Ytdyklly

Does there exist a countable set of connected proper smooth $mathbb{C}$-schemes such that any connected proper smooth $mathbb{C}$-scheme admits a $mathbb{C}$-immersion into one of them?

I am just posting my comment as one answer. So long as you are only asking about schemes (rather than complex analytic spaces), you can avoid the hard analysis from the previous MathOverflow answer. The argument below sketches this. The main additional detail beyond Hilbert scheme techniques is a strong variant of Chow's Lemma.

Chow's Lemma. Let $X$ be a separated, finitely presented scheme over a field $k$, and let $Usubset X$ be a dense open subscheme that is a quasi-projective $k$-scheme. There exists a strongly projective morphism $nu:widetilde{X}to X$ such that $widetilde{X}$ is a quasi-projective $k$-scheme and such that the restriction of $nu$ over $U$ is an isomorphism.

There may be an earlier source, but the source that I know is the following article.

MR0308104 (46 #7219)

Raynaud, Michel; Gruson, Laurent

Critères de platitude et de projectivité. Techniques de "platification'' d'un module.

Invent. Math. 13 (1971), 1–89.

Finally, the very last step of the argument requires Nagata compactification.

Nagata compactification Every separated, finite type $k$-scheme is isomorphic to a dense open subscheme of a proper $k$-scheme.

Definition. For every separated morphism that restricts as an isomorphism over a dense open in the target, the isomorphism locus is the maximal open subscheme of the target over which the morphism is an isomorphism. For a separated, flat, finite type morphism of schemes, $$rho:mathcal{X}to B,$$ a Chow covering is an ordered $n$-tuple (for some integer $ngeq 0$) of pairs of $B$-morphisms, $$(nu_ell:widetilde{mathcal{X}}_ell to mathcal{X}, e_ell:widetilde{mathcal{X}}_ell to mathbb{P}^m_B),$$ where each $e_ell$ is a locally closed immersion of flat $B$-schemes and where the morphisms $nu_ell$ are strongly projective morphisms whose isomorphism loci give an open covering of $mathcal{X}$.

Corollary. Every separated, finite type $k$-scheme has a Chow covering.

Proof. Since $X$ is quasi-compact, there exists a finite covering of $X$ by open affine subschemes. By Chow's Lemma, for each open affine, there exists a strongly projective $k$-morphism from a quasi-projective $k$-scheme to $X$ whose isomorphism locus contains that open affine. QED

Hypothesis. The field $k$ has characteristic $0$.

Definition. A smooth Chow covering is a Chow covering such that every $B$-scheme $widetilde{mathcal{X}}_ell$ is smooth over $B$.

Corollary. Every smooth, separated, quasi-compact $k$-scheme has a smooth Chow covering.

Proof. This follows from the previous corollary and Hironaka's Theorem. QED

Notation. For every smooth Chow covering over $B$, denote by $widetilde{mathcal{X}}$ the disjoint union of the $B$-schemes $widetilde{mathcal{X}}_ell$. Denote by $nu:widetilde{mathcal{X}}to mathcal{X}$ the unique $B$-morphism whose restriction to each component $widetilde{mathcal{X}}_ell$ equals $nu_ell$. Denote by $mathcal{Y}$ the disjoint union over all ordered pairs $(j,ell)$ with $1leq j,ellleq n$ of the closed subscheme of the product, $$mathcal{Y}_{j,ell}:= widetilde{mathcal{X}}_jtimes_{mathcal{X}} widetilde{mathcal{X}}_ell subseteq widetilde{mathcal{X}}_jtimes_{B} widetilde{mathcal{X}}_ell,$$ together with its two projections, $$text{pr}_{1,(j,ell)}:mathcal{Y}_{j,ell}to widetilde{mathcal{X}}_j, text{pr}_{2,(j,ell)}:mathcal{Y}_{j,ell}to widetilde{mathcal{X}}_ell.$$ Denote the disjoint union of these morphisms by $$text{pr}_1:mathcal{Y} to widetilde{mathcal{X}}, text{pr}_2:mathcal{Y}to widetilde{mathcal{X}}.$$ For every $ell=1,dots,n$, denote the diagonal morphism by $$delta_ell:widetilde{mathcal{X}}_ell to mathcal{Y}_{ell,ell}.$$ Denote the disjoint union of these morphisms by $$delta:widetilde{mathcal{X}}to mathcal{Y}.$$ For every $(j,ell)$, denote the involution of $B$-schemes that transposes the factors by $$sigma_{j,ell}:mathcal{Y}_{j,ell}to mathcal{Y}_{ell,j}.$$ Denote the disjoint union of these involutions by the involution, $$sigma:mathcal{Y}to mathcal{Y}, text{pr}_2circ sigma = text{pr}_1, text{pr}_1circ sigma = text{pr}_2.$$
For every ordered triple $(j,ell,r)$ with $1leq j,ell,rleq n$, denote by $$c_{j,ell,r}:mathcal{Y}_{j,ell}times_{widetilde{mathcal{X}}_ell} mathcal{Y}_{ell,r} to mathcal{Y}_{j,r},$$ the morphism induced by the first and final projections. Denote the disjoint union of these morphisms by $$c:mathcal{Y}times_{text{pr}_2,widetilde{mathcal{X}},text{pr}_1} mathcal{Y} to mathcal{Y}.$$

Definition. For every $k$-scheme $B$, a Chow descent predatum over $B$ is a collection of projective $B$-schemes and morphisms of $B$-schemes, $$((pi_ell:widetilde{mathcal{X}}_ell to B, ((text{pr}_{1,(j,ell)},text{pr}_{2,(j,ell)}):mathcal{Y}_{j,ell}hookrightarrow widetilde{mathcal{X}}_jtimes_{B}widetilde{mathcal{X}}_ell)_{j,ell}, (delta_ell:widetilde{mathcal{X}}_ellto Y_{ell,ell})_ell, (sigma_{j,ell}:Y_{j,ell}to Y_{ell,j})_{j,ell}, (c_{j,ell,r}:Y_{j,ell}times_{text{pr}_2,widetilde{X}_ell,text{pr}_1} Y_{ell,r}to Y_{j,r})_{j,ell,r})$$ together with a collection of locally closed immersions of $B$-schemes, $$e_ell:widetilde{mathcal{X}}_ellhookrightarrow mathbb{P}^m_B.$$ The emph{isomorphism locus} is the maximal open subscheme $U_ell$ of $widetilde{X}_ell$ on which the closed immersion $delta_ell$ is an open immersion.

Constraint axioms for a Chow datum. As the $1$-truncation of the simplicial scheme induced by $nu$, there are many constraints satisfied by the Chow descent predatum of a Chow covering. Taken together, these constraints define a Chow descent datum.

Constraint 1. For every $(j,ell)$, the $text{pr}_1$-inverse image in $Y_{j,ell}$ of $U_j$ is an open subscheme whose closed complement in $Y_{j,ell}$ equals its total inverse image under $text{pr}_2$ of its closed image in $widetilde{X}_ell$. Denote by $widetilde{X}_{ell,j}$ the open complement in $widetilde{X}_ell$ of this closed image.

Constraint 2. For each $ell=1,dots,n$, the collection of open subschemes $$(widetilde{X}_{ell,j})_{j=1,dots,n},$$
form an open covering of $widetilde{X}_ell$. Stated differently, the $n$-fold intersection of the closed complements is empty.

Constraint 3. Each projection $$text{pr}_{2,(j,ell)}:Y_{j,ell} to widetilde{X}_ell,$$ restricts to an isomorphism over $widetilde{X}_{ell,j}$. Thus, the inverse image of $widetilde{X}_{ell,j}$ under this isomorphism equals the graph of a unique $k$-morphism, $$nu_{ell,j}:widetilde{X}_{ell,j} to U_j.$$

Constraint 4. Denote by $U_{ell,j}$ the intersection of $U_ell$ and the $nu_{ell,j}$-inverse image of $U_j$ in $widetilde{X}_{ell,j}$. For each triple $(j,ell,r)$, the inverse image in $U_{ell,j}$ under $nu_{ell,j}$ of $U_{j,r}$ equals $U_{ell,j}cap U_{ell,r}$. Denote this open by $U_{ell,j,r}$. Also, the inverse image in $widetilde{X}_{ell,j}$ under $nu_{ell,j}$ of $U_jcap widetilde{X}_{j,r}$ equals the inverse image in $widetilde{X}_{ell,r}$ under $nu_{ell,r}$ of $U_rcap widetilde{X}_{r,j}$. Denote this open by $widetilde{X}_{ell,j,r}$.

Constraint 5. On the open $U_{ell,j,r}$, the composition $nu_{j,r}circ nu_{ell,j}$ equals $nu_{ell,r}$. Similarly, on the open $widetilde{X}_{ell,j,r}$, the composition $nu_{j,r}circ nu_{ell,j}$ equals $nu_{ell,r}$.

Effectivity Proposition. Every Chow descent datum is effective, i.e., it is isomorphic to the Chow descent datum of a Chow covering.

Proof Altogether, these constraints are the descent conditions for the Zariski descent datum of $k$-schemes, $$((U_ell)_ell, (U_{ell,j}subset U_ell)_{ell,j}, (nu_{ell,j}:U_{ell,j}to U_j)_{ell,j})$$ to glue to a $k$-scheme $X$, together with the descent conditions for each Zariski descent datum of $k$-morphisms, $$(widetilde{X}_{ell,j}to U_j)_j,$$ to glue to a $k$-morphism $$nu_ell:widetilde{X}_ell to X.$$ QED

Constraint 6. The morphisms $sigma$ and $c$ equal the natural morphisms induced by this effective descent datum.

Nota bene. Since we have already incorporated the glueing constraints above, Constraint 6, and even the data of the morphisms $sigma$ and $c$, are extraneous. However, from the point of view of a $1$-truncation of the simplicial scheme of the Chow covering, it seems best to include this data as part of the definition.

Constructibility Proposition. For every separated, finite type $k$-scheme $B$, for every Chow descent predatum over $B$, there exists a finite collection of locally closed subschemes of $B$ (with the induced reduced structure) whose set of geometric points are precisely those geometric points of $B$ where the Chow descent predatum is a Chow descent datum.

Proof sketch. Each of these constraint conditions involves equalities or inclusions of closed subsets of $widetilde{X}_ell$, it involves a morphism being an isomorphism, or it involves certain associated morphisms to a self-fiber product factoring through the diagonal. Pass to flattening stratifications of the base for the relevant closed subschemes, etc. Then each of these conditions defines a locally closed subscheme of the base. QED

Thus, we can form a countable collection of families of Chow data satisfying the constraint conditions in the "usual way". For every smooth Chow covering, for the associated Chow descent predatum,$$(([widetilde{X}_ell])_{ell},([Y_{j,ell}])_{j,ell},([delta_ell])_{ell},([sigma_{j,ell}])_{j,ell}, ([c_{j,ell,r}])_{j,ell,r}),$$ the first two parts of the datum are points in appropriate Hilbert schemes. The last three parts are points in appropriate Hom schemes.

Altogether, there are countably many ordered pairs $(n,m)in mathbb{Z}_{geq 0}times mathbb{Z}_{geq 0}$. For each such pair, there are countably many ordered $n$-tuples of Hilbert polynomials for the closed subschemes $widetilde{X}_ell$ of $mathbb{P}^m_k$. For each such ordered $n$-tuple of Hilbert polynomials, for the associated $n$-fold product of Hilbert schemes parameterizing $(widetilde{X}_ell)_{ell=1,dots,n}$, there are countably many $n^2$-tuples of Hilbert polynomials for the closed subschemes $Y_{j,ell}$. For each such tuple, there is a further relative Hilbert scheme parameterizing closed subschemes $Y_{j,ell}$ in $widetilde{X}_jtimes_{text{Spec} mathbb{C}}widetilde{X}_ell$. Then there are countably many components of each Hom scheme for the morphisms $delta_ell$, $sigma_{j,ell}$ and $c_{j,ell,r}$. For each of these countably many quasi-projective $k$-scheme parameterizing a Chow descent predatum, there are finitely many locally closed subschemes (with reduced structures) parameterizing those Chow descent predata that are Chow descent data (i.e., satisfy the constraint conditions). Over this countable disjoint union of quasi-projective $k$-schemes, we glue together the isomorphism loci to form a morphism $mathcal{X}to B$ and the $B$-morphisms $nu_ell$. By the existence of Chow coverings, every proper $k$-scheme occurs as a fiber over a $k$-point of one of the schemes $B$ in the countable collection.

In the end, for each of the countably many data, $iin I$, of $(m,n)$ and all of the relevant Hilbert polynomials, there is a quasi-projective $mathbb{C}$-scheme (with its induced reduced scheme structure), $B_i$, and a datum, $$(pi_i:mathcal{X}_i to B_i, (widetilde{mathcal{X}}_{i,ell} subset mathcal{X}_itimes_{text{Spec} mathbb{C}} mathbb{P}^m_{mathbb{C}})_{ell})$$ of a smooth, proper morphism $pi_i$ and a collection of closed subschemes $widetilde{mathcal{X}}_{i,ell}$ such that each projection, $$widetilde{mathcal{X}}_{i,ell} to B_itimes_{text{Spec} mathbb{C}}mathbb{P}^m_{mathbb{C}},$$ is a closed immersion of a smooth $B_i$-scheme into projective space over $B_i$, and such that the projections, $$(widetilde{mathcal{X}}_{i,ell}to mathcal{X}_i)_{ell=1,dots,n}$$ define a smooth Chow covering. Every smooth, proper $mathbb{C}$-scheme occurs as a fiber of some $pi_i$.

Finally, by applying Hironaka's Theorem to each $B_i$, we can assume that each $B_i$ is a smooth quasi-projective $k$-scheme. The effect is that every $mathcal{X}_i$ is itself a separated, quasi-compact, smooth $k$-scheme. It certainly is not proper. However, we can apply Nagata compactification and Hironaka's Theorem (for a third time) to realize $mathcal{X}_i$ as a dense open subscheme of a proper, smooth $k$-scheme.