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Complex Projective Contact Manifolds

Final Report Summary - CONTACT MANIFOLDS (Complex Projective Contact Manifolds)

The project's main objectives are to study contact Fano manifolds. These are very special objects in complex algebraic geometry of great importance because of their relations with quaternion-Kaehler manifolds and thus with the last unsolved case from the famous 50-year old list of Berger of all the possible holonomy types of Riemannian manifolds. LeBrun-Salamon conjecture expects that there is no non-homogeneous contact Fano manifold and by consequence no compact non-homogeneous quaternion-Kaehler manifold. We use the techniques of minimal rational curves, Legendrian subvarieties, representation theory of Lie groups, and Moishezon spaces to approach this problem.

Further objectives are related to toric geometry and secant varieties. We pose the problem of coordinate description of maps between toric varieties. We investigate secant varieties of the Segre products, of Veronese reembeddings and provide an answer to a 14-year-old question of Eisenbud. Furthermore, we initiate the study of secant varieties to Lagrangian Grassmannians.

The researcher has undergone training in Riemannian geometry and quaternion-Kaehler manifolds, in rational curves on Fano varieties, in exterior differential systems, in representation theory of Lie groups and Lie algebras and in homogeneous varieties. Furthermore, the researcher improved his skills in toric geometry and computational and algorithmic algebra and in Hilbert schemes of points and Gorenstein algebras.

The researcher proved that any contact Fano manifold shares much of the structure of a homogeneous manifold. Specifically, the Killing form, the Lie algebra grading and some part of the Lie bracket can be constructed in terms of the geometry of a contact Fano manifold, and thus these notions extend to arbitrary contact Fano manifolds. This is further supporting evidence for the LeBrun-Salamon conjecture. The techniques used in the article involve contact lines, as well as their chains, and varieties of minimal rational tangents. A very detailed understanding of the representation theory involved in the homogeneous cases was necessary for obtaining the result.

In a collaborative work with Thomas Peternell, the researcher proved that any compact complex contact manifold (with mild and reasonable assumptions) of dimension three that has second Betti number equal to one must be a projective space. This result generalises an old result of Ye for projective three folds, and is the first step in the search of non-homogeneous examples, which might not be projective.

Either proving or disproving the LeBrun-Salamon conjecture would be a big advance to Riemannian geometry, and would give a final response to a 50-year-old problem. The two above results constitute a progress in this direction and it is very much appreciated.

Toric geometry is an important branch of algebraic geometry, where potentially complicated algebraic varieties are represented by relatively simple combinatorial objects, such as lattice polytopes. Toric varieties are important, not only on their own, but also they are ambient spaces where other objects live. In the project, we have finished the work of describing algebraic maps between toric varieties in their homogeneous coordinates and wrote a computer code that allows explicit computations using these descriptions.

There is a huge interest in classical and recent questions about the secant varieties to certain (usually homogeneous) varieties. This interest is motivated by numerous applications to statistics, signal processing and other areas of science. The concepts of rank and border-rank of tensors, symmetric tensors or skew-symmetric tensor, are naturally interpreted in geometric terms. Embeddings of Segre products of projective spaces, Veronese varieties and Pluecker Grassmannians, and their secant varieties are among the most intensively studied problems in algebraic geometry. In the project (collaborative work with Joseph Landsberg), we investigate secant varieties of the Segre products and suggest a new point of view on them. We classify points in low dimensions and low ranks. We also study secant varieties of Veronese reembeddings and investigate a 14-year old question of Eisenbud (collaborative works with Weronika Buczynska, Adam Ginensky, and Joseph Landsberg). The questions concerns defining equations of secant varieties to high degree Veronese reembeddings. We presented an answer to this question in terms of smoothability of finite Gorenstein schemes. We also investigated minimal decompositions of monomials into sums of powers of linear forms.

Furthermore, we have calculated the dimensions of some secant varieties to Lagrangian Grassmannian (joint work with Ada Boralevi).