Riemann's existence theorem for algebraic curves

Intuition [Beginner]

A compact Riemann surface is the answer to a question about complex analysis: where does a multivalued algebraic function $w(z)$ become single-valued? Riemann's surfaces solve the question topologically — by replacing $\mathbb{C}$ with a closed surface dressed in just enough geometry to ask whether a function is holomorphic. Algebraic curves, by contrast, are the answer to a question in algebra: what are the solution sets of polynomial equations $P(x,y)=0$ in two variables? Riemann's existence theorem says these two questions describe the same objects. Every compact Riemann surface is, in a way you can pin down precisely, the same thing as a smooth projective algebraic curve.

Three different routes lead to this conclusion. The first is to find enough holomorphic functions on the surface to embed it inside projective space; once it sits there, polynomials cut it out, and it is algebraic. The second is to look at the meromorphic functions: pick a non-constant one, call it $f$, then build a second function $g$ that satisfies a polynomial relation $P(f,g)=0$. The surface is recovered as the smooth model of the curve carved out by $P$. The third route is the converse of the previous unit on branched coverings: a branched cover of the sphere by combinatorial data is automatically algebraic.

Why bother? Because the same object is now available through three different vocabularies. Topology, complex analysis, and algebra each have their own tools, and now those tools all apply to the same surface. This is the foundation on which the entire bridge between Riemann surfaces and algebraic geometry rests.

Visual [Beginner]

A schematic shows a compact Riemann surface (a torus) on the left and a plane curve $y^2=x^3-x$ on the right, with an arrow labelled "biholomorphic" between them. The torus carries a marked basis of cycles; the plane curve carries a few highlighted points. A second arrow runs upward from the torus to a copy of ${\mathbb{P}}^3$, showing the surface embedded as a smooth curve cut out by polynomials.

The two pictures are the same object viewed through two vocabularies.

Worked example [Beginner]

Take the elliptic curve presented analytically as the quotient $X=\mathbb{C}/\Lambda$ for $\Lambda =\mathbb{Z}+\tau \mathbb{Z}$ a lattice with $\mathrm{Im}(\tau )>0$. As a compact Riemann surface, $X$ is a torus of genus $1$. The Weierstrass $\wp$-function $\wp (z;\Lambda )$ and its derivative ${\wp }^{\prime }(z;\Lambda )$ are meromorphic functions on $X$.

These two functions satisfy a polynomial relation $$ (\wp')^2 ;=; 4 \wp^3 - g_2(\Lambda) \wp - g_3(\Lambda), $$ where $g_2(\Lambda )$ and $g_3(\Lambda )$ are explicit numerical constants computed from the lattice (the Eisenstein values). Set $x=\wp (z)$ and $y={\wp }^{\prime }(z)$. The map $z\mapsto (x(z),y(z))$ realises $X$ as the smooth projective closure of the affine plane curve $$ y^2 ;=; 4 x^3 - g_2 x - g_3. $$

Every point of the torus, including the lattice point $z=0$ (where $\wp$ has a pole of order $2$), maps to a unique point on the projective curve, including the point at infinity $[0:1:0]$ in ${\mathbb{P}}^2$. The map is a biholomorphism.

What this tells us: an analytic torus and a plane cubic are the same object. The $j$-invariant of the lattice $\Lambda$ — a transcendental quantity built from $g_2,g_3$ — equals the $j$-invariant of the cubic, an algebraic combination of its coefficients. Two languages, one curve.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Set-up. Let $X$ be a compact Riemann surface of genus $g$. Write $\mathcal{M}(X)$ for the field of meromorphic functions on $X$, $\mathcal{O}(X)$ for the ring of holomorphic functions on $X$, and ${\Omega }^1(X)=H^0(X,{\Omega }_X^1)$ for the space of holomorphic $1$-forms. By compactness, $\mathcal{O}(X)=\mathbb{C}$ — every global holomorphic function on a compact Riemann surface is constant.

Definition (analytification). A smooth projective algebraic curve $C/\mathbb{C}$ is a smooth one-dimensional projective variety over $\mathbb{C}$. Its analytification $C^{\mathrm{an}}$ is the underlying complex-analytic space, with the induced complex structure: take the complex points of $C$ as a set, give them the Hausdorff topology induced from ${\mathbb{P}}^N(\mathbb{C})$ via any closed immersion $C\hookrightarrow {\mathbb{P}}^N$, and equip the result with the sheaf of holomorphic functions inherited from ${\mathbb{P}}^N(\mathbb{C})$. The analytification is a compact Riemann surface.

Definition (Riemann's existence theorem statement). Riemann's existence theorem asserts that the analytification functor $$ \mathrm{Curves}_{\mathbb{C}}^{\mathrm{sm,proj}} ;\xrightarrow{;(-)^{\mathrm{an}};}; \mathrm{CompactRiemannSurfaces}, $$ sending a smooth projective algebraic curve $C/\mathbb{C}$ to its analytification $C^{\mathrm{an}}$ and a morphism $f:C_1\to C_2$ of curves to the induced holomorphic map $f^{\mathrm{an}}:C_1^{\mathrm{an}}\to C_2^{\mathrm{an}}$, is an equivalence of categories. Concretely:

1. Essential surjectivity. Every compact Riemann surface $X$ is biholomorphic to the analytification $C^{\mathrm{an}}$ of some smooth projective algebraic curve $C/\mathbb{C}$.
2. Full faithfulness. For any two smooth projective curves $C_1,C_2$ over $\mathbb{C}$, the map ${\mathrm{Hom}}_{\mathbb{C}}(C_1,C_2)\to {\mathrm{Hom}}_{\mathrm{hol}}(C_1^{\mathrm{an}},C_2^{\mathrm{an}})$ is a bijection — every holomorphic map between analytifications is the analytification of a unique algebraic morphism.

Equivalent formulation (function-field). The functor $C\mapsto K(C)$, sending a smooth projective curve to its function field, lifts to an equivalence between the opposite of the category of smooth projective curves over $\mathbb{C}$ and the category of finitely generated field extensions of $\mathbb{C}$ of transcendence degree $1$. Composed with analytification, this delivers an equivalence between compact Riemann surfaces and such field extensions, sending $X$ to $\mathcal{M}(X)$ (necessarily finitely generated of transcendence degree $1$ over $\mathbb{C}$).

Equivalent formulation (Proj construction). For a compact Riemann surface $X$ and a holomorphic line bundle $L$ on $X$ of degree $d\ge 2g+1$, the bundle $L$ is very ample: the global sections embed $X$ as a smooth closed subvariety of $\mathbb{P}(H^0(X,L{)}^{\ast })$. The image is cut out by polynomials, and the homogeneous coordinate ring $R(X,L):={⨁}_{n\ge 0}{H}^0(X,L^{\otimes n})$ recovers $X$ as $\mathrm{Proj}(R(X,L))$.

Counterexamples to common slips

• Compactness is essential. The open unit disc $\{|z|<1\}$ is a non-compact Riemann surface and is not biholomorphic to any algebraic curve. The Riemann-mapping-theorem dichotomy (${\mathbb{P}}^1$, $\mathbb{C}$, disc) shows the disc is its own analytic class; no algebraic curve over $\mathbb{C}$ has it as analytification.
• Smoothness is essential on the algebraic side. A nodal plane curve $y^2=x^2(x+1)$ is an algebraic curve but its analytification is not a Riemann surface (the node is a singular point); the theorem requires smooth projective curves on the algebraic side.
• The functor is not just bijective on objects but fully faithful on morphisms. It is not enough that every Riemann surface arises analytically from a curve; the data of holomorphic maps and algebraic morphisms also coincide, and that is the content of GAGA on top of essential surjectivity.
• The theorem is special to dimension $1$ (in this clean form). Chow's theorem provides the higher-dimensional analogue — every compact complex submanifold of ${\mathbb{P}}^N$ is algebraic — but compact complex manifolds in general are not all projective (e.g., complex tori of dimension $\ge 2$ that fail the Riemann bilinear conditions); the analytic-algebraic equivalence holds only for projective complex manifolds, of which curves are automatically a member.

Key theorem with proof [Intermediate+]

Theorem (Riemann's existence theorem, function-field route). Every compact Riemann surface $X$ is biholomorphic to the analytification of a smooth projective algebraic curve over $\mathbb{C}$. Equivalently, the field $\mathcal{M}(X)$ of meromorphic functions on $X$ is a finitely generated field extension of $\mathbb{C}$ of transcendence degree $1$, and $X$ is recovered as the unique smooth projective curve with function field $\mathcal{M}(X)$.

Proof. Three steps: (i) construct a non-constant meromorphic function $f$ on $X$; (ii) show $\mathcal{M}(X)/\mathbb{C}(f)$ is a finite algebraic field extension, hence $\mathcal{M}(X)=\mathbb{C}(f)[g]$ for some $g$ satisfying a polynomial relation $P(f,g)=0$; (iii) recover $X$ as the smooth projective model of the affine plane curve $\{P=0\}$.

Step 1 — existence of a non-constant meromorphic function. Riemann-Roch on a compact Riemann surface, combined with Serre duality (cf. 06.04.04), gives for any divisor $D$ on $X$ $$ \dim H^0(X, \mathcal{O}(D)) - \dim H^0(X, \mathcal{O}(K - D)) ;=; \deg D - g + 1, $$ where $K$ is the canonical divisor of degree $2g-2$. For $D$ effective with $\mathrm{deg}D\ge 2g-1$, the second term vanishes (since $\mathrm{deg}(K-D)<0$ forces $H^0(X,\mathcal{O}(K-D))=0$), so $\dim H^0(X,\mathcal{O}(D))=\mathrm{deg}D-g+1\ge g\ge 1$. Pick any point $p\in X$ and set $D=(2g-1)p$; then $H^0(X,\mathcal{O}((2g-1)p))$ contains a non-constant section, equivalently a non-constant meromorphic function $f$ with poles only at $p$ of total order at most $2g-1$. Viewed as a holomorphic map $f:X\to {\mathbb{P}}^1$, $f$ is non-constant and has finite degree $n:=\mathrm{deg}f\le 2g-1$ ^{[Forster §16]}.

Step 2 — finiteness of $\mathcal{M}(X)/\mathbb{C}(f)$. The pullback $f^{\ast }:\mathcal{M}({\mathbb{P}}^1)\to \mathcal{M}(X)$ identifies $\mathbb{C}(f)\cong \mathcal{M}({\mathbb{P}}^1)=\mathbb{C}(z)$ as a subfield of $\mathcal{M}(X)$. The extension $\mathcal{M}(X)/\mathbb{C}(f)$ is generated by the values that meromorphic functions on $X$ take on a generic fibre $f^{-1}(z_0)$; the fibre has cardinality $n=\mathrm{deg}f$, so any meromorphic $h$ on $X$ satisfies the polynomial relation ${\prod }_{i=1}^n(T-h(y_i))=0$ in $\mathbb{C}(f)[T]$ where $y_1,\dots ,y_n$ are the preimages of a generic $z_0$ (a Galois-theoretic argument; the symmetric functions of the $h(y_i)$ are meromorphic functions of $z_0$, hence in $\mathbb{C}(f)$). Thus every meromorphic $h$ on $X$ is algebraic over $\mathbb{C}(f)$ of degree at most $n$, and $[\mathcal{M}(X):\mathbb{C}(f)]\le n$. The reverse inequality is the primitive element theorem applied at a non-Weierstrass point; $\mathcal{M}(X)=\mathbb{C}(f)[g]$ for some $g$ with $[\mathcal{M}(X):\mathbb{C}(f)]=n$.

Step 3 — recovery of $X$ as a smooth projective curve. Let $P(z,w)\in \mathbb{C}(z)[w]$ be the minimal polynomial of $g$ over $\mathbb{C}(f)$. Clearing denominators, $P$ becomes a polynomial in $\mathbb{C}[z,w]$, irreducible and of degree $n$ in $w$. The affine plane curve $C_0:=\{(z,w)\in {\mathbb{C}}^2:P(z,w)=0\}$ has function field $\mathbb{C}(z)[w]/(P)=\mathcal{M}(X)$. Take its projective closure ${\overline{C}}_0\subset {\mathbb{P}}^2$ and the smooth projective model $C:=\widetilde{{\overline{C}}_0}$ (the normalisation), well-defined for an irreducible algebraic curve. The function field of $C$ equals the function field of ${\overline{C}}_0$, which equals $\mathcal{M}(X)$. The identification $C\to X$ at the level of points is given by the rational map $C⇢X$ extending the assignment $(z,w)\mapsto$ the unique $y\in X$ with $f(y)=z$ and $g(y)=w$; the rational map extends uniquely to a morphism by smoothness of $X$ and properness of $C$, and the morphism is an isomorphism because the function fields match and both curves are smooth projective. Composing with the analytification $C\to C^{\mathrm{an}}$ yields the biholomorphism $C^{\mathrm{an}}\cong X$. $\square$

The full faithfulness of analytification — every holomorphic map between analytifications comes from a unique algebraic morphism — is the GAGA half of the equivalence and is treated at Master tier.

Bridge. The function-field route above produces the algebraic curve from analytic data; the very-ample embedding route produces it from a sufficiently positive line bundle and Riemann-Roch directly, without passing through a non-constant function first; the branched-cover route inverts the previous unit (cf. 06.02.02) by combining the topological / monodromy classification with GAGA. Each route picks a different generating set of meromorphic functions to embed $X$ into projective space — coordinates of a high-degree line bundle, or two functions $f,g$ with the polynomial relation $P(f,g)=0$, or the coordinates of a Belyi-style cover. The same algebraic curve emerges, by the universal property of normalisation. This builds toward the full Master-tier statement of GAGA, the Belyi reformulation over $\overline{\mathbb{Q}}$, and the higher-dimensional Chow theorem; it appears again in the variation-of-Hodge-structure picture (cf. 06.08.02) and in the moduli-theoretic identification ${\mathcal{M}}_g$ of compact Riemann surfaces of genus $g$ with the moduli of smooth projective curves of genus $g$.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

Mathlib does not currently formalise the analytification functor, the equivalence with smooth projective curves, or the GAGA comparison theorem. A target signature, in Lean 4 / Mathlib syntax, sketches the statement:

[object Promise]

The proof depends on names not yet in Mathlib: the analytification functor on schemes locally of finite type over $\mathbb{C}$, the GAGA comparison theorem $H^q(X,\mathcal{F})\cong H^q(X^{\mathrm{an}},{\mathcal{F}}^{\mathrm{an}})$ for coherent $\mathcal{F}$ on projective $X$, the very-ampleness criterion for line bundles of degree $\ge 2g+1$ on a curve of genus $g$, the function-field-of-curves equivalence between smooth projective curves and finitely generated transcendence-degree-$1$ extensions of $\mathbb{C}$, and the meromorphic-function-field $\mathcal{M}(X)$ on a compact Riemann surface as a finitely generated extension of $\mathbb{C}$. Each is a candidate Mathlib contribution; the unit ships with lean_status: none.

Advanced results [Master]

The GAGA principle. Serre's Géométrie algébrique et géométrie analytique (Ann. Inst. Fourier 6, 1956) ^{[Serre 1956]} establishes that on a complex projective variety $X$, the analytification functor on coherent sheaves is an equivalence of abelian categories, and that the cohomology of any coherent algebraic $\mathcal{F}$ on $X$ matches the cohomology of its analytification ${\mathcal{F}}^{\mathrm{an}}$ on $X^{\mathrm{an}}$. The key inputs are: $X^{\mathrm{an}}$ is a compact complex space with finite-dimensional cohomology of coherent sheaves (Cartan-Serre 1953); the analytification of an algebraic coherent sheaf is coherent in the analytic category (Oka coherence theorem); and a bootstrap argument from $\mathcal{O}(d)$ on ${\mathbb{P}}^N$ to general projective $X$ via twisting and exact sequences. GAGA is the bridge that turns Riemann's analytic existence statement into the categorical equivalence between projective varieties and their analytifications: smooth projective curves over $\mathbb{C}$ and compact Riemann surfaces are equivalent categories, with morphisms being either algebraic morphisms or holomorphic maps.

The function-field-of-curves equivalence. For any field $k$, the contravariant functor sending a smooth projective integral curve $C$ over $k$ to its function field $K(C)$ is an equivalence of categories between smooth projective curves and the opposite of the category of finitely generated field extensions of $k$ of transcendence degree $1$. The inverse is the normalisation: given $K/k$ finitely generated of transcendence degree $1$, build the unique smooth projective curve $C/k$ with $K(C)=K$ by taking the integral closure of $k[t]$ in $K$ for any transcendence basis $t$ and gluing the resulting affine pieces. Combined with Riemann's existence theorem, this gives a triple equivalence of categories $$ \mathrm{CompactRiemannSurfaces} ;\simeq; \mathrm{SmoothProjectiveCurves}_{\mathbb{C}}^{\mathrm{op}} ;\simeq; \big{\text{f.g. field extensions of } \mathbb{C} \text{ of transcendence degree } 1\big}^{\mathrm{op}}. $$ The first equivalence is analytification (Riemann existence + GAGA), the second is function field. Composing, every compact Riemann surface $X$ is uniquely determined by its meromorphic function field $\mathcal{M}(X)$ as a finitely generated transcendence-degree-$1$ field extension of $\mathbb{C}$, and conversely every such extension arises.

Belyi's theorem and dessins d'enfants. Belyi 1979 ^{[Belyi 1979]} proved that a smooth projective curve $C$ over $\mathbb{C}$ is defined over the algebraic closure $\overline{\mathbb{Q}}\subset \mathbb{C}$ iff $C$ admits a Belyi map — a non-constant morphism $\beta :C\to {\mathbb{P}}^1$ branched only over $\{0,1,\infty \}$. The "iff" combines Riemann's existence theorem (every compact Riemann surface is algebraic) with a refinement: the moduli of finite covers of ${\mathbb{P}}^1\setminus \{0,1,\infty \}$, branched only at those three points, is combinatorial (classified by the free group $F_2$ on two generators, or equivalently by dessins d'enfants — bipartite ribbon graphs embedded in the cover). Combinatorial moduli have no transcendental parameters, so a Belyi-coverable curve is automatically defined over $\overline{\mathbb{Q}}$. Conversely, an explicit construction (iterated polynomial post-compositions, due to Belyi) builds a Belyi map from any rational function on a $\overline{\mathbb{Q}}$-curve. Grothendieck's Esquisse d'un programme (1984) interpreted the absolute Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ as acting faithfully on dessins, opening a combinatorial / Galois-theoretic route to study $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ directly.

Chow's theorem and the higher-dimensional analogue. Chow 1949 proved that every closed complex analytic subvariety of ${\mathbb{P}}^N$ is the analytification of a unique closed algebraic subvariety. The proof is local: a coherent analytic sheaf supported on a closed complex submanifold of ${\mathbb{P}}^N$ has finite-dimensional global sections (Cartan-Serre vanishing), and the multiplication maps $H^0({\mathbb{P}}^N,\mathcal{O}(d))\otimes H^0({\mathbb{P}}^N,\mathcal{O}(e))\to H^0({\mathbb{P}}^N,\mathcal{O}(d+e))$ become surjective for $d,e$ large; the image of these maps determines the homogeneous ideal of the subvariety algebraically. For a smooth projective complex curve, Chow's theorem is the algebraic identification of the very-ample-embedded image and is one input to the very-ample-embedding route to Riemann's existence theorem. The higher-dimensional analogue: every compact complex submanifold of ${\mathbb{P}}^N$ is a smooth projective algebraic variety. Compact complex manifolds in general are not all projective — complex tori of dimension $\ge 2$ failing the Riemann bilinear conditions provide examples — so the analytic-algebraic equivalence is special to the projective case.

The very-ample-embedding route, rigorously. Take $X$ a compact Riemann surface of genus $g$ and $L$ a holomorphic line bundle on $X$ of degree $d\ge 2g+1$. By Riemann-Roch and Serre duality, $\dim H^0(X,L)=d-g+1$ and the global sections separate every pair of points and tangent direction. The complete linear system $|L|$ gives a holomorphic embedding ${\iota }_L:X\hookrightarrow \mathbb{P}(H^0(X,L{)}^{\ast })={\mathbb{P}}^{d-g}$. By Chow's theorem, ${\iota }_L(X)$ is the analytification of a unique smooth projective algebraic curve $C\subset {\mathbb{P}}_{\mathbb{C}}^{d-g}$. Equivalently, $C=\mathrm{Proj}R(X,L)$ for the homogeneous coordinate ring $R(X,L):={⨁}_{n\ge 0}{H}^0(X,L^{\otimes n})$, finitely generated over $\mathbb{C}$ by the very-ample $H^0(X,L)$. The Hodge-theoretic content lies in the graded structure of $R(X,L)$: each summand has the dimension predicted by Riemann-Roch, and the multiplication is forced by the holomorphic structure of $X$.

Klein's discovery (1883) of the algebraicity of compact Riemann surfaces. Felix Klein, in Vorlesungen über Riemannsche Flächen and earlier papers (early 1880s), first stated the theorem in modern form: every compact Riemann surface comes from an algebraic curve. Klein's argument used the topology of fundamental polygons and the existence of meromorphic functions with prescribed pole behaviour. Riemann's 1857 paper had asserted the result on the basis of Dirichlet's principle, which Weierstrass criticised in 1870 for lacking a rigorous existence proof for the relevant minimisers. Hilbert's 1900 paper ^{[Hilbert 1900]} supplied the missing existence step in a refined form. Weyl 1913 ^{[Weyl 1913]} reorganised the entire foundations of Riemann surfaces around the Hodge-theoretic existence of harmonic forms, providing the modern proof of Riemann's existence theorem via the Hodge decomposition on a compact Riemann surface; the analogue on compact Kähler manifolds (Hodge 1941 ^{[Hodge 1941]}) extends the harmonic-form approach to higher dimensions and feeds directly into the GAGA framework via the analytic-de-Rham comparison.

Synthesis. Three perspectives converge on a single result. The analytic perspective: $X$ a compact Riemann surface produces $\mathcal{M}(X)$ as a finitely generated transcendence-degree-$1$ extension of $\mathbb{C}$ via Riemann-Roch and Serre duality, and the smooth projective curve associated to $\mathcal{M}(X)$ recovers $X$ topologically and complex-analytically. The algebraic perspective: a finitely generated transcendence-degree-$1$ extension $K/\mathbb{C}$ produces a unique smooth projective curve $C/\mathbb{C}$ via normalisation, whose analytification is a compact Riemann surface. The categorical perspective: GAGA assembles these into an equivalence of categories. Every theorem about compact Riemann surfaces — the genus-degree relationship, Hurwitz formula, Brill-Noether theory, the moduli space ${\mathcal{M}}_g$ of curves, the Schottky problem — is simultaneously a theorem about smooth projective algebraic curves over $\mathbb{C}$, and conversely. Riemann's existence theorem builds toward the variation of Hodge structure on the Jacobian (cf. 06.08.02), the moduli space of compact Riemann surfaces (cf. [alg-geom.moduli-of-riemann-surfaces]), and the period mapping into Siegel space; it appears again in the higher-dimensional setting through Hodge theory, GAGA, and Chow's theorem; it underlies Belyi's combinatorial reformulation of curves over $\overline{\mathbb{Q}}$ and the Grothendieck-Teichmüller programme; and it is the foundational lemma on which the bridge between complex analysis and algebraic geometry rests. Each of the four downstream applications cited here uses Riemann's existence theorem at its first step, and none of them would type-check without it.

Full proof set [Master]

Lemma (compact Riemann surfaces have a non-constant meromorphic function). For every compact Riemann surface $X$ of genus $g$ and every point $p\in X$, $\dim H^0(X,\mathcal{O}((2g-1)p))\ge g$, hence $X$ admits a non-constant meromorphic function with poles only at $p$.

Proof. Riemann-Roch on $X$ for the divisor $D=(2g-1)p$: $$ \dim H^0(X, \mathcal{O}(D)) - \dim H^0(X, \mathcal{O}(K - D)) ;=; \deg D - g + 1 ;=; g. $$ The divisor $K-D$ has degree $2g-2-(2g-1)=-1<0$, so $H^0(X,\mathcal{O}(K-D))=0$. Thus $\dim H^0(X,\mathcal{O}(D))=g\ge 1$ for $g\ge 1$. The constants are a one-dimensional subspace, so for $g\ge 1$ there exists a non-constant section, i.e. a meromorphic function with poles only at $p$ of total order at most $2g-1$. For $g=0$, $X={\mathbb{P}}^1$ has the meromorphic function $z$ already. $\square$

Lemma (very-ampleness of high-degree line bundles). For $X$ a compact Riemann surface of genus $g$ and $L$ a holomorphic line bundle on $X$ of degree $d\ge 2g+1$, $L$ is very ample.

Proof. The very-ampleness criterion: $L$ separates points and tangent directions iff for every effective divisor $D$ of degree $2$ on $X$, $\dim H^0(X,L\otimes \mathcal{O}(-D))=\dim H^0(X,L)-2$.

By Riemann-Roch, $\dim H^0(X,L)=d-g+1$, since $\mathrm{deg}(K\otimes L^{-1})=2g-2-d\le -3<0$ forces $H^0(X,K\otimes L^{-1})=0$. Similarly, $\dim H^0(X,L\otimes \mathcal{O}(-D))=(d-2)-g+1=(d-g+1)-2$, since $\mathrm{deg}(K\otimes L^{-1}\otimes \mathcal{O}(D))=2g-2-d+2\le -1<0$ also forces $H^0(X,K\otimes L^{-1}\otimes \mathcal{O}(D))=0$. The dimension drops by exactly $2$, so $L$ is very ample ^{[Forster §17]}. $\square$

Theorem (Riemann's existence theorem, function-field route, as proved at Intermediate tier). $\square$

Theorem (Riemann's existence theorem, very-ample-embedding route). Every compact Riemann surface $X$ of genus $g$ is biholomorphic to the analytification of a smooth projective algebraic curve over $\mathbb{C}$.

Proof. Pick any holomorphic line bundle $L$ on $X$ of degree $d\ge 2g+1$; for instance $L=\mathcal{O}(D)$ for an effective divisor $D$ of degree $d$ on $X$, or $L=K^{\otimes 3}$ on $X$ for $g\ge 2$ (the tricanonical bundle has degree $3(2g-2)=6g-6\ge 2g+1$ for $g\ge 2$). By the previous lemma, $L$ is very ample; the global sections embed $X$ as a closed complex submanifold $X\hookrightarrow {\mathbb{P}}^{d-g}$. By Chow's theorem ^{[Hartshorne App. B]}, the embedded image is the analytification of a unique smooth projective algebraic curve $C\subset {\mathbb{P}}_{\mathbb{C}}^{d-g}$. Equivalently, $C=\mathrm{Proj}{⨁}_{n\ge 0}{H}^0(X,L^{\otimes n})$ as a graded ring. The biholomorphism $C^{\mathrm{an}}\cong X$ is the analytification of the canonical $X\hookrightarrow {\mathbb{P}}^{d-g}$. The genus $g=0,1$ cases are handled separately: $g=0$ gives $X={\mathbb{P}}^1$ via $L=\mathcal{O}(1)$, and $g=1$ gives an elliptic curve via $L=\mathcal{O}(3p)$ for any $p\in X$ (degree $3=2g+1$). $\square$

Corollary (full faithfulness of analytification, GAGA route). For smooth projective algebraic curves $C_1,C_2$ over $\mathbb{C}$, the natural map ${\mathrm{Hom}}_{\mathbb{C}}(C_1,C_2)\to {\mathrm{Hom}}_{\mathrm{hol}}(C_1^{\mathrm{an}},C_2^{\mathrm{an}})$ is a bijection.

Proof. GAGA on the projective complex variety $C_1\times C_2$ ^{[Serre 1956]}: the analytification functor is fully faithful and essentially surjective on coherent sheaves, and in particular induces a bijection between closed algebraic subvarieties of $C_1\times C_2$ and closed analytic subvarieties of $(C_1\times C_2{)}^{\mathrm{an}}$. The graph of an algebraic morphism $f:C_1\to C_2$ is a closed algebraic subvariety ${\Gamma }_f\subset C_1\times C_2$ projecting isomorphically to $C_1$; the graph of a holomorphic map $f^{\mathrm{an}}:C_1^{\mathrm{an}}\to C_2^{\mathrm{an}}$ is a closed analytic subvariety ${\Gamma }_{f^{\mathrm{an}}}\subset (C_1\times C_2{)}^{\mathrm{an}}$ also projecting isomorphically. The GAGA bijection identifies ${\Gamma }_{f^{\mathrm{an}}}={\Gamma }_f^{\mathrm{an}}$ for a unique ${\Gamma }_f$, hence $f^{\mathrm{an}}=f^{\mathrm{an}}$ for a unique algebraic $f$. $\square$

Theorem (function-field-of-curves equivalence). The functor $C\mapsto K(C)$ from smooth projective integral curves over $\mathbb{C}$ to finitely generated field extensions of $\mathbb{C}$ of transcendence degree $1$ is a contravariant equivalence of categories.

Proof. The functor is fully faithful: a morphism $f:C_1\to C_2$ of smooth projective curves induces a homomorphism $f^{\ast }:K(C_2)\to K(C_1)$, and conversely every $\mathbb{C}$-algebra homomorphism $K(C_2)\to K(C_1)$ comes from a unique rational map $C_1⇢C_2$ (defined on a Zariski-dense open subset by the field-theoretic data); the rational map extends uniquely to a morphism by smoothness of $C_1$ and properness of $C_2$ (the valuative criterion of properness applied to the local rings of $C_1$). Essential surjectivity: given $K/\mathbb{C}$ finitely generated of transcendence degree $1$, pick a transcendence basis $\{t\}$ and write $K=\mathbb{C}(t)[{\alpha }_1,\dots ,{\alpha }_m]$ for finitely many algebraic generators. The integral closure of $\mathbb{C}[t]$ in $K$ is a Dedekind domain $R$, finitely generated as a $\mathbb{C}$-algebra; $\mathrm{Spec}R$ is a smooth affine curve with function field $K$. Glue the analogous construction at $t=\infty$ to obtain a smooth projective curve $C$ with $K(C)=K$. The construction is functorial in $K$ (up to canonical isomorphism). $\square$

Corollary (Riemann existence as a triple equivalence). The composition $$ \mathrm{CompactRiemannSurfaces}^{\mathrm{op}} ;\xrightarrow{;\mathcal{M};}; \big{\text{f.g. field extensions of } \mathbb{C} \text{ of transcendence degree } 1\big} $$ is an equivalence of categories, sending a compact Riemann surface to its meromorphic function field.

Proof. Compose Riemann's existence theorem (analytification equivalence) with the function-field-of-curves equivalence. The intermediate object is the smooth projective curve $C$ with $C^{\mathrm{an}}\cong X$ and $K(C)=\mathcal{M}(X)$; the equality $K(C)=\mathcal{M}(X)$ holds because GAGA identifies $K(C)$, the algebraic function field, with the meromorphic function field of $C^{\mathrm{an}}$, which equals $\mathcal{M}(X)$ via the biholomorphism. $\square$

Theorem (Belyi 1979). A smooth projective curve $C$ over $\mathbb{C}$ is defined over $\overline{\mathbb{Q}}$ iff $C$ admits a Belyi map $\beta :C\to {\mathbb{P}}^1$ branched only over $\{0,1,\infty \}$.

Proof sketch. The "$\Leftarrow$" direction: a Belyi-coverable curve $C$ is determined by combinatorial data on ${\mathbb{P}}^1\setminus \{0,1,\infty \}$ (a finite-index subgroup of the free group on two generators, by Riemann's existence theorem applied to the punctured base). Combinatorial data have no transcendental moduli, hence $C$ is defined over the field of algebraic numbers $\overline{\mathbb{Q}}$. The "$\Rightarrow$" direction is Belyi's iterative construction: starting with any non-constant rational function $f:C\to {\mathbb{P}}^1$ with branch locus contained in $\overline{\mathbb{Q}}$, repeatedly post-compose with polynomial maps moving branch points to $\{0,1,\infty \}$ while preserving the $\overline{\mathbb{Q}}$-structure. The algorithm terminates after finitely many steps; the result $\beta ={\pi }_k\circ \cdots \circ {\pi }_1\circ f$ is a Belyi map. The full proof appears in Belyi 1979 ^{[Belyi 1979]}. $\square$

Theorem (Chow 1949). Every closed complex analytic subvariety of ${\mathbb{P}}^N(\mathbb{C})$ is the analytification of a unique closed algebraic subvariety of ${\mathbb{P}}_{\mathbb{C}}^N$.

Proof sketch. Reduce by induction on dimension to the case of an irreducible closed analytic subvariety $V\subset {\mathbb{P}}^N$ of pure codimension $r$. The line bundle ${\mathcal{O}}_{{\mathbb{P}}^N}(d)$ has $\dim H^0({\mathbb{P}}^N,\mathcal{O}(d))=\left(\genfrac{}{}{0ex}{}{N+d}{N}\right)$, while $\dim H^0(V,{\mathcal{O}}_V(d))\le C\cdot d^{N-r}$ (Hilbert-polynomial growth on $V$). The kernel of the restriction map $H^0({\mathbb{P}}^N,\mathcal{O}(d))\to H^0(V,{\mathcal{O}}_V(d))$ has dimension $\ge \left(\genfrac{}{}{0ex}{}{N+d}{N}\right)-C\cdot d^{N-r}$, growing as $d^N-O(d^{N-r})$, hence non-zero for $d$ large. A homogeneous polynomial of degree $d$ vanishing on $V$ provides an algebraic equation. Iterating, the homogeneous ideal $I(V)$ of $V$ contains finitely generated polynomial data, sufficient to define $V$ algebraically. The full proof is in Hartshorne ^{[Hartshorne App. B]} or Griffiths-Harris ^{[Griffiths-Harris Ch. 2 §3]}. $\square$

Connections [Master]

• Branched coverings of Riemann surfaces 06.02.02. The previous unit identifies the category of finite branched covers of a compact Riemann surface $X$ with conjugacy classes of finite-index subgroups of ${\pi }_1(X\setminus B,x_0)$, the topological half of Riemann's existence theorem. The current unit completes the picture: combined with GAGA and the function-field-of-curves equivalence, these branched covers are also classified by finite separable extensions of the algebraic function field $K(C)=\mathcal{M}(X)$. The triple equivalence "branched covers — finite-index subgroups of ${\pi }_1$ — finite extensions of the function field" is the master statement, and the present unit supplies its algebraic vertex.

• Serre duality on a curve 06.04.04. Serre duality is the analytic-cohomological input to Riemann-Roch on a compact Riemann surface, and through Riemann-Roch supplies the dimension counts that drive both the function-field route and the very-ample-embedding route to Riemann's existence theorem. Without the duality $H^1(X,L)\cong H^0(X,K\otimes L^{-1}{)}^{\ast }$, the dimension computations of $H^0(X,L)$ for high-degree $L$ are not available, and the existence theorem cannot be proved.

• Riemann-Roch on compact Riemann surfaces 06.04.01. The Riemann-Roch formula $\dim H^0(X,\mathcal{O}(D))-\dim H^0(X,\mathcal{O}(K-D))=\mathrm{deg}D-g+1$ is the dimension-counting tool that produces non-constant meromorphic functions, gives the very-ampleness criterion for high-degree line bundles, and computes the multiplication tables in the homogeneous coordinate ring. Riemann's existence theorem rests on Riemann-Roch in two of its three classical proof routes.

• Hurwitz formula 04.04.02. A non-constant holomorphic map $f:C_1^{\mathrm{an}}\to C_2^{\mathrm{an}}$ between two compact Riemann surfaces is identified by Riemann's existence theorem with an algebraic morphism of smooth projective curves; the Hurwitz formula then computes the genus of $C_1$ from the genus of $C_2$, the degree of $f$, and the ramification data, all of which are now available algebraically. The Hurwitz formula and Riemann's existence theorem together give a complete numerical and categorical description of finite morphisms of curves.

• Variation of Hodge structure on the Jacobian 06.08.02. The Hodge structure on $H^1(X^{\mathrm{an}};\mathbb{C})=H^{1,0}(X)\oplus H^{0,1}(X)$ is an analytic invariant of a compact Riemann surface, but Riemann's existence theorem reorganises it as the Hodge structure on $H^1(C;\mathbb{C})$ for the corresponding algebraic curve, and the variation along moduli is the variation of Hodge structure on the universal Jacobian. Without Riemann's existence theorem, the Hodge structure has no algebraic-geometric content; with it, the Hodge structure becomes the bridge to algebraic-de-Rham cohomology and Galois representations.

• Schottky problem 06.06.08. The Schottky problem identifies which principally polarised abelian varieties $\tau \in {\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ are Jacobians of compact Riemann surfaces. Riemann's existence theorem ensures the Jacobi locus ${\mathcal{J}}_g$ is the same as the algebraic Schottky locus inside the moduli space ${\mathcal{A}}_g$; without the analytic-algebraic identification, ${\mathcal{J}}_g$ would be an analytic locus disconnected from algebraic-geometric methods.

• Riemann surface 06.03.01. The current unit's Master-tier statement upgrades the foundational Riemann-surface definition into an algebraic-geometric structure. The smooth projective curve $C$ associated to a compact Riemann surface $X$ is intrinsic, functorial, and unique up to isomorphism, and Riemann's existence theorem turns the analytic invariants of $X$ (genus, Hodge structure, automorphism group, branching of meromorphic maps) into algebraic invariants of $C$.

• Genus of a Riemann surface 06.03.02. The genus of a compact Riemann surface $X$ equals the genus of the corresponding smooth projective curve $C$ as defined by either $\dim H^0(C,{\Omega }_C^1)=g$ or the topological Euler characteristic $\chi (C^{\mathrm{an}})=2-2g$. Riemann's existence theorem makes the equality of these two invariants — analytic and algebraic — automatic.

• Stein Riemann surfaces 06.09.01. A non-compact Riemann surface is Stein, and the analytic-algebraic equivalence fails: Stein surfaces have plenty of holomorphic functions but typically no embedding into projective space, so they are not analytifications of algebraic curves. The compactness hypothesis in Riemann's existence theorem is essential, and the contrast with the Stein case explains why.

Historical & philosophical context [Master]

Bernhard Riemann announced the existence theorem for compact Riemann surfaces in his 1857 Theorie der Abel'schen Functionen (J. Reine Angew. Math. 54) ^{[Riemann 1857]}. His argument used Dirichlet's principle: the existence of a meromorphic function with prescribed pole behaviour reduces to the existence of a real-valued function minimising a Dirichlet-type energy integral. Karl Weierstrass criticised the principle in 1870 for assuming, without proof, that the minimising function exists; the variational problem is well-defined but the minimiser need not lie in any well-defined function space without further hypotheses. Hermann Schwarz and others worked on partial fixes throughout the 1870s and 1880s. The rigorous existence proof for Dirichlet's principle came from David Hilbert in 1900 ^{[Hilbert 1900]}, and Felix Klein in the early 1880s formalised the algebraic-curve identification in modern form.

Hermann Weyl 1913 Die Idee der Riemannschen Fläche ^{[Weyl 1913]} reorganised the theory of Riemann surfaces around the existence of harmonic forms and the Hodge decomposition on a compact Riemann surface, providing the modern proof of Riemann's existence theorem free of Dirichlet-principle complications. W. V. D. Hodge 1941 ^{[Hodge 1941]} extended the harmonic-form framework to compact Kähler manifolds, recovering Riemann's theorem on curves as the dimension-$1$ case and establishing the analytic foundation that Serre 1956 ^{[Serre 1956]} would algebraicise as GAGA. The categorical equivalence of analytic and algebraic geometry on projective complex varieties — the view that Riemann's existence theorem is really a statement about functors — emerges in Serre's paper and is the framework in which the theorem is taught today. Chow's theorem (Chow 1949) provided the higher-dimensional analogue of the very-ample-embedding step. Belyi's 1979 paper ^{[Belyi 1979]} extracted the surprising arithmetic content: a curve over $\mathbb{C}$ is defined over $\overline{\mathbb{Q}}$ iff it admits a Belyi map. Grothendieck's 1984 Esquisse opened the dessins-d'enfants programme, which connects Riemann's analytic existence theorem to the Galois group $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ via combinatorial graph data on Belyi-coverable curves.

Bibliography [Master]

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