Applications of special real geometry in Kähler geometry

• Applicant: Dr. David Lindemann
• Subject Area: Mathematics
• Term: since 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 470013099

The geometric structure of projective special real manifolds, that is hypersurfaces contained in positive level sets of hyperbolic polynomials with positive definite centro-affine fundamental form, appear naturally in both multiple fields of mathematics and theoretical physics. The main goal of this project in the context of a Walter Benjamin fellowship is the study of their applications to the geometry of Kähler cones, in particular in the context of toric varieties, and in quaternionic Kähler geometry. While the first point is a largely unexplored field, there are already some important known results in the applications of special real geometry in quaternionic Kähler geometry. For example, the first known explicit examples of complete locally inhomogeneous non-compact quaternionic Kähler manifolds were constructed with the help of the so-called supergravity q-map. This construction concretely uses the geometric properties of the underlying projective special real manifolds. In this part of the project we will study applications of new results in special real geometry in this setting. The other main goal of this project, that is to better understand the geometry of Kähler cones of toric varieties, is motivated by a geometric flow called the Kähler Ricci flow. More specifically, we will study its possible limit behaviour in the time-incomplete cases using new results about the asymptotic geometry of projective special real manifolds. Furthermore we expect to obtain first results about which projective special real manifolds can be realised as hypersurfaces in Kähler cones.

• DFG Programme: WBP Fellowship
• International Connection: Denmark
• Host: Professor Dr. Andrew Francis Swann