The Euler characteristic of the structure sheaf is one minus the genus

Given: Full AC, a compact Riemann surface $X$ of topological genus $g$, and the compatible metrics on $X$, ${\mathcal{O}}_X$, and $K$ required by the Hodge inputs.

Full AC is used by the surface-classification, universal-coefficient, derived-sheaf-cohomology, long-exact-sequence, finiteness, and Hodge inputs. Its consequences ${\mathrm{A}\mathrm{C}}_{\omega }$ and DC supply the hypotheses of the real de Rham comparison and the smooth-module acyclicity argument (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), AC implies DC implies countable choice).

The genus $g$ is topological. For $g\ge 1$, the polygonal model has one vertex, $2g$ one-cells, and one two-cell attached by ${\prod }_{i=1}^g{a}_i{b}_i{a}_i^{-1}{b}_i^{-1}$; for $g=0$, the model is $S^2$ (Genus and Euler characteristic of a compact Riemann surface, Topological classification of compact Riemann surfaces, Polygonal schemas and paired boundary edges).

With coefficients in a field $F$, for $g\ge 1$ the cellular chain groups are $C_2=F$, $C_1=F^{2g}$, $C_0=F$, and both boundary maps vanish: each one-cell begins and ends at the sole vertex, and each generator has exponent sum zero in the attaching commutator word. For $g=0$, the sphere's CW model has one zero-cell and one two-cell, with zero boundary maps. Cellular homology computes singular homology (Cellular homology, Cellular homology computes singular homology).

For $F=\mathbb{R}$ or $\mathbb{C}$, the singular chain complex $C_{\ast }(X;F)$ is free over the PID $F$ and its dual cochain complex is the singular cochain complex with coefficients in $F$. The universal-coefficient exact sequence has zero Ext term because every $F$-module is free, so $H_{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}^q(X;F)\cong {\mathrm{Hom}}_F(H_q(X;F),F)$ (Singular simplices and singular chain groups with coefficients, Singular cochain complex with coefficients, The universal coefficient theorem for cohomology over a PID).

Under ${\mathrm{A}\mathrm{C}}_{\omega }$, the real de Rham comparison identifies real de Rham cohomology with continuous real singular cohomology. Integration identifies top-degree real de Rham cohomology of a closed connected oriented surface with $\mathbb{R}$ (De rham cohomology, De Rham vector-space comparison with continuous singular cohomology, Top de Rham cohomology of a closed connected oriented manifold is real).

Complex-valued smooth forms are the complexification of real-valued smooth forms. Since the exterior derivative is real-linear, kernels, images, and cohomology commute with this scalar extension by the unique real-plus-imaginary decomposition.

Closed smooth forms of positive degree are locally exact (Closed differential forms are locally exact).

The sheaves of smooth complex-valued $k$-forms are modules over the sheaf of real smooth functions; under full AC they are acyclic in positive sheaf-cohomology degrees (A smooth differential $k$-form, Sheaves of smooth-function modules are cohomologically acyclic).

The constant sheaf ${\mathbb{C}}_X$ is the sheaf of locally constant complex-valued functions (The constant sheaf is the sheaf of locally constant functions).

Sheaf-sequence exactness is stalkwise, and a short exact sequence of sheaves gives a long exact sequence in sheaf cohomology (Exact sequences of sheaves, A sequence of abelian sheaves is exact exactly when it is exact on every stalk, Long exact sequence of sheaf cohomology).

In a holomorphic coordinate, the coordinate formula for $d$ and the Cauchy–Riemann equations give $dF=F^{\mathrm{\prime }}(z)dz$ for a holomorphic function $F$. Such functions have local holomorphic primitives, and a holomorphic function with zero derivative on a connected domain is constant (The exterior derivative of a function is its differential, Complex differentiability is equivalent to real total differentiability together with a complex-linear derivative, with ${\mathrm{\partial }}_{\bar{z}}f=0$, or with the Cauchy–Riemann equations, Every complex analytic function has a primitive on a neighbourhood of each point, A holomorphic function with zero derivative on a domain is constant).

A holomorphic atlas gives the underlying smooth surface, and $K={\mathrm{\Lambda }}^{1,0}{T}^{\ast }X$ is its canonical holomorphic line bundle. Compatible metrics on $X$ and on each holomorphic line bundle exist by averaging smooth real metrics with the complex structures; this existence uses ${\mathrm{A}\mathrm{C}}_{\omega }$, supplied by full AC (Riemann surfaces and holomorphic atlases, Holomorphic functions are real analytic and smooth in their two real coordinates, Smooth manifolds and their smooth charts, Holomorphic line bundles and meromorphic sections on a Riemann surface, Hermitian metric and $L^2$ pairing on a compact Riemann surface).

The global holomorphic sections and degree-one sheaf cohomology of ${\mathcal{O}}_X(D)$ are finite-dimensional, and $\chi ({\mathcal{O}}_X(D))=\mathrm{\ell }(D)-i(D)$ (Finite-dimensionality of the cohomology of a divisor on a compact Riemann surface).

The global Dolbeault resolution identifies sheaf cohomology with smooth Dolbeault cohomology in degrees $0,1$ and gives $H^q(X,{\mathcal{O}}_X(E))=0$ for $q\ge 2$, without a finite-cover hypothesis (Cech--Dolbeault comparison for holomorphic line bundles on a compact Riemann surface).

For a holomorphic Hermitian line bundle $G$ on compact $X$, the Dolbeault groups are finite-dimensional with unique harmonic representatives, and the perfect complex-bilinear Hodge pairing gives $H^{0,1}(X,G{)}^{\ast }\cong H^0(X,K\otimes G^{\ast })$. In particular, for $G={\mathcal{O}}_X$ and $G=K$, their dual holomorphic spaces are $H^0(X,K)$ and $H^0(X,{\mathcal{O}}_X)$ respectively (Dolbeault cohomology of a compact riemann surface is finite dimensional, Harmonic star duality for line bundle valued dolbeault cohomology).

The zero-divisor bundle is canonically trivial, identifying ${\mathcal{O}}_X(0)$ with the structure sheaf ${\mathcal{O}}_X$ (Divisors, principal divisors and canonical divisors on a Riemann surface, The holomorphic line bundle associated to a divisor).

The local maximum-modulus principle says that if the modulus of a holomorphic function on a domain has an interior local maximum, then the function is constant (Local maximum modulus principle).