Grant: Discovery Grants Program - Individual

My proposed research focuses primarily on symmetries of complex geometric objects. The physicist, H. v. Helmholtz, considered spaces for which neighbourhoods of any two points are equivalent; i.e., the spaces are ``homogeneous''. Examples abound in mathematics, e.g., spheres (under rotations), planes (under translations), sets of vector subspaces of a fixed dimension of a given vector space (projective spaces and, more generally, flag manifolds), open orbits in flag manifolds, say of real forms, etc. One basic object we study are complex manifolds (spaces modelled locally on complex vector spaces) that have enough transformations to map any given point to any other. These symmetries form a group and we mostly restrict to settings when this is a Lie group, as there is an extensive theory concerning their structure. More precisely, suppose G is a complex Lie group and H is a closed complex subgroup of G. We seek to understand connections between the structure of the pair (G,H) and the geometric and complex analytic properties of the complex homogeneous space G/H, e.g., obstructions to the existence of objects like analytic hypersurfaces and plurisubharmonic functions on the space G/H. Another question is to determine the shape, i.e., in the complex setting this is called the Levi curvature, of the boundary of flag domains - these arise as the open orbits of certain real subgroups of some complex groups G in the context of complex homogeneous spaces G/H that are flag manifolds. An elucidation of the fine structure of K3-surfaces that admit finite groups of certain special types of symmetries is an area where we will make further significant contributions.