Ordinary differential forms measure oriented infinitesimal volume. On a real smooth manifold, a $k$-form only remembers its total degree $k$. Complex geometry needs a finer measurement. A complex coordinate $z_i=x_i+i y_i$ has two natural infinitesimal covectors, $dz_i$ and $d\bar z_i$, and these behave differently under holomorphic coordinate changes. A $(p,q)$-form is the part of a differential form with $p$ holomorphic covector factors and $q$ antiholomorphic covector factors.

This distinction is structural. The equation $\bar\partial f=0$ is the coordinate-free form of holomorphicity, Dolbeault cohomology is graded by pairs $(p,q)$, and Hermitian metrics are naturally encoded by real $(1,1)$-forms. Thus $(p,q)$-forms are a basic language for complex manifolds, several complex variables, and complex versions of de Rham cohomology.

## Definition

The basic purpose of the definition is to separate a complex-valued form by the kind of complex covectors it uses. A form of type $(p,q)$ has exactly $p$ holomorphic covector factors and exactly $q$ antiholomorphic covector factors at every point. Here $(T^*M)^{1,0}$ denotes the bundle of holomorphic cotangent directions, locally spanned by the coordinate covectors $dz_i$, while $(T^*M)^{0,1}$ denotes the antiholomorphic cotangent directions, locally spanned by the $d\bar z_i$. The exterior power $\Lambda^p$ means that $p$ such covectors are wedged alternately, the [tensor product](/page/Tensor%20Product) combines the holomorphic and antiholomorphic parts over $\mathbb C$, and a section is a smooth choice of such an alternating covector at every point of $M$.

[definition: Differential Form of Type $(p,q)$] Let $M$ be a complex manifold of complex dimension $n$. For integers $p,q$ with $0\le p,q\le n$, a smooth $(p,q)$-form on $M$ is a smooth section of the complex vector bundle \begin{align*} \Lambda^{p,q}T^*M=\Lambda^p (T^*M)^{1,0}\otimes_{\mathbb C}\Lambda^q (T^*M)^{0,1}. \end{align*} The space of smooth $(p,q)$-forms on $M$ is denoted by \begin{align*} \Omega^{p,q}(M)=\Gamma\left(\Lambda^{p,q}T^*M\right). \end{align*} [/definition]

On $\mathbb C^2$, the coordinate form $dz_1\wedge d\bar z_2$ is already a $(1,1)$-form: it uses one holomorphic covector and one antiholomorphic covector. By contrast, $dz_1\wedge dz_2$ has type $(2,0)$, and $d\bar z_1\wedge d\bar z_2$ has type $(0,2)$. This local count is the fibrewise rule encoded by the bundle notation above.

[example: Reading the Bidegree] On $\mathbb C^2$, the form \begin{align*} z_1\,dz_1\wedge d\bar z_2 \end{align*} has type $(1,1)$ because its coefficient is a smooth function and its wedge factor contains one $dz$ covector and one $d\bar z$ covector. The coefficient does not change the bidegree; only the covector factors do. [/example]

This definition is compact because it refers to the holomorphic and antiholomorphic cotangent bundles. The rest of this section unpacks those ingredients, starting from the real cotangent bundle underneath the complex manifold.

### Holomorphic and Antiholomorphic Cotangent Directions

The cotangent bundle of the underlying smooth manifold is real, while the symbols $dz_i$ and $d\bar z_i$ are complex-valued covectors. To make the notation intrinsic, the real cotangent spaces are first enlarged by allowing complex scalar coefficients. The complex structure then separates the enlarged space into holomorphic and antiholomorphic parts.

[definition: Complexified Cotangent Space] Let $M$ be a smooth manifold. The complexified cotangent space at $x\in M$ is \begin{align*} T_x^*M\otimes_{\mathbb R}\mathbb C. \end{align*} The complexified cotangent bundle is \begin{align*} T^*M\otimes_{\mathbb R}\mathbb C=\bigsqcup_{x\in M}\left(T_x^*M\otimes_{\mathbb R}\mathbb C\right). \end{align*} [/definition]

The complexified cotangent space alone does not know which covectors should count as complex-linear and which should count as conjugate-linear. Without an intrinsic split, a local expression involving $dz_i$ and $d\bar z_i$ would depend on the chosen chart rather than on the complex structure of $M$.

The next definition supplies the missing intrinsic split: it singles out the two coordinate-stable subspaces that will later provide the two indices in a $(p,q)$-form. The key point is that holomorphic coordinate changes preserve the distinction between differentials of the coordinates and differentials of their complex conjugates. Thus the spans of the $dz_i$ and the $d\bar z_i$ are not just chart artifacts; they are the intrinsic covector directions that measure holomorphic and antiholomorphic variation separately.

[definition: Holomorphic and Antiholomorphic Cotangent Spaces] Let $M$ be a complex manifold of complex dimension $n$, and let $x\in M$. The holomorphic cotangent space $(T_x^*M)^{1,0}$ is the complex vector subspace of $T_x^*M\otimes_{\mathbb R}\mathbb C$ spanned in any holomorphic coordinate chart $(U,\varphi)$ with coordinates $(z_1,\ldots,z_n)$ by the covectors $dz_1,\ldots,dz_n$. The antiholomorphic cotangent space $(T_x^*M)^{0,1}$ is the complex vector subspace of $T_x^*M\otimes_{\mathbb R}\mathbb C$ spanned in the same chart by the covectors $d\bar z_1,\ldots,d\bar z_n$. [/definition]

Holomorphic coordinate changes express each new $dw_i$ as a complex linear combination of the old $dz_j$, and each new $d\bar w_i$ as a complex linear combination of the old $d\bar z_j$. This is why the two spaces above are intrinsic. Once those two spaces are available, the next step is to build alternating covectors with a prescribed number of factors of each type.

[definition: Covector of Type P Q] Let $M$ be a complex manifold of complex dimension $n$, and let $x\in M$. For integers $p,q$ with $0\le p,q\le n$, a $(p,q)$-covector at $x$ is an element of \begin{align*} \Lambda^p (T_x^*M)^{1,0}\otimes_{\mathbb C}\Lambda^q (T_x^*M)^{0,1}. \end{align*} The [vector space](/page/Vector%20Space) of all $(p,q)$-covectors at $x$ is denoted by $\Lambda_x^{p,q}M$. [/definition]

A single covector at one point is not enough for calculus. Differential operators and integration require a smoothly varying covector at every point; that is why the primary definition used sections of the bundle $\Lambda^{p,q}T^*M$.

### Local Coordinate Formula

If $p<0$, $q<0$, $p>n$, or $q>n$, the convention is $\Omega^{p,q}(M)=\{0\}$. The total degree of a $(p,q)$-form is $p+q$, but the bidegree carries additional information that is invisible on the underlying real manifold. To compute with these forms, the abstract section definition must be translated into coordinates.

[definition: Local Expression of Form of Type P Q] Let $M$ be a complex manifold of complex dimension $n$, and let $(U,\varphi)$ be a holomorphic chart with coordinates $(z_1,\ldots,z_n)$. A smooth $(p,q)$-form $\alpha\in\Omega^{p,q}(M)$ has a local expression on $U$ of the form \begin{align*} \alpha|_U=\sum_{|I|=p,\,|J|=q}\alpha_{I J}\,dz_I\wedge d\bar z_J, \end{align*} where $I=(i_1<\cdots<i_p)$, $J=(j_1<\cdots<j_q)$, $dz_I=dz_{i_1}\wedge\cdots\wedge dz_{i_p}$, $d\bar z_J=d\bar z_{j_1}\wedge\cdots\wedge d\bar z_{j_q}$, and each coefficient $\alpha_{I J}:U\to\mathbb C$ is smooth. [/definition]