Let $X$ be a compact complex manifold and let $M\subset X$ be a smooth compact orientable real hypersurface which separates $X$ into two open complex manifolds $\Omega^+$ and $\Omega^-$, with

\begin{align*} X\setminus M=\Omega^+\sqcup\Omega^-,\qquad \partial\Omega^+=\partial\Omega^-=M. \end{align*}

For $\sigma\in\{+,-\}$, let $H^{0,q}(\overline{\Omega^\sigma})$ denote Dolbeault cohomology computed from smooth $(0,q)$-forms on $\overline{\Omega^\sigma}$ whose coefficients extend smoothly to the boundary, with the usual $\bar\partial$ operator up to $M$. Let $H_b^{0,q}(M)$ denote the smooth Kohn-Rossi cohomology of the induced CR structure on $M$. Then the restriction maps from $X$ to the two sides, the difference of the two boundary restrictions to $M$, and the connecting homomorphism defined by extending a boundary class to the two sides fit into a natural long exact sequence

\begin{align*} \cdots \longrightarrow H^{0,q}(X) \longrightarrow H^{0,q}(\overline{\Omega^+})\oplus H^{0,q}(\overline{\Omega^-}) \longrightarrow H_b^{0,q}(M) \longrightarrow H^{0,q+1}(X) \longrightarrow \cdots . \end{align*}

The middle arrow sends a pair of smooth up-to-boundary Dolbeault classes to the Kohn-Rossi class of the boundary jump of their tangential restrictions, using the two boundary orientations with opposite signs.