Canonical bundle ($K_X$)

The canonical bundle of a complex $n$-manifold is the holomorphic line bundle $K_X={\bigwedge }^n(T^{1,0}X{)}^{\ast }$.

If a meromorphic top form has at most a simple pole along a smooth complex analytic hypersurface $Y=\{z=0\}$, write $\alpha =(dz/z)\wedge \beta$ with $\beta$ holomorphic. Its residue is $\beta {|}_Y$, a holomorphic section of the canonical bundle $K_Y$. A change of defining coordinate $t=az$, with $a$ holomorphic and nonzero, changes $dt/t$ by the holomorphic form $da/a$ and leaves this restriction invariant. Placing the normal factor first fixes the sign convention.

For a smooth hypersurface $Y=\{f=0\}\subset {\mathbb{CP}}^n$ of degree $n+1$, let $E=\sum z_i{\partial }_{z_i}$ and $\Omega =d{z}_0\wedge \cdots \wedge d{z}_n$. The meromorphic form $({\iota }_E\Omega )/f$ descends to projective space because its scaling weights cancel and it is horizontal. Its Poincaré residue is a nowhere-zero section of $K_Y$. On the chart $z_j=1$, with remaining coordinates $t_1,\dots ,t_n$ in original-index order and $F=f{|}_{z_j=1}$, it is $(-1{)}^{j+r-1}(d{t}_1\wedge \cdots \widehat{d{t}_r}\cdots \wedge d{t}_n)/F_{t_r}$ wherever $F_{t_r}\ne 0$. Thus $K_Y\cong {\mathcal{O}}_Y$.

A section of the cube of the canonical bundle, locally $c=\varphi (z)d{z}^3$ with holomorphic coefficient. The flat coordinates obtained by integrating cube roots have transitions $w\mapsto \zeta w+b$, ${\zeta }^3=1$; general real-linear deformations do not preserve these as holomorphic transitions.

A section of the square of the canonical bundle, locally $q=\varphi (z)d{z}^2$; its coefficient changes by the square of the coordinate derivative. A holomorphic quadratic differential has holomorphic coefficient, and its flat metric of a quadratic differential is $|q{|}^{1/2}$.

For $q=\varphi (z)d{z}^2$, the length element is $|\varphi (z){|}^{1/2}|dz|$, locally Euclidean in flat coordinates. Zeros give cone points, not punctures. Its infimal length in a homotopy class need not determine the hyperbolic length of that class.

The area of the flat metric of a quadratic differential is ${\int }_X|q|$. Multiplying $q$ by a scalar $a$ multiplies this area by $|a|$ and lengths by $|a{|}^{1/2}$. It is preserved by the SL2R action on differentials.

A surface with local Euclidean charts whose changes of coordinate are $w\mapsto \pm w+b$, outside finitely many cone points. A nonzero holomorphic quadratic differential defines such a surface; a zero of order $m$ has cone angle $(m+2)\pi$.

A quadratic differential with holomorphic local coefficients. If it is nonzero, its zeros have total order $4g-4$ on a compact genus-$g$ Riemann surface. Away from its zeros, a nonzero differential has flat coordinates whose changes of coordinate have the form $w\mapsto \pm w+c$.

On a Riemann surface, a holomorphic differential form is a holomorphic section of the canonical bundle. In a local complex coordinate $z$ it has the form $f(z)dz$ with $f$ holomorphic.

If a local holomorphic $r$-differential is $z^{m}u(z)d{z}^r$, where $u(0)\ne 0$, its zero order is $m$. The order is unchanged by a holomorphic coordinate change with nonzero derivative. For a holomorphic one-form the cone angle is $2\pi (m+1)$, and for a holomorphic quadratic differential it is $(m+2)\pi$.

For a nonvanishing local $m$-differential $\varphi (z)d{z}^m$, choose $w=\int {\varphi }^{1/m}dz$. Then the differential is $d{w}^m$. Different root choices give $w\mapsto \zeta w+b$, ${\zeta }^m=1$. For $m=1,2$ the permitted linear parts are translations or signs, producing translation surfaces and half-translation surfaces.

A section $\omega$ of the canonical bundle of a Riemann surface, locally $\omega =\varphi (z)dz$ with holomorphic $\varphi$. A nonzero form on a compact genus-$g$ surface has total zero order $2g-2$. Its integral supplies flat coordinates away from zeros.

The locus of nonzero holomorphic one-forms with a fixed list of zero multiplicities $k_1,\dots ,k_r$, summing to $2g-2$. The SL2R action on differentials preserves this list. For example $\mathcal{H}(2)$ and $\mathcal{H}(1,1)$ are distinct genus-two strata.

On a compact complex surface, the claim also holds without a Kähler metric. For a holomorphic one-form $\alpha$, let $\beta =d\alpha$. It is a closed holomorphic two-form, and the Stokes theorem gives $\int \beta \wedge \bar{\beta }=\int d(\alpha \wedge \bar{\beta })=0$. Positivity of $\beta \wedge \bar{\beta }$ forces $\beta =0$. Functions are constant on each connected component, and holomorphic two-forms are closed by dimension.

Closed holomorphic one-forms inject into de Rham cohomology: an exact one would have a global holomorphic primitive and hence vanish. If a holomorphic class equals an antiholomorphic class, their difference is $df$, and $\partial \bar{\partial }f=0$; compact pluriharmonic rigidity makes both forms vanish. Their direct sum gives the lower bound. The sheaf of closed holomorphic one-forms exact sequence gives $0\to H^0({\Omega }^1)\to H^1(\mathbb{C})\to H^1(\mathcal{O})$, yielding the upper bound and $h^{1,0}\le h^{0,1}$. These inequalities do not require a Kähler manifold.

A holomorphic $(p,0)$ form is annihilated by $\bar{\partial }$ and, by bidegree, by ${\bar{\partial }}^{\ast }$. The Kähler Laplacian identity makes it harmonic for $d$, hence $d$-closed on a compact Kähler manifold.

This complex has the sheaves ${\Omega }_M^p$ of holomorphic differential forms and differential $\partial$. By the holomorphic Poincaré lemma, it resolves the complex constant sheaf.

On a complex manifold, ${\Omega }_X^p$ is the sheaf of holomorphic degree-$p$ forms. It is the kernel of the Dolbeault operator on ${\mathcal{A}}^{p,0}$. The Dolbeault theorem computes its sheaf cohomology using smooth forms.

A differential form of type (p, q) of type $(p,0)$ is holomorphic when its local coefficients in holomorphic coordinates are holomorphic functions, equivalently $\bar{\partial }\alpha =0$. It need not be $d$-closed on every complex manifold. Top-degree holomorphic forms are closed by bidegree. On a compact complex surface, holomorphic forms of all degrees are closed, even without a Kähler metric.

On a compact complex manifold of complex dimension $n$, a holomorphic p-form on a complex manifold of degree $n$ is $d$-closed. The form $i^{n^2}\alpha \wedge \bar{\alpha }$ is a nonnegative volume density, positive wherever $\alpha \ne 0$: in holomorphic coordinates it equals $2^n|f{|}^{2}d{x}_1\wedge d{y}_1\wedge \cdots \wedge d{x}_n\wedge d{y}_n$ for $\alpha =fd{z}_1\wedge \cdots \wedge d{z}_n$. If $\alpha =d\beta$, its integral would vanish by the Stokes theorem, contradicting this positivity. No Kähler metric is needed.

The sheaf is ${\mathcal{Z}}_M^1=\ker (\partial :{\Omega }_M^1\to {\Omega }_M^2)$. The holomorphic Poincaré lemma gives $0\to \underline{\mathbb{C}}\to {\mathcal{O}}_M\stackrel{\partial }{\to }{\mathcal{Z}}_M^1\to 0$. On a compact complex surface its long exact sequence in sheaf cohomology yields $0\to H^0(M,{\Omega }_M^1)\to H^1(M,\mathbb{C})\to H^1(M,{\mathcal{O}}_M)$.

On a star-shaped holomorphic coordinate neighbourhood, a closed holomorphic differential form of positive degree has a holomorphic primitive. Radial integration gives the homotopy operator. This proves exactness of the holomorphic de Rham complex on stalks.

For a closed holomorphic one-form $\sum b_{k}d{z}_k$ on a polydisc, the displayed coordinate-path integral is a holomorphic function. Differentiating in $z_j$ gives the $k=j$ endpoint term; for $k>j$ use ${\partial }_j{b}_k={\partial }_k{b}_j$ and integrate that derivative. The resulting sum telescopes to $b_j(z)$. Thus $\partial F=\sum b_{k}d{z}_k$, giving an explicit degree-one proof of the holomorphic Poincaré lemma.

If $f(x,y)=0$ has a smooth projective completion of degree $d$ and $f_y\ne 0$, then its holomorphic differentials have basisThe equality $f_{x}dx+f_{y}dy=0$ gives the equivalent expressions $-x^i{y}^{j}dy/f_x$.

For a smooth hypersurface $X$ in a complex manifold $Y$,

For a degree-$d$ smooth surface $X\subset {\mathbb{P}}^3$ and a smooth rational curve $C$ of degree $a$, the canonical bundle is $K_X={\mathcal{O}}_X(d-4)$. The adjunction formula gives $(K_X+C)\cdot C=-2$, hence $C^2=-2-(d-4)a$. In particular a line has self-intersection $2-d$.

For regular homogeneous equations of degrees $d_1,\dots ,d_r$ in ${\mathbb{P}}^n$, the normal bundle of a regular zero locus is ${⨁}_i\mathcal{O}(d_i){|}_V$. The Euler sequence gives $K_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$, hence $K_V=\mathcal{O}({\sum }_i{d}_i-n-1){|}_V$. Equations with unreduced multiplicities do not have this conclusion for the underlying reduced submanifold.

For a smooth complex submanifold $V\subset M$, the tangent exact sequence gives $\det TM{|}_V=\det TV\otimes \det N_V$. Dualizing yields $K_V=K_M{|}_V\otimes \det N_V$. This form of adjunction formula applies in any codimension.