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

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 $\mathrm{K}\mathrm{e}\mathrm{r}\gamma$ 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 $\mathcal{O}({\bigwedge }^k\mathrm{K}\mathrm{e}\mathrm{r}\gamma )$ and $$\mathcal{O}$(L\otimes \bigwedge^{k-1} {\rm Ker}, \gamma)as sheaves of {\it$\mathbb{C}$$-module}. The theorem induces the short exact sequence of cohomology of holomorphic $k$-vectors, and we obtain vanishing theorems for the cohomology of $\mathcal{O}({\bigwedge }^k\ker \gamma )$.