Simple spaces elucidate complicated structures
Toric geometry, a branch of algebraic geometry, makes use of relatively simple discrete structures to represent potentially complicated algebraic varieties. The latter are important as ambient spaces where other objects live. The 'Complex projective contact manifolds' (Contact manifolds) project is working to classify complex projective contact Fano manifolds and quaternion-Kähler manifolds with positive scalar curvature. The EU-funded project also aims to classify smooth sub-varieties of projective space with specific properties such as smooth duality. This refers to geometric transformations, where points are replaced by lines and lines by points, without altering the object's properties of incidence or subset meeting points. Having detailed a number of objectives to guide their work, researchers are working to further differentiate between the geometric properties of quaternion-Kähler manifolds and the algebro-geometric properties of complex contact Fano manifolds. Experimental results have so far proved that any contact Fano manifold shares much of the structure of a homogeneous manifold. That is, the geometry of a contact Fano manifold can be used to construct various other algebraic notions such as the Killing form (symmetric bilinear form), the Lie algebra grading, and some part of the Lie bracket. Team members have succeeded in describing algebraic maps between toric varieties and their homogeneous coordinates. They have also written a computer code that enables the execution of explicit computations on the strength of these descriptions. In ongoing work, project partners are investigating secant varieties of the Segre products with the aim of suggesting a new point of view on them. The Contact manifolds project expects to achieve further extensions of results obtained on contact manifolds and Legendrian varieties.