Let $M$ be a complex manifold of dimension $n$. That is, in each point of $M$ we have a complex chart $\varphi:U\longrightarrow \mathbb{C}^n$, which locally parameterizes $U\subset M$ by means of $n$ complex coordinates $z^1$, ... , $z^n$. From a differentiable point of view, $\mathbb{C}^n\simeq\mathbb{R}^{2n}$, so $M$ has also a structure of (real) differentiable manifold of dimension $2n$, locally described by the coordinates $x^1$, $y^1$, ... , $x^n$, $y^n$, with $z^j=x^j+\mathrm{i} y^j$. Associated to this structure we have the real tangent space, that may be seen as the set of derivations acting on real-valued functions:

and its dual space:

If we'd like to work with the set of derivations acting on complex-valued functions, there's only need to complexify the previous space:

and likewise for the dual space:

But for the last two spaces another base will be more useful

respect to with the following decompositions appear:

These are well-defined independently of the charts, because the derivations $\left\{\dfrac{\partial}{\partial \bar{z}^1},...,\dfrac{\partial}{\partial \bar{z}^n}\right\}$ are annihilated on holomorphic functions. This decomposition reaches the space of forms:

where $\Lambda^{p,q}(M)$ is generated by the forms with $p$ holomorphic terms and $q$ antiholomorphic terms $\omega(z)=\eta(z)\mathrm{d} z^{j_1}\wedge\cdots\wedge\mathrm{d} z^{j_p}\wedge\mathrm{d}\bar{z}^{k_1}\wedge\cdots\wedge \mathrm{d}\bar{z}^{k_q}$.

We also have the following differential operators:

that is,

These operators satisfy

Indeed,

$\partial^2$ adds two holomorphic terms, $\bar{\partial}^2$ add two antiholomorphic terms and $\partial\bar{\partial}+ \bar{\partial}\partial$ adds a term of each type, so each term has to be identically zero.

There exists a canonical isomorphism

given by the equality of derivations with respect to holomorphic functions. Multiplying by $\mathrm{i}$ in $T^{1,0}M$ induces, together with $\theta$, the following linear map:

called almost complex structure.