Journal of Symplectic Geometry

Volume 18 (2020)

Number 3

H-principle for complex contact structures on Stein manifolds

Pages: 733 – 767



Franc Forstnerič (Faculty of Mathematics and Physics, University of Ljubljana, Slovenia; and Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia)


In this paper we introduce the notion of a formal complex contact structure on an odd dimensional complex manifold. Our main result is that every formal complex contact structure on a Stein manifold, $X$, is homotopic to a holomorphic contact structure on a Stein domain $\Omega \subset X$ which is diffeotopic to $X$. We also prove a parametric h‑principle in this setting, analogous to Gromov’s h‑principle for contact structures on smooth open manifolds. On Stein threefolds we obtain a complete homotopy classification of formal complex contact structures. Our method furnishes a parametric h‑principle for germs of holomorphic contact structures along totally real submanifolds of class $\mathscr{C}^2$ in any complex manifold.

The author is supported by the research program P1-0291 and grants J1-7256 and J1-9104 from ARRS, Republic of Slovenia.

Received 19 November 2018

Accepted 16 July 2019

Published 30 July 2020