Quiver full-flag incidence has a smooth projective compactification

For each vertex $i$, forgetting the repeated colors identifies the flag data of colors in $V_i$ with a single full flag of $V_i$, so ${\mathcal{F}}_{\nu }$ is identified with ${\prod }_i\mathrm{Flag}(V_i)$: a graded increasing flag with successive colors ${\nu }_1,\dots ,{\nu }_m$ is exactly a choice, for every occurrence of $i$ in $\nu$, of a strictly increasing chain of subspaces of $V_i$ ending at $V_i$. The full flag variety of a $\beta$-dimensional space $W$ is the closed incidence locus of successive inclusions of tautological subspaces inside ${\prod }_{d=1}^{\beta -1}\mathrm{Gr}(d,W)$: over the product of Grassmannians the composite of the tautological monomorphism $S_d\hookrightarrow W$ with the tautological quotient $W\twoheadrightarrow W\mathrm{/}{S}_{d^{\mathrm{\prime }}}$ for $d<d^{\mathrm{\prime }}$ vanishes exactly on the pairs of subspaces with $S_d\subseteq S_{d^{\mathrm{\prime }}}$, a closed condition. Each Grassmannian is projective by [F2] and [F3], finite products of nonempty projective varieties are projective by [F4], so $\mathrm{Flag}(W)$ is a nonempty projective algebraic set. It is irreducible: $\mathrm{GL}(W)$, the nonvanishing-determinant open of a matrix affine space, is irreducible and maps surjectively to the flag set by transporting a fixed flag, since every flag admits an adapted basis. Thus it is a projective variety; empty products and $\beta \in \{0,1\}$ give a point.

Successively choosing a line in the remaining quotient exhibits $\mathrm{Flag}(W)$ as an iterated projective-space bundle with relative dimensions $\beta -1,\beta -2,\dots ,1$: the step adding the next subspace has fibre the projective space of lines in the quotient by the previously chosen subspace, and the standard charts of [F2] trivialize each step algebraically. Hence $\mathrm{Flag}(W)$ is smooth of dimension $\beta (\beta -1)\mathrm{/}2$, and the chart computations give algebraic local bases adapted to the tautological flag, in which the flag is the span of the first $r$ basis vectors. Applied factorwise this makes ${\mathcal{F}}_{\nu }={\prod }_i\mathrm{Flag}(V_i)$ a smooth projective variety of dimension ${\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2$, covered by charts carrying an adapted local basis of $V$.

On such an adapted-basis chart, label each basis vector $e_p$ by its occurrence $p$ in $\nu$, so $F^r$ is spanned by the $e_p$ with $p\le r$. Stability says that the coefficient of $e_q$ in $x_h{e}_p$ vanishes whenever $q>p$; a loopless arrow has distinct source and target colors, so $q=p$ never occurs, and the free coefficients are exactly the pairs with ${\nu }_p=i$, ${\nu }_q=j$, $q<p$ for an arrow $h:i\to j$, in the number $r_{\nu }={\sum }_{h:i\to j}\mathrm{\#}\{(p,q):{\nu }_p=i,{\nu }_q=j,q<p\}$ of the Statement. These defining equations are linear in the representation coordinates with $r_{\nu }$ independent free coordinates, so on the chart the incidence is the product of the chart with a linear subspace of $E_V$; changes of adapted basis preserve the conditions and act linearly on the coordinates, so the local kernels glue to a vector subbundle $W_{\nu }\subseteq E_V\times {\mathcal{F}}_{\nu }$ with locally free quotient, whose total space is exactly $X_{\nu }$. Consequently $\dim X_{\nu }=r_{\nu }+{\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2$ by step 2.1, and the action of $g\in G_V$ on matrices and on flags by translation preserves the incidence equations, so $W_{\nu }$ is $G_V$-equivariant.

By step 1.1 the flag variety ${\mathcal{F}}_{\nu }$ is a closed subvariety of a product of projective spaces, hence of some ${\mathbb{P}}^N$, and $E_V\times {\mathcal{F}}_{\nu }$ is closed in the relative projective space ${\mathbb{P}}_{E_V}^N$. Since $X_{\nu }\subseteq E_V\times {\mathcal{F}}_{\nu }$ is closed by step 3.1, the map ${\pi }_{\nu }$ is a closed immersion followed by that projection, hence projective, and proper by [F5], which consumes the assumed AC as recorded in [A1]. No paving or semismallness hypothesis enters this argument.

Definition of the compactification: ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(U)=\mathrm{Proj}\mathrm{Sym}(U^{\vee })$ is the instance of the quotient convention of [F1] for the dual bundle, so on a local frame it is again a projective-space bundle with homogeneous coordinates the coefficients of a vector. The inclusion $W_{\nu }\oplus \mathcal{O}\hookrightarrow (E_V\oplus \mathbb{C})\times {\mathcal{F}}_{\nu }$ of vector bundles over ${\mathcal{F}}_{\nu }$ induces a closed immersion of line projectivizations ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(W_{\nu }\oplus \mathcal{O})\hookrightarrow {\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(E_V\oplus \mathbb{C})\times {\mathcal{F}}_{\nu }$ over ${\mathcal{F}}_{\nu }$, because on a local frame the equations are the same homogeneous linear equations with the extra coordinate $t$ unconstrained. Composing with the projection to ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(E_V\oplus \mathbb{C})$ gives the $G_V$-equivariant map ${\bar{\pi }}_{\nu }$, which is projective; its source is projective by its displayed closed embedding into the product of projective varieties [F4], and smooth because [F1] gives local products of the smooth base ${\mathcal{F}}_{\nu }$ with ${\mathbb{P}}^{r_{\nu }}$, including $r_{\nu }=0$. The affine chart $t\mathrm{\ne }0$ of the target consists of the lines spanned by vectors $(w,1)$ with $w\in E_V$, and its inverse image under the closed immersion consists of the $(w,1)$ with $w\in W_{\nu ,F}$, so as a set and scheme-theoretically it is exactly $X_{\nu }$, with ${\bar{\pi }}_{\nu }(w,F)=w={\pi }_{\nu }(w,F)$ there.

For the finite associated version fix $N_i\ge {\beta }_i$ and put $B_N={\prod }_{{\beta }_i>0}\mathrm{Gr}({\beta }_i,N_i)$ with tautological subbundles $S_i\subseteq {\mathcal{O}}^{N_i}$ for ${\beta }_i>0$, setting $S_i=0$ for ${\beta }_i=0$, and $E_N={⨁}_{h:i\to j}\mathcal{H}om(S_i,S_j)$ over $B_N$. Let ${\mathcal{F}}_{\nu ,N}$ be the relative full flag variety over $B_N$ whose points over $(S_i{)}_i$ are the increasing full graded flags of ${⨁}_i{S}_i$ with successive colors ${\nu }_r$, recording the terminal ${\beta }_i$-plane $S_i\subseteq {\mathbb{C}}^{N_i}$. Recording all intermediate subspaces and the terminal $S_i$ realizes this variety as the product, over ${\beta }_i>0$, of the closed inclusion loci in ${\prod }_{d=1}^{{\beta }_i}\mathrm{Gr}(d,N_i)$; hence it is projective by [F3] and [F4]. Over a Grassmannian frame chart it is a product of that chart with ${\prod }_i\mathrm{Flag}({\mathbb{C}}^{{\beta }_i})$, so by [F2] and step 2.1 it is smooth and irreducible, of dimension $s_N+f$, where $s_N={\sum }_i{\beta }_i(N_i-{\beta }_i)$ and $f={\sum }_i{\beta }_i({\beta }_i-1)\mathrm{/}2$. The adapted-basis computation of step 3.1 applies verbatim to $E_N$ and to the chains inside the $S_i$, so the associated incidence is the total space of a vector subbundle $W_{\nu ,N}\subseteq E_N{\times }_{B_N}{\mathcal{F}}_{\nu ,N}$ of the same rank $r_{\nu }$.

Absolute projectivity of the target needs no general compactification theorem. Arrows with a zero-dimensional endpoint contribute the zero Hom bundle and may be omitted. Put $D={⨂}_{{\beta }_i>0}\det S_i$ (the trivial line for the empty product); for each remaining arrow, $S_i^{\vee }\otimes \det S_i\cong {\bigwedge }^{{\beta }_i-1}{S}_i$, so $\mathcal{H}om(S_i,S_j)\otimes D\cong {\bigwedge }^{{\beta }_i-1}{S}_i\otimes S_j\otimes {⨂}_{l\mathrm{\ne }i}\det S_l$, and $D$ itself is a tensor product of the $\det S_l$. Each $S_i\subseteq {\mathcal{O}}^{N_i}$ is a subbundle with locally free quotient, hence so are its exterior powers, and tensor products of such subbundles are again subbundles of finite trivial bundles with locally free quotients; therefore $(E_N\oplus \mathcal{O})\otimes D$ is a subbundle of a finite trivial bundle ${\mathcal{O}}^M$. Since $D$ is a line bundle, twisting identifies ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}((E_N\oplus \mathcal{O})\otimes D)={\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(E_N\oplus \mathcal{O})$, which is thus a closed subvariety of the smooth projective $B_N\times {\mathbb{P}}^{M-1}$ by [F1] and [F4], hence projective. It is smooth by its projective-space-bundle charts over smooth $B_N$, not merely by closedness.

The incidence inclusion of step 3.1 embeds ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(W_{\nu ,N}\oplus \mathcal{O})$ closed in ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(E_N\oplus \mathcal{O}){\times }_{B_N}{\mathcal{F}}_{\nu ,N}$ over $B_N$, exactly as in step 4.2 with base $B_N$. The fiber product is closed in the product of the two smooth projective varieties, being the inverse image of the closed diagonal of $B_N$, so the source is projective by step 5.1, and smooth by its projective-space-bundle charts over the smooth ${\mathcal{F}}_{\nu ,N}$ of step 4.3. The induced map ${\bar{\pi }}_{\nu ,N}$ is projective: its graph is closed in the product of its projective source with the target, since the target has closed diagonal, and a projective embedding of the source gives a closed immersion of the graph into a relative projective space over the target. Thus it is a projective map between smooth projective varieties. Its inverse image of the affine chart $t\mathrm{\ne }0$ is, by the same unique-representative argument as in step 4.2, the total space of $W_{\nu ,N}$ over ${\mathcal{F}}_{\nu ,N}$, mapped to $E_N$ by $(w,b,F)\mapsto w$, which is the associated incidence map. The empty dimension, zero arrow and arbitrary finite multiplicity cases are included throughout, since they only remove factors or arrows from the constructions above.