H-principle for complex contact structures on Stein manifolds

Abstract

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 methods also furnish a parametric h-principle for germs of holomorphic contact structures along totally real submanifolds of class ${\mathcal{C}}^2$ in arbitrary complex manifolds.

J. Symplectic Geom., to appear

References in corpus (2)

• Stein surfaces as open subsets of C^2
• Darboux charts around holomorphic Legendrian curves and applications

Cited by in corpus (1)

• Mergelyan approximation theorem for holomorphic Legendrian curves