Smooth projective complex curve with formal-coordinate bundle

A smooth projective complex curve with specified projective presentation is a nonempty compact connected Riemann surface $X$, together with an integer $N\ge 1$ and a holomorphic embedding $\iota :X\hookrightarrow {\mathbb{C}\mathbb{P}}^N$ whose image is the common zero set in ${\mathbb{C}\mathbb{P}}^N$ of a family of homogeneous polynomials on ${\mathbb{C}}^{N+1}$ and is locally the graph of a holomorphic map from an open subset of $\mathbb{C}$ to ${\mathbb{C}}^{N-1}$ in an affine projective chart. Here ${\mathbb{C}\mathbb{P}}^N:=({\mathbb{C}}^{N+1}\setminus \{0\})\mathrm{/}{\mathbb{C}}^{\times }$ is the set of complex lines with the quotient topology; its standard complex charts are $U_i=\{[z_0:\cdots :z_N]:z_i\mathrm{\ne }0\}$ with coordinates $z_j\mathrm{/}{z}_i$ for $j\mathrm{\ne }i$. A holomorphic embedding means an injective holomorphic map with injective differential that is a homeomorphism onto its image. The embedding is included as a presentation of the input; no algebraization or comparison theorem is asserted.

For each $p\in X$, let ${\mathcal{O}}_{X,p}$ be the ring of holomorphic germs at $p$, let ${\mathfrak{m}}_p$ be its ideal of germs vanishing at $p$, and set . The formal-coordinate bundle is the following inverse system of holomorphic principal bundles and its inverse limit. For $k\ge 1$ let $G_k$ be the group of substitutions $a_{1}t+\cdots +a_k{t}^k$ modulo $t^{k+1}$ with $a_1\mathrm{\ne }0$, using the right-composition product; its complex manifold is ${\mathbb{C}}^{\times }\times {\mathbb{C}}^{k-1}$. Put ${\mathrm{Coord}}_X^{(k)}:={\coprod }_{p\in X}{\mathrm{Iso}}_{\mathbb{C}\text{-alg}}({\mathcal{O}}_{X,p}\mathrm{/}{\mathfrak{m}}_p^{k+1},\mathbb{C}[t]\mathrm{/}(t^{k+1}))\mathrm{.}$ A holomorphic chart $z:U\to z(U)\subseteq \mathbb{C}$ supplies ${\tau }_{z,p}^{(k)}$ by Taylor expansion at $z(p)$: a germ $f$ is sent to the degree-$k$ Taylor polynomial of $f\circ z^{-1}$ in $t$ about $z(p)$. The trivialization $U\times G_k\to {\mathrm{Coord}}_X^{(k)}{\mathrm{\mid }}_U$ sends $(p,g)$ to ${\varphi }_g^{-1}\circ {\tau }_{z,p}^{(k)}$. Give ${\mathrm{Coord}}_X^{(k)}$ the complex-manifold structure from these charts. On an overlap $w=h(z)$ put ${\rho }_p(u):=h(z(p)+u)-h(z(p))$ modulo $u^{k+1}$. Taylor substitution gives ${\tau }_{w,p}^{(k)}={\varphi }_{{\rho }_p}^{-1}{\tau }_{z,p}^{(k)}$, so the fiber coordinates satisfy $g_w={\rho }_p^{-1}{g}_z$. These transition maps are holomorphic: their finite jet coefficients are holomorphic functions of $p$, and inversion uses only division by powers of the nonzero linear coefficient. Taylor substitution also gives the cocycle on triple overlaps. Right multiplication by $G_k$ in each trivialization is compatible with these transitions, and is free and transitive on each fiber.

Truncation defines holomorphic maps ${\mathrm{Coord}}_X^{(k+1)}\to {\mathrm{Coord}}_X^{(k)}$ over $X$. Define with the inverse-limit topology; its specified pro-holomorphic structure means precisely this inverse system of finite-jet bundles. Locally it is , with . Its fiber at $p$ is ${\mathrm{Iso}}_{\mathbb{C}\text{-alg}}^{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}}({\widehat{\mathcal{O}}}_{X,p},\mathbb{C}\mathrm{⟦}t\mathrm{⟧})$: compatible truncated isomorphisms and their inverses pass to inverse limits in both directions. Its right action on ring isomorphisms is $\tau \cdot \rho ={\varphi }_{\rho }^{-1}\circ \tau$, as defined above. No finite-dimensional manifold structure is asserted on the inverse limit, and no global coordinate section is selected.

For the family clauses, a complex manifold of dimension $m$ means a Hausdorff second-countable space equipped with a covering atlas of homeomorphisms to open subsets of ${\mathbb{C}}^m$ whose transition maps are holomorphic; a map is holomorphic when its coordinate expressions are holomorphic. Let $S$ and $\mathcal{X}$ be complex manifolds, $\pi :\mathcal{X}\to S$ a proper holomorphic submersion (inverse images of compact sets are compact, and its complex differential $d\pi$ is surjective at every point), with every fiber $X_s:={\pi }^{-1}(s)$ a nonempty compact connected Hausdorff second-countable topological $2$-manifold without boundary, equipped with a holomorphic atlas to open subsets of $\mathbb{C}$. This family clause uses the underlying analytic curves and does not assert a relative projective embedding or an algebraization theorem. A pointed family has a finite ordered tuple of holomorphic sections $s_i:S\to \mathcal{X}$ of $\pi$ with pairwise disjoint images. Let ${\mathcal{I}}_i$ be the ideal sheaf of $s_i(S)$ and define the complete ${\mathcal{O}}_S$-algebra sheaf along the section by with augmentation ideal ${\widehat{\mathcal{I}}}_i$ the kernel of restriction to ${\mathcal{O}}_S$. For an open $W\subseteq S$, a relative formal coordinate is an isomorphism of complete ${\mathcal{O}}_W$-algebra sheaves ${\tau }_i:{\widehat{\mathcal{O}}}_{\mathcal{X},s_i(W)}\stackrel{\sim }{\to }{\mathcal{O}}_W\mathrm{⟦}{t}_i\mathrm{⟧}$ satisfying ${\tau }_i({\widehat{\mathcal{I}}}_i)=t_i{\mathcal{O}}_W\mathrm{⟦}{t}_i\mathrm{⟧}$. The right action is by ${\mathrm{Aut}}_{{\mathcal{O}}_W}{\mathcal{O}}_W\mathrm{⟦}{t}_i\mathrm{⟧}$, the group of continuous ${\mathcal{O}}_W$-algebra substitutions with zero constant term and invertible linear coefficient, as defined by the base-algebra version of the preceding definition. If nonempty, the relative-coordinate fiber is a torsor under this group by composition; the corresponding fiber over a point $s\in S$ is a torsor under $\mathrm{Aut}\mathcal{O}$. No local or global nonemptiness or global-section assertion is made for a family.

This definition supplies individual curves, formal-coordinate fibers, and the disjoint-section convention for families. It does not define or assert existence of a moduli stack of stable curves or its boundary strata.

Riemann surfaces and holomorphic atlases

Formal-coordinate change group Aut O

A formal-coordinate fibre is an Aut O torsor

A holomorphic function equals its Taylor series throughout the largest centred disc in its domain