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 defined by the short exact normal sequence

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

Let $D$ be the first-order analytic germ whose local algebra is $\mathbb C[\varepsilon]/(\varepsilon^2)$. An embedded first-order deformation of $Y$ in $X$ over $D$ means a closed complex subspace $\mathcal Y\subset X\times D$ that is flat over $D$, together with an identification of its central fibre $\mathcal Y\times_D \{0\}$ with $Y\subset X$. Let $\operatorname{Def}^{\mathrm{emb}}_{Y/X}(D)$ denote the set of such deformations modulo equality as closed subspaces of $X\times D$ preserving the central fibre identification. Its vector-space structure is induced locally in adapted coordinates by addition and scalar multiplication of the first-order normal coefficients. Equivalently, if $(z_1,\dots,z_n,w_1,\dots,w_r)$ are adapted holomorphic coordinates in which $Y$ is given by $w_1=\cdots=w_r=0$, then a deformation is locally represented by equations $w_a=\varepsilon f_a(z)$ with holomorphic functions $f_a$ on $Y$. A first-order reparametrisation of a parametrized graph is an automorphism of $Y\times D$ inducing the identity on the central fibre; it changes the associated first-order section of $i^*T_X$ by an element of $di(T_Y)$ and hence does not change the embedded closed subspace or its normal coefficient. Then there is a natural vector-space isomorphism

\begin{align*} \operatorname{Def}^{\mathrm{emb}}_{Y/X}(D)\cong H^0(Y,N_{Y/X}). \end{align*}