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The Schwarz lemma states that a holomorphic function $f:\mathbb{D}\to \mathbb{D}$ with $f(0)=0$ satisfiesIf equality holds in the first inequality at any nonzero point, or in the derivative inequality, then $f(z)=\lambda z$ for a constant $|\lambda |=1$. Conversely those rotations give equality.

To prove it, the function $g(z)=f(z)/z$ extends holomorphically at zero with value $f^{\prime }(0)$, by the power series of $f$. On $|z|=r<1$, $|g(z)|\le 1/r$. The maximum modulus principle gives this same bound inside that circle. For each fixed $z$, let $r\uparrow 1$ to conclude $|g(z)|\le 1$. This proves both inequalities. Equality in either makes $g$ attain its maximum modulus at an interior point, so the maximum modulus principle makes $g$ a constant of modulus one. This proves the equality cases as well.

For the classification of automorphisms of the unit disk, choose $a\in \mathbb{D}$ and defineDirect expansion givesThus $T_a$ is a biholomorphism of the disk and sends $a$ to zero. If $F$ is any conformal map of the disk onto itself, take $a=F^{-1}(0)$ and put $h=F\circ T_a^{-1}$. Both $h$ and $h^{-1}$ fix zero. The Schwarz lemma gives $|h(z)|\le |z|$ and, applied to $h^{-1}$ at $h(z)$, $|z|\le |h(z)|$. Equality holds, so $h(z)=e^{i\theta }z$. Consequently all the holomorphic conformal automorphisms are preciselyEvery displayed map is indeed a conformal automorphism by the verified inverse and a rotation. Here conformal maps have the usual holomorphic, orientation-preserving meaning; if orientation reversal is allowed, their complex conjugates give the antiholomorphic alternatives.

The divisor determines the bundle through its local equations. Choose an open cover on which $V$ has a reduced holomorphic defining equation $f_i$, using $f_i=1$ on sets disjoint from $V$. Because $V$ is a complex submanifold of codimension of a submanifold one, its ideal is locally generated by a coordinate; consequently $f_i/f_j$ is a nowhere-zero holomorphic function on each overlap. Inside the sheaf of meromorphic functions on a complex manifold, consider the locally free sheaf of rank oneThese local descriptions agree on overlaps. With $e_i=1/f_i$, its transition functions of a vector bundle are specified byThe ratios satisfy the Čech cocycle condition, and the associated holomorphic line bundle associated to a divisor is $[V]$. The section $1=f_i{e}_i$ is holomorphic and vanishes to order one along $V$. Replacing $f_i$ by $u_i{f}_i$, with $u_i$ a holomorphic unit, replaces the frame by $u_i^{-1}{e}_i$ without changing the subsheaf of meromorphic functions. Refining the cover changes no local sections either. This proves choice independence up to isomorphism rather than merely producing a bundle for one cover.

For a complex manifold of dimension $n$, the canonical bundle isIt is the holomorphic line bundle of holomorphic top-degree differential forms. To compute it on Complex projective space, let $S=\mathcal{O}(-1)$ be the complex tautological line bundle, so that $S^{\ast }=[H]=\mathcal{O}(1)$ is the hyperplane line bundle. At a line $\ell \subset {\mathbb{C}}^{n+1}$, the tangent space is $\mathrm{Hom}(\ell ,{\mathbb{C}}^{n+1}/\ell )$. Tensoring the tautological quotient sequence by $S^{\ast }$ gives the Euler sequence on complex projective spaceTaking the top exterior power gives $\det T^{1,0}{\mathbb{P}}^n=\mathcal{O}(n+1)$; taking its dual bundle yieldsHere a negative tensor power means the corresponding positive tensor power of the dual bundle.

The proposed sections over $U$ form a sheaf: compatible maps into $L$ glue uniquely, and their projections remain $\varphi$. Addition and multiplication by holomorphic functions are defined in the fibres of $L$, making this a sheaf of ${\mathcal{O}}_X$-modules. If $e_i$ is a holomorphic local frame of $L$ on $W_i\subset Y$, every section over $U\subset {\varphi }^{-1}(W_i)$ has the unique expressionThus the sheaf is a locally free sheaf of rank one. Its transition functions of a vector bundle are those of $L$ composed with $\varphi$, so the corresponding pullback vector bundle is precisely ${\varphi }^{\ast }L$. This proves invertibility locally and compatibility globally.

For the local blowup of a complex manifold at a point, work on the projective chart $Y_i\ne 0$ and writeAll the incidence equations reduce to $x_j=u{t}_j$. Therefore $(u,(t_j{)}_{j\ne i})$ gives a holomorphic coordinate chart isomorphic to ${\mathbb{C}}^n$. On an overlap with $Y_{\ell }\ne 0$,which is a biholomorphism where $t_{\ell }\ne 0$. The incidence subset is closed in the product of two Hausdorff spaces and is covered by these charts; hence it is a complex manifold of dimension $n$. In each chart the exceptional fibre is $u=0$, so it is a smooth complex submanifold, with its induced projective coordinates givingAway from the origin the inverse of the projection is $x\mapsto (x,[x])$, which is holomorphic; the projection is therefore a biholomorphism outside $E$.

To find the canonical bundle, pull back the nowhere-zero top form on ${\mathbb{C}}^n$. In the chart above, its Jacobian determinant givesThe sign depends only on the ordering of the coordinates. This is a section of $K_X$ with divisor on a complex manifold exactly $(n-1)E$: it has that vanishing order in every exceptional chart and no zeros elsewhere. A nonzero meromorphic section of a holomorphic line bundle with divisor $D$ identifies its holomorphic line bundle with $[D]$, by sending the local generator $1/f_i$ of $\mathcal{O}(D)$ to the nowhere-zero frame obtained by dividing the section by $f_i$. ConsequentlyThis also includes $n=1$, when the local map is an isomorphism and the exponent is zero.

For a general complex manifold, choose a coordinate neighbourhood $U$ of $P$, with $P$ sent to $0$, and replace $U$ by the inverse image of this coordinate neighbourhood in the local blowup of a complex manifold at a point. Glue this space to $M\setminus \{P\}$ along $U\setminus \{P\}$ using the preceding biholomorphism. The resulting charts give a complex manifold $\tilde{M}$ and a holomorphic map $\varphi$; the exceptional fibre is ${\mathbb{P}}^{n-1}$ and the map is a biholomorphism off that fibre. For completeness, the local projection is proper because its inverse image over a compact subset is closed in that subset times compact Complex projective space. This ensures the gluing is Hausdorff: separate points with distinct images downstairs, and points over $P$ inside the local blowup. The atlas is second countable as well.

The top exterior power of the differential defines a global holomorphic bundle map ${\varphi }^{\ast }{K}_M\to K_{\tilde{M}}$. Equivalently, it is a section of $K_{\tilde{M}}\otimes ({\varphi }^{\ast }{K}_M{)}^{-1}$. The same local Jacobian determinant has vanishing order $n-1$ on $E$, and the differential is invertible elsewhere. Applying the holomorphic line bundle associated to a divisor construction gives the canonical bundle formula for a point blowup

The weights of $E_4^3$ and $\Delta$ agree, so their ratio is invariant under the modular group. Nonvanishing of the modular discriminant on the half-plane makes the ratio holomorphic there. At an elliptic point, an invariant holomorphic function has a power series in the quotient coordinate, so the ratio descends holomorphically. At the unique modular cusp its expansion isThus it is a meromorphic function on the compact Riemann surface with precisely one pole, of order one. The degree of a nonconstant meromorphic map to the sphere equals its total pole order. It therefore has degree one, hence is a biholomorphism:This is the single-pole criterion for a spherical coordinate.

The matrix $\left(\begin{array}{cc}0& -1\\ 1& 1\end{array}\right)$ fixes $\omega$. Weight-four transformation gives $E_4(\omega )=(\omega +1{)}^4{E}_4(\omega )$, and $(\omega +1{)}^4\ne 1$. Hence $E_4(\omega )=0$ and $j(\omega )=0$. Since $j$ is a degree-one coordinate, this is a simple zero on the quotient surface. Pulled back to the half-plane it has order three, by the analytic ramification index in part (a).

Two holomorphic function elements represent the same germ of a holomorphic function at $z$ if they agree on some neighbourhood of $z$. For a holomorphic $f$ on an open $D\subseteq G$, set $U(f,D)=\{[f{]}_z:z\in D\}$. These sets form a basis of a topology on the space of germs of holomorphic functions: every germ lies in one, and if a germ lies in two, equality of the representatives on a smaller neighbourhood supplies a basis of a topology set in their intersection. Declaring arbitrary unions of basis of a topology sets open therefore gives a topology.

Different base points are separated by disjoint small discs in $G$. If two different germs have the same base point, choose representatives on one connected small disc. Their sheets are disjoint: any common germ elsewhere on that disc would force equality throughout the disc by the identity theorem. This proves the Hausdorff property.

The projection $\pi ([f{]}_z)=z$ maps each sheet $U(f,D)$ homeomorphically onto $D$. Use these maps as complex charts. On overlaps their transition functions are the identity, hence holomorphic; thus each connected component is a Riemann surface, and $\pi$ is a local biholomorphism, an analytic germ projection. For completeness, second countability of a component follows by following continuation chains through a countable basis of a topology of rational discs: starting at one germ, finite chains of such discs give only countably many continuation sheets, by uniqueness on overlaps, and they cover every germ in that component.

A complete analytic function consists of all germs reachable from one initial germ by analytic continuation. The supplied path-lifting equivalence identifies these with its path component. Since the germ space is locally path connected, connected components and path components agree. Thus its continuation surface is exactly one component ${\mathcal{G}}_{\mathcal{F}}$.

A local biholomorphism need not evenly cover the whole base; the germ projection is an analytic covering in this local sense. The examples below distinguish a genuine regular covering map from a surjective nonregular projection, with the meaning of regular stated explicitly.

Equip the closed Riemann surface with its uniformizing hyperbolic metric. This assumption matters: arbitrary metrics can have an annular region filled by parallel simple closed geodesics. On a hyperbolic surface, a nontrivial free homotopy class has exactly one closed geodesic, and a contractible closed geodesic cannot exist. Consequently distinct disjoint simple closed geodesics are essential and nonparallel.

Cut along $k$ such geodesics. Every component has negative Euler characteristic, since disk and annulus components are excluded. If a component has genus $h$ and $b$ boundary components, its complexity is $3h-3+b$. This is nonnegative for the possible components, with equality precisely for a pair of pants. Summing over the cut components givesFor completeness, if there are $n$ components, Euler additivity gives $\sum h=g+n-k-1$, while $\sum b=2k$, which proves the identity. Hence $k\le 3g-3$. If any component has positive complexity, it contains an essential nonperipheral simple closed curve. Its unique geodesic representative stays in that component and is disjoint from the existing boundary geodesics, contradicting maximality. A maximal collection therefore cuts the surface into pairs of pants and hasThe second count follows because each pair of pants has Euler characteristic $-1$. Conversely, any topological pants decomposition can be straightened to its disjoint geodesic representatives, so this maximum is attained.

The pants decomposition graph has one vertex for each pair of pants and one edge for each glued cuff. Two cuffs on the same pant give a loop; multiple edges are allowed. It is connected and trivalent, with $V=2g-2$, $E=3g-3$ and first Betti number $E-V+1=g$. For genus three one possibility is the complete graph $K_4$. Another has vertices $C,D,E,F$, double edges $CE$ and $DF$, and single edges $CD$ and $EF$. These are nonisomorphic graphs: the second has multiple edges while $K_4$ has none.

There is an explicit diffeomorphism between the represented surfaces. In the $K_4$ decomposition, unite the pants labelled $A,B$ across their shared cuff. This union is a four-holed sphere, whose exterior cuffs lead twice to $C$ and twice to $D$. Replace the interior cuff by a simple curve separating the two $C$ cuffs from the two $D$ cuffs. The new pants $E,F$ then have double adjacencies to $C,D$ respectively, while their shared cuff gives $EF$ and the untouched original cuff gives $CD$. This is an elementary move of a pants decomposition. We have simply recut the same oriented surface, so the identity on that surface is the desired diffeomorphism between its two descriptions. The graphs do not make this evident because they record a chosen pants decomposition, which the diffeomorphism need not preserve; they also omit all lengths and twists.

A hyperbolic pair of pants is uniquely determined up to isometry by its three positive boundary lengths. Cutting along the three perpendicular seams yields two congruent right-angled hexagons, with alternating sides half the cuff lengths; hyperbolic hexagon identities determine the remaining sides. To recover the surface, choose each of the $3g-3$ cuff lengths ${\ell }_i>0$, and for each gluing choose the signed displacement ${\tau }_i$ between the seam feet. For a fixed marking and decomposition these are the Fenchel–Nielsen coordinates:Unmarked gluing displacement at one cuff is periodic modulo ${\ell }_i$; the real lift of ${\tau }_i$ remembers the marking, and a full Dehn twist changes it by ${\ell }_i$. To classify unmarked surfaces globally, take the quotient by the mapping class group, allowing changes of pants decomposition as well; if orientation is forgotten, use the extended mapping class group. Lengths alone do not determine the surface: the twists change the geometry of crossing geodesics.

Fix an oriented reference surface $S_g$. Its Teichmüller space ${\mathcal{T}}_g$ consists of pairs $(X,f)$, where $X$ is a Riemann surface and $f:S_g\to X$ is an orientation-preserving marking. Two pairs are equivalent when an orientation-preserving biholomorphism $h:X\to Y$ makes $h\circ f$ isotopic to the other marking. For $g\ge 2$, the uniformization theorem identifies these with marked hyperbolic metrics. The space is a complex manifold of complex dimension $3g-3$, and Fenchel–Nielsen coordinates identify its underlying real manifold with ${\mathbb{R}}^{6g-6}$. For the analytic argument below, the essential property is that these are real-analytic coordinates and the length ${\ell }_{\gamma }(X)$ of every fixed closed curve class is a real-analytic function on Teichmüller space. The length and twist coordinates are not themselves holomorphic coordinates.

The Wolpert generic spectral rigidity theorem says that, for each $g\ge 2$, there is a closed proper real-analytic subset ${\mathcal{E}}_g\subset {\mathcal{T}}_g$, of positive codimension, such that if $X\notin {\mathcal{E}}_g$ and $Y$ has the same Laplace-Beltrami operator spectrum as $X$, then $X,Y$ are isometric. Equivalently, a generic compact hyperbolic surface is determined by its unmarked length spectrum, with multiplicities. “Isometric” allows reversal of orientation: an unoriented spectrum cannot distinguish a surface from its complex conjugate.

Two major proof steps explain both parts of this conclusion. First is the analytic reduction to finitely many possible spectral matchings locally. The Selberg trace formula identifies the Laplacian spectrum with the unmarked length spectrum. A finite collection of marked length functions determines the length and twist parameters: include pants cuffs and enough transverse curves to determine each twist and its sign. For each assignment of those curves to lengths in a putative partner, the equalities of length functions are real-analytic equations. Uniform length cutoffs on a neighbourhood, bounded geometry of isospectral partners and the discreteness of the length spectrum give local finiteness of the possibilities; finite-length determination of the full spectrum allows the remaining conditions to be reduced to finitely many analytic equations. Thus this is a locally finite analytic problem, not an uncontrolled union of zero sets indexed by infinitely many arbitrary permutations. The role of this step is to put exceptional isospectral coincidences into real-analytic loci and ultimately to give a closed analytic exceptional set.

Second is the geometric recognition of the matchings that can persist on an open set. Such a matching produces identities between length functions as the length and twist parameters vary. Pinching a pants cuff makes its hyperbolic collar wide: by the collar lemma, any crossing curve acquires length tending to infinity, whereas disjoint curves can stay bounded. Varying cuffs and twists separately, together with the hyperbolic trace and hexagon identities for neighbouring pants, identifies which persistent matches preserve simple curves, disjointness and the gluing data. The resulting persistent transformation is induced by a mapping class, possibly with orientation reversal, and so it only gives an isometry of the underlying surfaces. Every other local matching imposes a nonidentical real-analytic equation, whose zero set has positive codimension. Removing these proper exceptional loci proves generic rigidity. The geometric step is indispensable: analyticity alone would not exclude an entire open family of distinct surfaces with the same unmarked length spectrum.

A related useful consequence is that a fixed hyperbolic surface has only finitely many isometry classes of isospectral partners. The common length spectrum fixes the systole, so the partners lie in a compact thick part of moduli space. If distinct partners converged, choose local markings and a finite determining collection of curves. Their lengths converge, but belong to the same discrete length spectrum, so eventually every one of these finitely many lengths is constant. The determining collection then forces the marked metrics to coincide, a contradiction. This compactness-and-discreteness argument proves finiteness; the additional geometric recognition step above is what strengthens it to the generic uniqueness asserted by Wolpert's theorem.

Let ${\pi }_i:{\mathbb{C}}^{g_i}\to X_i$ be the quotient covering maps. Since the source universal cover is simply connected, the lifting criterion for a covering space gives a lift $F:{\mathbb{C}}^{g_1}\to {\mathbb{C}}^{g_2}$ of $f\circ {\pi }_1$. The lift is a holomorphic map because ${\pi }_2$ is locally a biholomorphism.

For every $\lambda \in {\Lambda }_1$, the difference $F(z+\lambda )-F(z)$ belongs to the discrete Euclidean lattice ${\Lambda }_2$. It is a continuous function of $z$ on a connected space, hence is constant. Differentiating shows that each coefficient of $DF$ is ${\Lambda }_1$-periodic. Such a coefficient is bounded on a compact fundamental parallelepiped, and periodicity bounds it on all of ${\mathbb{C}}^{g_1}$. The several-variable Liouville theorem, obtained by applying the one-variable theorem on coordinate lines, makes every coefficient constant.

Consequently $DF=A$ for a constant complex-linear map $A$ and $F(z)=Az+x$. The period differences become $A\lambda \in {\Lambda }_2$. Thus the affine lift of a holomorphic map between complex tori givesEvery holomorphic map between complex tori is a homomorphism followed by a translation. The translation vector $x=F(0)$ is determined modulo ${\Lambda }_2$, while $A$ is the derivative of any lift and is unaffected by that ambiguity.

Fix $z_0\in \mathcal{U}$. We prove the Riemann mapping theorem in this case by maximizing a normalized derivative. Choose $a\notin \mathcal{U}$. Since $\mathcal{U}$ is a simply connected domain and $z-a$ never vanishes, it has a holomorphic square root $g$. This $g$ is injective, and $g(\mathcal{U})$ is disjoint from $-g(\mathcal{U})$: equality up to sign would first force the original points to coincide. Choose $b\in g(\mathcal{U})$ and $r>0$ with $B(b,r)\subset g(\mathcal{U})$. The ball $B(-b,r)$ is omitted, so $1/(g+b)$ is a bounded univalent function. Scaling and composing with an automorphism of the unit disk produces an injective holomorphic function $\varphi :\mathcal{U}\to \mathbb{D}$ with $\varphi (z_0)=0$ and ${\varphi }^{\prime }(z_0)>0$.

Let $\mathcal{F}$ be the family of all such normalized univalent functions. A Cauchy estimate in a small disk about $z_0$ bounds their derivatives, so $M={\sup }_{\varphi \in \mathcal{F}}{\varphi }^{\prime }(z_0)$ is finite and positive. Choose a maximizing sequence in this normal family. Montel theorem gives a subsequence converging uniformly on compact subsets to $\varphi$. Its derivative at $z_0$ is $M>0$. The maximum modulus principle puts its image in $\mathbb{D}$, and Hurwitz's theorem implies that a nonconstant limit of injective holomorphic functions is injective. Thus the maximum is attained.

If $a_1\in \mathbb{D}$ is omitted, then $a_1\ne 0$. Put $T_a(w)=(w-a)/(1-\overline{a}w)$. A holomorphic square root $v$ of $T_{a_1}\circ \varphi$ exists, is injective, and takes values in $\mathbb{D}$. Compose $v$ with $T_{v(z_0)}$ and a rotation to normalize it. Writing $s=|a_1|\in (0,1)$, the new derivative has magnitudea contradiction. Therefore $\varphi$ maps onto $\mathbb{D}$. The Cayley transform between the half-plane and disk now givesIts inverse is holomorphic by the holomorphic inverse function theorem, so this is a biholomorphism. The proper-subset hypothesis was used to choose $a\notin \mathcal{U}$; the whole complex plane cannot be mapped this way, by Liouville theorem.

Use the Gaussian curvature $-1$ normalizationof the Poincaré half-plane model. This is a complete conformal metric, invariant under real Möbius transformations. Its quotient by the free effective principal congruence subgroup is again complete: a geodesic lifts to the complete covering plane and extends there for all time. Transporting the quotient metric through the biholomorphism supplies the required complete conformal metric.

A nonconstant closed geodesic corresponds to a hyperbolic Möbius transformation $A$ of $\Gamma (2)$, and its hyperbolic translation length is $2\mathrm{arcosh}(|\mathrm{tr}A|/2)$. Indeed its eigenvalues have magnitudes $\lambda ,{\lambda }^{-1}$ with $\lambda >1$, and its axis quotient has length $2\log \lambda$. For $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ in $\Gamma (2)$, $bc\equiv 0(\mathrm{mod}4)$ implies $ad\equiv 1(\mathrm{mod}4)$. The odd numbers $a,d$ therefore have the same residue modulo four, andA hyperbolic Möbius transformation must have $|\mathrm{tr}A|>2$, so its smallest possible absolute trace is six. It is attained byConsequentlyThis element is primitive: a proper power would have a shorter root represented by a hyperbolic Möbius transformation, contradicting the trace bound. No essential simple closed geodesic exists on the three-punctured sphere, since every essential simple loop is peripheral and corresponds to a parabolic Möbius transformation. The minimizing closed geodesic is therefore nonsimple. Finally, the PDF leaves the numerical Gaussian curvature unspecified: the displayed length uses Gaussian curvature $-1$; for Gaussian curvature $-{\kappa }^2$, it is divided by $\kappa$.

Use the closed genus $g\ge 2$ version of Teichmüller's uniqueness theorem. Let $f_0:X\to Y$ be a Teichmüller map: for a nonzero holomorphic quadratic differential $q$, normalized by ${\int }_X|q|=1$, and $0<k_0<1$,The value at a zero of $q$ is irrelevant to the Beltrami coefficient, which is defined almost everywhere. For every quasiconformal map $f:X\to Y$ preserving orientation and homotopic to $f_0$, the conclusion isThe normalized $q$ is also uniquely determined when $k_0>0$.

We use the following analytic input, stated with its hypotheses. The Reich–Strebel inequality says that if $f_0$ has the displayed Beltrami coefficient and a competitor $f$ has the same target and homotopy class, then, for $\mu ={\mu }_f$,This is the standard fundamental inequality for an integrable holomorphic quadratic differential; it also holds on a finite-type punctured surface with the homotopy fixing the punctures. Its formulation is given in Gardiner and Hu, §5. We quote this analytic inequality as the lecture result used in the proof.

Put $k=\Vert \mu {\Vert }_{\infty }$. Pointwise, away from the isolated zeros of $q$,Integrating against the probability density function $|q|$ proves $K_0\le K(f)$. If $K(f)=K_0$, all these inequalities are equalities almost everywhere. The strictly increasing last function forces $|\mu |=k_0$ almost everywhere; equality in the triangle inequality then forces $\mu q/|q|=k_0$. ThereforeTwo quasiconformal maps with the same Beltrami coefficient differ by postcomposition with a biholomorphism, by the local chain rule for the Beltrami equation. Consequently $h=f\circ f_0^{-1}$ is a biholomorphism from $Y$ to itself homotopic to the identity.

For completeness, such a biholomorphism is the identity when $g\ge 2$. Apply the uniformization theorem and choose the lift of the homotopy to $\mathbb{H}$ starting at the identity. Its endpoint lift $\tilde{h}$ commutes with every deck transformation. It is a real Möbius transformation. The compact quotient has no parabolic Möbius transformations in its deck group, and freeness excludes elliptic Möbius transformations. Two distinct hyperbolic axes exist: a discrete free group preserving just one axis would be cyclic, contradicting the fundamental group of a closed surface of genus at least two. Commutation with two deck transformations represented by hyperbolic Möbius transformations having distinct axes makes it fix their boundary endpoints individually; there are at least three such endpoints. A Möbius transformation fixing three points is the identity. Hence $h=\mathrm{id}$ and $f=f_0$.

If $q_1$ is another unit-area differential for the same map, then $|q_1|/q_1=|q|/q$, so $q_1/q$ is positive real wherever defined. This meromorphic function is constant by the open mapping theorem, and normalization makes the constant one. Without normalization, positive multiples of $q$ describe the same map. At $k_0=0$, uniqueness of the map still holds in genus at least two, but there is no distinguished $q$. In genus one, translations supply nontrivial biholomorphisms homotopic to the identity, so equality determines the map only up to those biholomorphisms; the genus hypothesis cannot be omitted.

An irreducible complex analytic hypersurface in a complex manifold $X$ is a closed irreducible analytic subset of pure complex codimension one. A local defining function of a complex analytic hypersurface $Y$ at $x$ is a holomorphic function $f$ on a neighbourhood $U$ such thatThe necessary local algebra is that the stalk ${\mathcal{O}}_{X,x}$ is a regular local ring, hence a unique factorization domain, and that the local branches of a hypersurface germ determine finitely many height-one prime ideals. Each is principal; the product of their generators gives $f$, and removing repeated factors makes it reduced. This also covers a globally irreducible hypersurface that has several local branches at a singular point.

A divisor on a complex manifold is a locally finite formal sum $D={\sum }_Y{n}_{Y}Y$ of irreducible analytic hypersurfaces with integer coefficients. On a sufficiently small $U_{\alpha }$, local defining functions give a meromorphic equation $f_{\alpha }$ for $D$. The quotients $f_{\alpha }/f_{\beta }$ are nowhere-zero holomorphic functions. Gluing frames byproduces the holomorphic line bundle associated to a divisor $[D]$, and $f_{\alpha }{e}_{\alpha }$ gives its canonical meromorphic section with divisor $D$.

The Euler sequence on complex projective spaceimplies $\det T{\mathbb{CP}}^n\cong \mathcal{O}(n+1)$. Taking the dual determinant yields the canonical bundle of complex projective spacewhere $H$ is a hyperplane divisor.

The hypotheses on the homogeneous polynomial $p$ say thatis a smooth projective hypersurface of degree $k$, so its divisor line bundle is $[V]\cong \mathcal{O}(kH)$. The adjunction formula givesThis is the canonical bundle of a smooth projective hypersurface.

Now fix an isomorphism $\Phi :[P]\to [Q]$ and regard $s_P$ and ${\Phi }^{-1}{s}_Q$ as holomorphic sections of the same holomorphic line bundle. They have no common zero because $P\ne Q$. Their homogeneous coordinates therefore define a well-defined holomorphic mapIn a local frame, the quotientis a meromorphic function with divisor $(f)=P-Q$. Thus $F^{-1}(0)=P$ and $F^{-1}(\infty )=Q$, both with multiplicity one. The degree of a holomorphic map $F$ is therefore one. A nonconstant degree-one holomorphic map between compact connected Riemann surfaces is a biholomorphism, so the displayed map is biholomorphic.

A nonconstant meromorphic function on a compact Riemann surface with exactly one simple pole gives a degree-one holomorphic map to the sphere. A degree-one map is a biholomorphism. The surface therefore has genus zero and the function is a global rational coordinate.

For a fixed closed oriented surface $S_g$, a point is a marked Riemann surface $(X,f:S_g\to X)$, modulo biholomorphisms intertwining the markings up to homotopy. For $g\ge 2$, Fenchel–Nielsen coordinates give real dimension $6g-6$.