Let $X$ be a complex manifold, let $i:Y\hookrightarrow X$ be a closed compact complex submanifold, and let $N_{Y/X}$ be the holomorphic normal bundle on $Y$ defined by the short exact normal sequence

\begin{align*} 0\to T_Y\to i^*T_X\to N_{Y/X}\to 0. \end{align*}

Let $D_{Y/X}$ denote the germ at the point $[Y]$ of the Douady space of compact complex subspaces of $X$, equivalently the local analytic germ representing embedded deformations of $Y$ in $X$ over Artinian local $\mathbb C$-algebras. Cohomology groups $H^j(Y,N_{Y/X})$ are sheaf cohomology groups of the sheaf of holomorphic sections of $N_{Y/X}$. If

\begin{align*} H^1(Y,N_{Y/X})=0, \end{align*}

then $D_{Y/X}$ is smooth at $[Y]$, and its Zariski tangent space at $[Y]$ is naturally isomorphic to $H^0(Y,N_{Y/X})$.