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