Algebraic Legendrian Varieties

TL;DR

This paper studies complex Legendrian subvarieties, especially in projective space, revealing their automorphism groups and constructing numerous new examples with diverse geometric properties.

Contribution

It determines the automorphism groups of Legendrian subvarieties and introduces a general construction method for generating many new examples in various dimensions.

Findings

Automorphism groups are fully characterized by sections of a line bundle.

Constructed a smooth toric surface as a Legendrian subvariety.

Produced a smooth quasihomogeneous Fano 8-fold with Legendrian embedding.

Abstract

Real Legendrian subvarieties are classical objects of differential geometry and classical mechanics and they have been studied since antiquity. However, complex Legendrian subvarieties are much more rigid and have more exceptional properties. The most remarkable case is the Legendrian subvarieties of projective space and prior to the author's research only few smooth examples of these were known. The first series of results of this thesis is related to the automorphism group of any Legendrian subvariety in any projective contact manifold. The connected component of this group (under suitable minor assumptions) is completely determined by the sections of the distinguished line bundle on the contact manifold vanishing on the Legendrian variety. Moreover its action preserves the contact structure. The second series of results is devoted to finding new examples of smooth Legendrian subvarieties of projective space. The contribution of this thesis is in three steps: First we find an example of a smooth toric surface. Next we find a smooth quasihomogeneous Fano 8-fold that admits a Legendrian embedding. Finally, we realise that both of these are special cases of a very general construction: a general hyperplane section of a smooth Legendrian variety, after a suitable projection, is a smooth Legendrian variety of smaller dimension. By applying this result to known examples and decomposable Legendrian varieties, we construct infinitely many new examples in every dimension, with various Picard rank, canonical degree, Kodaira dimension and other invariants.