Let $X$ be a compact complex analytic space with singular locus $X^{\mathrm{sing}}$. Suppose that $X\setminus X^{\mathrm{sing}}$ is a Riemann surface. If $X^{\mathrm{sing}} = \emptyset$, then $X$ is a compact Riemann surface and hence projective (i.e. $X$ admits a holomorphic embedding into some complex projective space).

In the general case where $X^{\mathrm{sing}}$ is not necessarily empty, does there exist an injective analytic mapping $j : X \to \mathbb{CP}^{N}$ to some projective space, such that $X$ is isomorphic to $j(X)$?


Yes, every proper 1-dimensional complex-analytic space $X$ admits a closed immersion (in the sense of locally ringed spaces over $\mathbf{C}$) into an analytic projective space and more specifically is the analytification of a 1-dimensional projective $\mathbf{C}$-scheme (uniquely determined up to unique isomorphism by the GAGA theorems). It also holds in the non-archimedean case in complete generality too.

Note that $X$ could be everywhere non-reduced, in which case $X^{\rm{sing}} = X$ (so $X - X^{\rm{sing}}$ is empty), so there is no need to mention $X^{\rm{sing}}$ at all.

Up to invoking GAGA cohomological comparison isomorphisms and GAGA algebraization of coherent analytic sheaves at appropriate steps, as well as Oka's coherence results for radicals and normalizations, the method is the same proof bootstrapping from the smooth case (a case that I assume you take as known, but see Remark 1 below) as for the proof of projectivity of arbitrary 1-dimensional proper schemes over a field. In particular, we do use some honest analytic input beyond the tools from the smooth case (no surprise, given the context for the question). The beautiful book "Coherent Analytic Sheaves" provides complete proofs of all analytic tools below (aside from the GAGA theorems, which are in Serre's paper).

In the argument below, I avoid all appeal to the analytic theory of ampleness, limiting ampleness considerations to the algebraic side. But finite-dimensionality of the space of global sections of coherent sheaves on $X$ is used all over the place (known by Cartan-Serre, and massively generalized by Grauert as proved near the end of the book "Coherent Analytic Sheaves", though in the present case can be deduced from the smooth case by standard coherence arguments due to finiteness of the normalization map for $X_{\rm{red}}$).

Remark 1: In Chapter VII, sections 1--2 of the book "Theory of Stein Spaces" one finds an elegant purely cohomological proof (using the "divisor exact sequence") that a connected compact Riemann surface $X$ admits (many) non-constant meromorphic functions. This argument uses as serious analytic input the finite-dimensionality of coherent sheaf cohomology. Thus, for such $X$ that provides a non-constant proper map $f:X \rightarrow \mathbf{CP}^1$, and such a map must have finite fibers and so is finite in the analytic sense. It therefore corresponds to a coherent sheaf of algebras on $\mathbf{CP}^1$, and that coherent sheaf of algebras is algebraic by GAGA, so that gives a proof of the projectivity of such smooth $X$ (since a scheme finite over a projective space is projective). Thus, the smooth case that we are taking as known can be proved in that way.

The merit of the argument in Remark 1 is that it applies essentially verbatim (up to minor adjustments to deal with non-rational points) to prove the projectivity in the non-archimedean case for proper rigid-analytic curves that are regular. Regularity is weaker than smoothness when the non-archimedean ground field $k$ is not perfect, but handling regularity is a crucial feature in practice since normalization of a reduced rigid-analytic curve is always regular but typically not smooth for imperfect ground fields $k$. Beware that for affinoid curves over $k$, geometric reducedness generally cannot be attained by killing nilpotents after a finite extension of $k$ when $p = {\rm{char}}(k)>0$ and $[k:k^p]$ is infinite, as can happen even for discretely-valued $k$ with awful residue field.

Remark 2: The method below is written so that it applies essentially verbatim (with only minor technical adjustments to deal with non-rational points, and replacing "smooth" with "regular") in the non-archimedean case (where analogues of all of the analytic inputs are known, largely by work of Kiehl) to reduce the projectivity (really even algebracity, by rigid-analytic GAGA) of proper rigid-analytic spaces of dimension 1 to the case of such spaces that are regular, which we have noted are handled by the method in Remark 1. In other words, the method we discuss also settles the non-archimedean case in complete generality. The desire to provide a method that applies also in the non-archimedean case partially governs the expository choices made below (for those who might care).

Now we finally take up the main task, to deduce projectivity for general $X$ from the smooth case. As we know from Oka's work, the radical $J$ of $\mathscr{O}_X$ is a coherent sheaf of ideals, so $J^n = 0$ for large $n$. Granting that the underlying reduced space $X_{\rm{red}} = V(J)$ is projective, to get the same for $X$ we consider $X_i = V(J^i)$ for $i = 1, 2, \dots, n$. Note that the closed immersion $X_{i+1} \hookrightarrow X_i$ is defined by a square-zero coherent ideal sheaf (i.e., $J^i/J^{i+1}$ is square-zero inside $O_X/J^{i+1}$) for $1 \le i < n$.

Assuming $X_{\rm{red}} = V(J)$ is algebraic, let's deduce the same for $X_i$ for every $1 \le i \le n$ by induction (so the general problem is brought down to the reduced case). If $n=1$ there is nothing to do, so assume $n>1$. Suppose $1 \le i < n$ with $X_i$ algebraic, and we shall deduce $X_{i+1}$ is algebraic. Since $J^i/J^{i+1}$ is a coherent sheaf on $V(J) = X_{\rm{red}}$, by the GAGA cohomological comparison for the algebraic $X_{\rm{red}}$ we have ${\rm{H}}^2(|X|, J^i/J^{i+1}) = 0$ by the algebraic theory. Hence, ${\rm{Pic}}(X_{i+1}) \rightarrow {\rm{Pic}}(X_i)$ is surjective due to arising from the exact sequence of sheaves $$0 \rightarrow J^i/J^{i+1} \rightarrow O_{X_{i+1}}^{\times} \rightarrow O_{X_i}^{\times} \rightarrow 1$$ (where the first map is $s \mapsto 1+s$). Thus, composing this surjections, the natural map ${\rm{Pic}}(X) \rightarrow {\rm{Pic}}(X_{\rm{red}})$ is surjective.

Letting $L_0$ be an ample line bundle on $X_{\rm{red}}$ (by our temporary hypothesis of projectivity in the reduced case), we can lift this to a line bundle $L$ on $X$. Coherent sheaf cohomology of $X$ vanishes beyond some finite degree (due to the existence of a finite Stein covering, as for any compact Hausdorff complex-analytic space, or by bootstrapping from the algebraicity of $X_{\rm{red}}$). By the same induction arguments through powers of the coherent radical which show that a line bundle on a proper noetherian scheme (over an affine base) is ample if its pullback to the underlying reduced scheme is ample, $L$ satisfies the cohomological criterion for ampleness (i.e., for any coherent sheaf $F$ on $X$ and all $m \gg_F 0$, $F \otimes L^{\otimes m}$ has vanishing higher cohomology and is generated by global sections). In particular, for any sufficiently large $N$, $L^{\otimes N}$ is generated by global sections and $\Gamma(X, L^{\otimes N}) \rightarrow \Gamma(X_{\rm{red}}, L_0^{\otimes N})$ is surjective.

Fix such an $N$, so the natural map $$f:X \rightarrow {\mathbf{P}}(\Gamma(X, L^{\otimes N})^{\ast})^{\rm{an}}$$ lifts the analogous map $$f_0:X_{\rm{red}} \rightarrow {\mathbf{P}}(\Gamma(X_{\rm{red}}, L_0^{\otimes N})^{\ast})^{\rm{an}}$$ between closed subspaces, and by GAGA the map $f_0$ is a closed immersion for large $N$ since $L_0$ is ample. But then $f$ is injective on underlying sets and hence is a proper morphism with finite fibers, so it is a finite morphism of complex-analytic spaces; i.e., it is classified by a coherent sheaf of algebras on the projective space target. By GAGA we can algebraize that coherent sheaf of algebras, so $X$ would be algebraic and hence projective; this just expresses that an analytic space finite (in the analytic sense) over a projective analytic space is itself projective (since a scheme finite over a projective scheme is projective).

That slog finally brings us down to the case when $X$ is reduced. We can ignore any isolated points on $X$, so we can assume all analytic irreducible components $Y_i$ of $X$ are 1-dimensional. Let $Y'_i \rightarrow Y_i$ be the finite analytic normalization map (thanks to Oka, or more classical methods, though 1-dimensional singularities can be incredibly nasty), so each $Y'_i$ is algebraic by the known smooth case. Let $Y' := \coprod Y'_i$, so $q:Y' \rightarrow X$ is the finite normalization and by reducedness of $X$ (and the analytic Nullstellensatz) the natural map $O_X \rightarrow q_{\ast}(O_{Y'})$ is an inclusion of coherent sheaves with coherent cokernel having finite support.

Let $x_i \in Y_i$ be a smooth point not on any other $Y_j$, so $x_i$ is a smooth point on $X$. Let $L = \otimes O(x_i)$, so $q^{\ast}(L)$ is ample on $Y'$ (by the algebraicity on $Y'$ and choice of the $x_i$'s). For any coherent $G$ on $X$, the natural map $G \rightarrow q_{\ast}(q^{\ast}(G))$ has coherent kernel and cokernel with finite support (isomorphism over $X^{\rm{sm}}$ that is the complement of a finite subset of $X$), so the natural map ${\rm{H}}^i(X, G) \rightarrow {\rm{H}}^i(Y', q^{\ast}(G))$ is an isomorphism for $i>0$. Hence, for any coherent $F$ on $X$, $F \otimes L^{\otimes m}$ has vanishing higher cohomology for large $m$ since the analogue holds on $Y'$ (where $L$ has ample pullback).

Thus, for any point $x \in X$, if $I_x$ denote the coherent ideal sheaf of $x$ we see that $\Gamma(X, L^{\otimes m}) \rightarrow L^{\otimes m}(x)$ is surjective for all large $m$ since ${\rm{H}}^1(X, I_x \otimes L^{\otimes m})=0$ for all large $m$ (depending on $x$). Applying to $x=x_i$ for each $i$, for all large enough $m$ we see that $\Gamma(X, L^{\otimes m})$ generates the fiber of $L^{\otimes m}$ at every $x_i$, so by coherence considerations if we pick a large enough $M$ then $\Gamma(X, L^{\otimes M})$ generates the stalks of $L^{\otimes M}$ away from an analytic set disjoint from all $x_i$'s. Such an analytic set must be finite (as every irreducible component contains some $x_i$ and is 1-dimensional), so for a big enough $M' \in M \mathbf{Z}$ we see that $L' := L^{\otimes M'}$ is generated by global sections.

It follows that we have a natural map $f:X \rightarrow {\mathbf{P}}(\Gamma(X, L')^{\ast})^{\rm{an}}$ that is injective on tangent spaces at each of the points $x_i \in X^{\rm{sm}}$, so $f$ has no positive-dimensional fibers (as such a fiber would have to contain some $Y_i$ for analyticity and dimension reasons, a contradiction by the tangential injectivity property at each $x_i$). Thus, $f$ has finite fibers, so by properness $f$ is a finite map in the analytic sense. Once again arguing with GAGA for coherent sheaves of algebras on projective space as we did above, $X$ is algebraic. Voila.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy