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 $\mathscr C^2$ in arbitrary complex manifolds.

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Stein 流形上复杂接触结构的 H 原理

在本文中，我们介绍了奇维复流形上形式复接触结构的概念。我们的主要结果是，Stein 流形 $X$ 上的每个正式复杂接触结构与 Stein 域 $\Omega\subset X$ 上的全纯接触结构同伦，后者与 $X$ 异位。我们还在此设置中证明了参数 h 原理，类似于 Gromov 用于光滑开放流形上的接触结构的 h 原理。在 Stein 三重上，我们获得了形式复杂接触结构的完整同伦分类。我们的方法还为在任意复杂流形中沿着类 $\mathscr C^2$ 的完全真实子流形的全纯接触结构的细菌提供了参数 h 原理。