Projectively deformable Legendrian surfaces

TL;DR

This paper introduces a new class of projective Legendrian surface deformations in complex projective space, characterizes surfaces with maximal deformation families, and constructs explicit examples involving blow-ups of the projective plane.

Contribution

It defines -deformations for Legendrian surfaces and provides a differential geometric characterization of maximally deformable surfaces, with explicit examples involving del Pezzo surfaces.

Findings

Maximal -deformable Legendrian surfaces are characterized geometrically.

Explicit examples include Legendrian maps from blown-up -surfaces.

Deformations are related to systems of cubics through specific points.

Abstract

Consider an immersed Legendrian surface in the five dimensional complex projective space equipped with the standard homogeneous contact structure. We introduce a class of fourth order projective Legendrian deformation called \emph{$\,\Psi$-deformation}, and give a differential geometric characterization of surfaces admitting maximum three parameter family of such deformations. Two explicit examples of maximally $\, \Psi$-deformable surfaces are constructed; the first one is given by a Legendrian map from $\, \PP^2$ blown up at three distinct collinear points, which is an embedding away from the -2-curve and degenerates to a point along the -2-curve. The second one is a Legendrian embedding of the degree 6 del Pezzo surface, $\, \PP^2$ blown up at three non-collinear points. In both cases, the Legendrian map is given by a system of cubics through the three points, which is a subsystem of the anti-canonical system.