Quiver Full Flag Projective Semisimplicity Examples

There are no arrows, so $E_V=0$, and the stability conditions of the incidence lemma Quiver full-flag incidence has a smooth projective compactification are vacuous; the lemma gives $X_{\nu }=\mathrm{Flag}(V)$ as the total space of the rank-zero subbundle, of dimension $n(n-1)\mathrm{/}2$, and the map ${\pi }_{\nu }$ is the constant map to the point $E_V=0$, which is proper since it is projective.

The semismall inequality on a target stratum $T$ with fibre dimension $d$ is $2d+\dim T\le \dim X_{\nu }$; here there is just the point stratum. The image of ${\pi }_{\nu }$ is the single point, and its single fiber is all of $X_{\nu }$, so the defect $2\dim (\text{fiber})+\dim (\text{image})-\dim X_{\nu }$ equals $2\cdot n(n-1)\mathrm{/}2+0-n(n-1)\mathrm{/}2=n(n-1)\mathrm{/}2$, which is positive for $n\ge 2$; hence this proper incidence map is not semismall. For $n=0$ or $n=1$ the flag space is a point and the defect is zero, so the non-semismall boundary is exactly $n\ge 2$, and a general quiver-flag decomposition proof must allow this case.

In either word the graded flag base is a single point, since the unique flag for $(i,j)$ is $0\subset V_i\subset V$ and the unique flag for $(j,i)$ is $0\subset V_j\subset V$. There are no arrows other than $i\to j$, and $\mathrm{Hom}(V_i,V_j)=\mathbb{C}$, so $E_V={\mathbb{C}}^h$, with coordinates the $h$ arrow entries. For the word $(i,j)$ the first line is $V_i$, and stability at $r=1$ forces $x_h(V_i)\subseteq F_j^1=0$, so every arrow coordinate vanishes and the subbundle rank is zero; for the word $(j,i)$ the first line is $V_j$, stability at $r=1$ reads $x_h(F_i^1)=x_h(0)=0\subseteq V_j$ and at $r=2$ reads $x_h(V_i)\subseteq V_j$, both automatic, so every arrow coordinate is free and the rank is $h$. These are the rank counts ${\sum }_{h:i\to j}\mathrm{\#}\{(p,q):{\nu }_p=i,{\nu }_q=j,q<p\}$ of the incidence lemma Quiver full-flag incidence has a smooth projective compactification, namely $0$ for $(i,j)$ and $h$ for $(j,i)$.

The lemma's compactification adjoins a trivial line to the incidence subbundle, so the line-projective sources are ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}(0\oplus \mathcal{O})$, a point over the flag point, and ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}({\mathbb{C}}^h\oplus \mathcal{O})$, and the corresponding maps to ${\mathbb{P}}_{\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}}({\mathbb{C}}^h\oplus \mathbb{C})={\mathbb{P}}^h$ are the point $[0:1]$ and the identity ${\mathbb{P}}^h\to {\mathbb{P}}^h$ respectively. The inverse image of the affine chart $t\mathrm{\ne }0$ is, exactly as in the lemma, the total space of the subbundle over the flag point: the point $x=0$ in the first case and all of ${\mathbb{C}}^h$ in the second. For $h=0$ both words give the two point maps, so arbitrary multiplicity including the zero case is covered.