Pointed analytic worldsheet and family

Use the supplied definition of a Riemann surface: a nonempty connected Hausdorff second-countable topological space with a holomorphic atlas of charts into open subsets of $\mathbb{C}$. A pointed compact analytic worldsheet is a tuple $(C;p_1,\dots ,p_N;z_1,\dots ,z_N)$ where $C$ is a compact Riemann surface, $N\ge 0$ is finite, the marked points $p_i\in C$ are pairwise distinct, and, for every $i\in \{1,\dots ,N\}$, the data specify an open neighborhood $U_i\subset C$, a radius $r_i>0$, and a biholomorphism $z_i:U_i\to D_{r_i}$ onto a disk $D_{r_i}=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<r_i\}$ with $z_i(p_i)=0$. The neighborhoods carrying local coordinates are pairwise disjoint. For a connected compact surface use the supplied topological genus $g$ and Euler characteristic $\chi =2-2g$. For a possibly disconnected compact fiber $C_b={\coprod }_{a=1}^{c_b}{C}_{b,a}$, use these invariants on each connected component; there are finitely many components because they are open and the fiber is compact. Define its total genus and Euler characteristic by $g_{\mathrm{t}\mathrm{o}\mathrm{t}}(C_b)={\sum }_{a}g(C_{b,a})$ and $\chi (C_b)={\sum }_a\chi (C_{b,a})=2c_b-2g_{\mathrm{t}\mathrm{o}\mathrm{t}}(C_b)$. An unqualified genus of a disconnected fiber means this total genus. Empty fibers, if present, have $c_b=g_{\mathrm{t}\mathrm{o}\mathrm{t}}(C_b)=\chi (C_b)=0$.

An analytic family of pointed compact worldsheets over a complex manifold $B$ is a tuple $(\pi :\mathcal{C}\to B;{\sigma }_1,\dots ,{\sigma }_N;z_1,\dots ,z_N)$ such that $\mathcal{C}$ and $B$ are complex manifolds, $\pi$ is a proper holomorphic submersion, every fiber $C_b={\pi }^{-1}(b)$ is a compact one-dimensional complex manifold without boundary (possibly disconnected, with each connected component carrying the supplied Riemann-surface structure), the ${\sigma }_i:B\to \mathcal{C}$ are holomorphic sections of $\pi$ with pairwise disjoint images, and, for every $i\in \{1,\dots ,N\}$, a local-coordinate germ $z_i$ is specified along ${\sigma }_i(B)$. Here proper means that inverse images of compact sets are compact, and submersion means that the complex differential $d\pi$ is surjective at every point. A vertical coordinate germ along ${\sigma }_i(B)$ means that locally over each base neighborhood $B_0\subseteq B$ there is a neighborhood $U_i\subseteq {\pi }^{-1}(B_0)$ for which $(z_i,\pi ):U_i\to D_{r_i}\times B_0$ is biholomorphic onto a neighborhood of $\{0\}\times B_0$ and $z_i\circ {\sigma }_i=0$. The family has no boundary; every connected component of each fiber used for conformal blocks meets a marked section.

Riemann surfaces and holomorphic atlases

Genus and Euler characteristic of a compact Riemann surface

Complex manifolds, holomorphic vector bundles and Hermitian metrics

The Riemann-surface supplier fixes that each fiber is a one-dimensional complex manifold without boundary. The complex-manifold supplier supplies the charts and holomorphic maps used for $B$, $\mathcal{C}$, and the total-space coordinates. The genus supplier is included because this page names the genus and Euler characteristic of compact fibers; it defines genus topologically and does not assert any analytic moduli-space result. This definition alone asserts neither the existence of a moduli space nor a Deligne–Mumford compactification.

Restriction to an open subset of $B$ and holomorphic base change preserve the analytic-family clauses, as verified in Stability of pointed analytic worldsheets under restriction and base change. That verification uses the analytic-family part of this definition; the imported genus convention's Axiom-of-Choice dependency is not used in that lemma.