Manifolds with Many Complex Structures

Quadratic Functions and 3-manifolds with Complex Spin Structures

We investigate the role played by quadratric functions in the topology of closed oriented 3-manifolds endowed with Spin-structures. In particular, we show how they naturally emerge from a Spin-refinement of the Goussarov-Habiro theory.

Manifolds With Many Hyperbolic Planes

We construct examples of complete Riemannian manifolds having the property that every geodesic lies in a totally geodesic hyperbolic plane. Despite the abundance of totally geodesic hyperbolic planes, these examples are not locally homogenous.

Poisson Structures on Complex Flag Manifolds Associated with Real Forms

For a complex semisimple Lie group G and a real form G0 we define a Poisson structure on the variety of Borel subgroups of G with the property that all G0-orbits in X as well as all Bruhat cells (for a suitable choice of a Borel subgroup of G) are Poisson submanifolds. In particular, we show that every non-empty intersection of a G0-orbit and a Bruhat cell is a regular Poisson manifold, and we ...

Complex structures on 4-manifolds with symplectic 2-torus actions

We apply the general theory for symplectic torus actions with symplectic or coisotropic orbits to prove that a 4-manifold with a symplectic 2-torus action admits an invariant complex structure and give an identification of those that do not admit a Kähler structure with Kodaira’s class of complex surfaces which admit a nowhere vanishing holomorphic (2, 0)-form, but are not a torus or a K3 surface.