Splitting theorem for sheaves of holomorphic $k$-vectors on complex contact manifolds

Abstract

A complex contact structure $\gamma$ is defined by a system of holomorphic local $1$-forms satisfying the completely non-integrability condition. The contact structure induces a subbundle $\mathrm{Ker}\gamma$ of the tangent bundle and a line bundle $L$. In this paper, we prove that the sheaf of holomorphic $k$-vectors on a complex contact manifold splits into the sum of $\mathcal{O}({\bigwedge }^k\mathrm{Ker}\gamma )$ and $\mathcal{O}(L\otimes {\bigwedge }^{k-1}\mathrm{Ker}\gamma )$ as sheaves of {\it $\mathbb{C}$-module}. The theorem induces the short exact sequence of cohomology of holomorphic $k$-vectors, and we obtain vanishing theorems for the cohomology of $\mathcal{O}({\bigwedge }^k\ker \gamma )$.