Exotic Holomorphic Engel Structures on $\mathbf {C}^4$

Abstract

A holomorphic Engel structure determines a flag of distributions $\mathcal {W}\subset \mathcal {D}\subset {\mathcal {E}}$. We construct examples of Engel structures on $\mathbf {C}^4$ such that each of these distributions is hyperbolic in the sense that it has no tangent copies of $\mathbf {C}$. We also construct two infinite families of pairwise non-isomorphic Engel structures on $\mathbf {C}^4$ by controlling the curves $f{:}\mathbf {C}\rightarrow \mathbf {C}^4$ tangent to $\mathcal {W}$. The first is characterised by the topology of the set of points in $\mathbf {C}^4$ admitting $\mathcal {W}$-lines and the second by a finer geometric property of this set. A consequence of the second construction is the existence of uncountably many non-isomorphic holomorphic Engel structures on $\mathbf {C}^4$.