Smooth-Projective Serre Duality and Flag-Variety Line Bundles — Examples

The flag variety as lines in ${\mathbb{C}}^2$ and its two charts. Let $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in G$. Since $g{e}_1=a{e}_1+c{e}_2$, the line $\mathbb{C}{e}_1$ is $g$-stable exactly when $c=0$; the matrices $\left(\begin{array}{cc}a& b\\ 0& a^{-1}\end{array}\right)$ with $a\in {\mathbb{C}}^{\times }$ and $b\in \mathbb{C}$ are exactly the products $\mathrm{diag}(a,a^{-1})u_{\alpha }(b\mathrm{/}a)$, hence form the subgroup $B=T⋉U_{\alpha }$, which is the upper triangular Borel of [F2], [F3] and [F5]. The action on lines is transitive: a nonzero vector $v=(v_1,v_2)$ with $v_1\mathrm{\ne }0$ equals $v_1(e_1+z{e}_2)$ with $z=v_2\mathrm{/}{v}_1$, and $u_{-\alpha }(z)e_1=e_1+z{e}_2$ by [F3], while $v=(0,v_2)=v_2{n}_{\alpha }{e}_1$ with $n_{\alpha }=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$. Hence $gB\mapsto \mathbb{C}g{e}_1$ is well defined, because $b\in B$ acts on $e_1$ by the scalar $a$ with $\mathrm{diag}(a,a^{-1})\in T$, and it is a bijection of $G\mathrm{/}B$ onto the set of lines: surjectivity is the transitivity just proved and injectivity follows because $\mathbb{C}{g}^{-1}{g}^{\mathrm{\prime }}{e}_1=\mathbb{C}{e}_1$ forces $g^{-1}{g}^{\mathrm{\prime }}\in B$. Under this bijection the $z$-chart $u_{-\alpha }(z)B$ consists of the lines $\mathbb{C}(e_1+z{e}_2)$ with $Z_1\mathrm{/}{Z}_0=z$ and the $s$-chart $u_{\alpha }(s)n_{\alpha }B$ consists of the lines $\mathbb{C}(s{e}_1+e_2)$ with $Z_0\mathrm{/}{Z}_1=s$, and $zs=1$ on the overlap; these are the two charts $U_0,U_{\mathrm{\infty }}$ of the statement and the chart description of [F6] and [F10]. Finally $G=B\bigsqcup B{n}_{\alpha }B$: for $c\mathrm{\ne }0$ direct multiplication gives $g=u_{\alpha }(a\mathrm{/}c)n_{\alpha }\left(\begin{array}{cc}c& d\\ 0& c^{-1}\end{array}\right)$, the first and third factors lying in $B$, and $c=0$ is exactly the case $g\in B$; so the two double cosets are disjoint and exhaust $G$.

The rank-one weights and pairings. By [F1] the root system of ${\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with the diagonal Cartan subalgebra is $\mathrm{\Phi }=\{\alpha ,-\alpha \}$ with $\alpha ={\epsilon }_1-{\epsilon }_2$, and the positive system is ${\mathrm{\Phi }}^{+}=\{\alpha \}$, so $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=1$; the coroot is ${\alpha }^{\vee }=2\alpha \mathrm{/}(\alpha ,\alpha )$ and the fundamental weight is characterised by ${\omega }_1({\alpha }^{\vee })=1$ [F13]. Evaluating the root on the diagonal torus gives $\alpha (\mathrm{diag}(t,t^{-1}))=t^2$, so by [F3] applied to ${\alpha }^{\vee }(u)=\mathrm{diag}(u,u^{-1})$ one has $u^{⟨\alpha ,{\alpha }^{\vee }⟩}=u^2$, that is $⟨\alpha ,{\alpha }^{\vee }⟩=2$; hence ${\omega }_1=\frac{1}{2}\alpha$ is the functional with ${\omega }_1(\mathrm{diag}(t,t^{-1}))=t$, it satisfies ${\omega }_1({\alpha }^{\vee })=1$ and generates $X^{\ast }(T)\cong \mathbb{Z}$, and the Weyl vector is $\rho =\frac{1}{2}{\sum }_{\beta \in {\mathrm{\Phi }}^{+}}\beta =\frac{1}{2}\alpha ={\omega }_1$; consequently $2\rho =\alpha =2{\omega }_1$ and $-2\rho =-\alpha =-2{\omega }_1$. Since [F3] gives $\lambda ({\alpha }^{\vee }(u))=u^{⟨\lambda ,{\alpha }^{\vee }⟩}$ for every $\lambda \in X^{\ast }(T)$, substituting $\lambda =m{\omega }_1$ yields $⟨m{\omega }_1,{\alpha }^{\vee }⟩=m$ for every $m\in \mathbb{Z}$.

The rank-one fibre is the whole flag variety, with the same two charts. By [F4] $P_{\alpha }=B\bigsqcup B{n}_{\alpha }B$ and by step 1.1 $G=B\bigsqcup B{n}_{\alpha }B$, so the two double cosets agree and $P_{\alpha }=G$; as a cross-check $\dim P_{\alpha }=\dim B+1$ with $\dim B=\dim T+\dim U=1+\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=2$ by [F2] and $\dim G=\dim \mathfrak{g}=\dim \mathfrak{h}+2\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=3$ by [F5], so $P_{\alpha }$ is a closed irreducible subgroup of the same dimension as the irreducible group $G$. The rank-one clause of [F6] therefore applies: the base $X_{\alpha }=G\mathrm{/}{P}_{\alpha }$ is a single point and the projection $f:X_B\to X_{\alpha }$ is the unique map to that point, with single fibre $F=P_{\alpha }[v_B]=G[v_B]=X_B$. By [F5] the stabilizer of $[v_B]$ is $B$ and $X_B$ is the orbit of $[v_B]$, so the two-chart description of the fibre in [F6] is the description of the whole flag variety computed in step 1.1; the identification $z\mapsto u_{-\alpha }(z)B$, $s\mapsto u_{\alpha }(s)n_{\alpha }B$ of the statement is thus the identification fixed in [F8], whose two charts $U_0,U_{\mathrm{\infty }}$ are those of the line description.

The tautological line subbundle and the sign of the convention. Glue the free rank-one ${\mathcal{O}}_{U_0}$-module generated by the section $(1,z)$ of ${\mathcal{O}}_{U_0}^{\oplus 2}$ to the free rank-one ${\mathcal{O}}_{U_{\mathrm{\infty }}}$-module generated by $(s,1)$ over the overlap by $(1,z)=z\cdot (s,1)$; by [F10] and the uniqueness part of [F12] this defines an invertible sheaf $\mathcal{T}$ on ${\mathbb{P}}^1$, whose frame relation $f_{\mathrm{\infty }}=t^{-1}{f}_0$ exhibits it as $\mathcal{O}(-1)$ under the convention of [F10]. It is the sheaf of sections of the line subbundle of ${\mathcal{O}}^{\oplus 2}$ spanned by the standard coordinates, so it carries the $G$-equivariant structure induced by the action of $G$ on ${\mathbb{C}}^2$, and over the fixed point $eB$ (the point $z=0$ in the $z$-chart) its fibre is $\mathbb{C}{e}_1$. There an element $b=\left(\begin{array}{cc}t& b_{12}\\ 0& t^{-1}\end{array}\right)\in B$ acts by $b\cdot e_1=t{e}_1$: the upper triangular unipotent factor fixes $e_1$ and the diagonal factor scales it by $t$, the character ${\omega }_1$ of step 1.2. By the fibre-character convention of [F7] the bundle ${\mathcal{L}}_{{\omega }_1}$ has fibre character $t^{-1}$ at $eB$, and the tautological line has fibre character $t={\omega }_1$; the frame relation $f_{\mathrm{\infty }}=t^{-1}{f}_0$ is therefore the two-chart shadow of the negative fundamental twist, not of $\mathcal{O}(1)$.

The change of frame of ${\mathcal{L}}_{m{\omega }_1}$. By [F7] the fibre of ${\mathcal{L}}_{\lambda }$ at a point $gB$ is $\{[g,v]:v\in \mathbb{C}\}$ with $(gb,v)\sim (g,\lambda (b{)}^{-1}v)$; hence for points $x$ of the two charts of step 1.1 the assignments $e_0(x)=[u_{-\alpha }(z(x)),1]$ and $e_{\mathrm{\infty }}(x)=[u_{\alpha }(s(x))n_{\alpha },1]$ are nowhere-vanishing sections, because the second coordinate is $1\mathrm{\ne }0$, so they are frames of ${\mathcal{L}}_{\lambda }$ on the two charts. Direct matrix multiplication in $G$ gives, for $z\in {\mathbb{C}}^{\times }$ and $s=z^{-1}$, $u_{\alpha }(s)n_{\alpha }=\left(\begin{array}{cc}{z}^{-1}& -1\\ 1& 0\end{array}\right)=\left(\begin{array}{cc}1& 0\\ z& 1\end{array}\right)\left(\begin{array}{cc}{z}^{-1}& -1\\ 0& z\end{array}\right)=u_{-\alpha }(z)\mathrm{diag}(z^{-1},z)u_{\alpha }(-z),$ and the right-hand factor $b_z:=\mathrm{diag}(z^{-1},z)u_{\alpha }(-z)$ lies in $B=T⋉U$ because $\mathrm{diag}(z^{-1},z)\in T$ and $u_{\alpha }(-z)\in U$ by step 1.1. Since the two lifts describe the same point of the flag variety, the equivalence relation of [F7] gives $e_{\mathrm{\infty }}(x)=[u_{-\alpha }(z(x))b_z,1]=[u_{-\alpha }(z(x)),b_z\cdot 1]=\lambda (b_z{)}^{-1}{e}_0(x)\mathrm{.}$ For $\lambda =m{\omega }_1$ the character value factors as $\lambda (b_z)=\lambda (\mathrm{diag}(z^{-1},z))\lambda (u_{\alpha }(-z))$; the second factor is $1$ because every character of $B$ is trivial on the unipotent radical [F2], and the first is $(z^{-1}{)}^{⟨m{\omega }_1,{\alpha }^{\vee }⟩}=z^{-m}$ by [F3] and $\mathrm{diag}(z^{-1},z)={\alpha }^{\vee }(z^{-1})$ together with step 1.2. Hence $\lambda (b_z{)}^{-1}=z^m$, and the two frames of ${\mathcal{L}}_{m{\omega }_1}$ are related over the overlap by $e_{\mathrm{\infty }}=t^m{e}_0$, where $t=z$ is the coordinate of $U_0$ by step 1.1.

The identity ${\mathcal{L}}_{m{\omega }_1}\cong \mathcal{O}(m)$ and the degree. By [F10] the twist $\mathcal{O}(m)$ is glued from the structure sheaves on $U_0$ and $U_{\mathrm{\infty }}$ with frames related by $e_{\mathrm{\infty }}=t^m{e}_0$, which is exactly the gluing relation found in step 2.3 for the frames of ${\mathcal{L}}_{m{\omega }_1}$; both sheaves are trivialized by these frames on the two charts, with induced overlap identification given in both cases by multiplication by the unit $t^{-m}$, so the uniqueness statement of [F12] gives an isomorphism ${\mathcal{L}}_{m{\omega }_1}\cong \mathcal{O}(m)$ compatible with the frames. Since $t^m$ is a unit on the overlap for every $m\in \mathbb{Z}$, this covers positive, zero and negative $m$, the case $m=0$ being ${\mathcal{L}}_0={\mathcal{O}}_{X_B}$ of [F7]; by [F11] the twist index is an isomorphism invariant, so the degree of ${\mathcal{L}}_{m{\omega }_1}$ under the fixed identification is well defined and equals $m$, in agreement with the general restriction formula ${\mathcal{L}}_{\lambda }{\mathrm{\mid }}_F\cong \mathcal{O}(⟨\lambda ,{\alpha }^{\vee }⟩)$ of [F8], which gives $⟨m{\omega }_1,{\alpha }^{\vee }⟩=m$ by step 1.2. In particular $\mathcal{O}(-1)\cong {\mathcal{L}}_{-{\omega }_1}$ and $\mathcal{O}(1)\cong {\mathcal{L}}_{{\omega }_1}$: step 2.2 computed the frame relation $f_{\mathrm{\infty }}=t^{-1}{f}_0$ for the tautological bundle, which is the case $m=-1$ of the relation just established, and the fibre-character computation of step 2.2 agrees with the character $t^{-m}=t$ of ${\mathcal{L}}_{-{\omega }_1}$ at $eB$ by [F7]; the sign convention of [F7] is therefore the one in which the fundamental weight ${\omega }_1$ corresponds to the positive twist, as required for the canonical value $m=-2$ below.

The canonical bundle. By [F9] the canonical bundle of the flag variety is ${\omega }_{G\mathrm{/}B}\cong {\mathcal{L}}_{-2\rho }$ with fibre at $eB$ the $B$-module ${\mathbb{C}}_{2\rho }$; by step 1.2 $-2\rho =-\alpha =-2{\omega }_1$, so ${\mathcal{L}}_{-2\rho }={\mathcal{L}}_{-2{\omega }_1}$ and step 3.1 with $m=-2$ gives ${\omega }_{G\mathrm{/}B}\cong {\mathcal{L}}_{-2{\omega }_1}\cong \mathcal{O}(-2)$ with frame relation $e_{\mathrm{\infty }}=t^{-2}{e}_0$. The two descriptions of the fibre agree: the fibre character of ${\mathcal{L}}_{-2{\omega }_1}$ at $eB$ is $t^{-m}=t^2$ by step 2.3 and [F7], while ${\mathbb{C}}_{2\rho }={\mathbb{C}}_{\alpha }$ is the line on which $b$ acts by $\alpha (b)=t^2$ by step 1.2, and this is the value $m=-2$ of the fibre-degree formula $⟨m{\omega }_1,{\alpha }^{\vee }⟩=m$ of step 1.2.

Wrap-up and the Axiom of Choice. Step 1.1 and step 2.1 prove (i): the identification of the flag variety with the projective line of lines, with the two-chart description matching $U_0,U_{\mathrm{\infty }}$, and the rank-one fibre $F=X_B$. Steps 2.2, 2.3 and 3.1 prove (ii): the change-of-frame computation gives $e_{\mathrm{\infty }}=t^m{e}_0$ for ${\mathcal{L}}_{m{\omega }_1}$, the gluing comparison gives ${\mathcal{L}}_{m{\omega }_1}\cong \mathcal{O}(m)$ for every $m\in \mathbb{Z}$, and the twist index gives the degree $m=⟨m{\omega }_1,{\alpha }^{\vee }⟩$ of step 1.2, with the sign checked against the tautological bundle. Step 4.1 proves (iii): $\rho ={\omega }_1$, $2\rho =\alpha =2{\omega }_1$ and ${\omega }_{G\mathrm{/}B}\cong {\mathcal{L}}_{-2\rho }\cong \mathcal{O}(-2)$ with fibre ${\mathbb{C}}_{2\rho }$. The Axiom of Choice [A1] is assumed in the statement and is inherited through the quotient, torsor, root-subgroup and sheaf-gluing suppliers cited above; the example itself selects nothing beyond the finitely many standard data $e_1,e_2$, the two chart coordinates and the matrices displayed in step 2.3. The quotient structure on $G\mathrm{/}B$ and the associated bundles ${\mathcal{L}}_{\lambda }$ used here are supplied by [F4], [F5], [F6] and [F7].

Root data and pairings. The flag manifold of $G=S{L}_3(\mathbb{C})$ has $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=3$ positive roots ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2$ and $\dim X_B=3$; the simple coroots are the diagonal matrices $h_{{\alpha }_1}=\mathrm{diag}(1,-1,0)$ and $h_{{\alpha }_2}=\mathrm{diag}(0,1,-1)$, which act on the coordinate functionals by $⟨{\epsilon }_i,{\alpha }_j^{\vee }⟩={\delta }_{i,j}-{\delta }_{i,j+1}$. With ${\omega }_1={\epsilon }_1$ and ${\omega }_2={\epsilon }_1+{\epsilon }_2$ this gives the four pairings $⟨{\omega }_1,{\alpha }_1^{\vee }⟩=1,⟨{\omega }_1,{\alpha }_2^{\vee }⟩=0,⟨{\omega }_2,{\alpha }_1^{\vee }⟩=1-1=0,⟨{\omega }_2,{\alpha }_2^{\vee }⟩=0+1=1,$ that is $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$, as required of the fundamental weights; also $\rho =\frac{1}{2}(2{\alpha }_1+2{\alpha }_2)={\alpha }_1+{\alpha }_2$, so $2\rho =2({\alpha }_1+{\alpha }_2)$ and $⟨2\rho ,{\alpha }_j^{\vee }⟩=2⟨{\alpha }_1+{\alpha }_2,{\alpha }_j^{\vee }⟩=2$ for $j=1,2$, using $⟨{\alpha }_1,{\alpha }_2^{\vee }⟩=⟨{\alpha }_2,{\alpha }_1^{\vee }⟩=-1$.

The flag variety as complete flags. The stabilizer in $G$ of the standard flag $E_1=\mathbb{C}{e}_1\subset E_2=\mathbb{C}{e}_1\oplus \mathbb{C}{e}_2$ is exactly the upper triangular subgroup $B$, and $G$ acts transitively on complete flags: given $0\mathrm{\ne }{v}_1\in L$ and $v_2\in H\setminus L$ one extends to a basis $v_1,v_2,v_3$ adapted to $L\subset H$, and after replacing $v_1$ by ${\lambda }^{-1}{v}_1$ for $\lambda =\det (v_1,v_2,v_3)$ the corresponding matrix lies in $S{L}_3$ and carries the standard flag to $L\subset H$; the replacement does not change $L$. Hence $G\mathrm{/}B\to \{L\subset H\}$, $gB\mapsto g(E_1\subset E_2)$, is a $G$-equivariant bijection.

The two parabolics are the two stabilizers. The subgroup of $G$ stabilising the plane $E_2$ consists of the block matrices $\left(\begin{array}{ccc}\ast & \ast & \ast \\ \ast & \ast & \ast \\ 0& 0& \ast \end{array}\right)$ of determinant one, a closed subgroup of dimension $6$ containing $B$ and the root group $U_{-{\alpha }_1}$; by [F1] the minimal parabolic $P_{{\alpha }_1}$ is an irreducible closed subgroup with $P_{{\alpha }_1}\mathrm{/}B\cong {\mathbb{P}}^1$, so $\dim P_{{\alpha }_1}=\dim B+1=8-3+1=6$, and a closed subgroup of the same dimension containing it equals it; hence $P_{{\alpha }_1}$ is the stabilizer of the plane $E_2$. The same count with the line $E_1$, whose stabilizer is the block subgroup of dimension $6$ containing $B$ and $U_{-{\alpha }_2}$, gives that $P_{{\alpha }_2}$ is the stabilizer of the line $E_1$. Consequently the projection $f_{{\alpha }_1}:X_B\to G\mathrm{/}{P}_{{\alpha }_1}$ forgets the line and keeps the plane, while $f_{{\alpha }_2}$ forgets the plane and keeps the line.

Both fibres are projective lines. Over a plane $H$ the fibre of $f_{{\alpha }_1}$ is the set of lines $L\subset H$, which is $\mathbb{P}H\cong {\mathbb{P}}^1$; over a line $L$ the fibre of $f_{{\alpha }_2}$ is the set of planes $H\supset L$, which corresponds to the set of lines in ${\mathbb{C}}^3\mathrm{/}L$, that is $\mathbb{P}({\mathbb{C}}^3\mathrm{/}L)\cong {\mathbb{P}}^1$. Both descriptions have exactly the two-chart shape $z\mapsto u_{-\alpha }(z)B$, $t\mapsto u_{\alpha }(t)n_{\alpha }B$ of [F1], with $z$ a coordinate on ${\mathbb{A}}^1={\mathbb{P}}^1\setminus \{\mathrm{\infty }\}$ and $t=z^{-1}$ on the other chart.

Fibre degrees. By [F2] the degree of ${\mathcal{L}}_{{\omega }_i}$ on the fibre of the projection attached to ${\alpha }_j$ is $⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij}$ by step 1.1; hence the fibre-degree pair of ${\mathcal{L}}_{{\omega }_1}$ is $(1,0)$ and that of ${\mathcal{L}}_{{\omega }_2}$ is $(0,1)$ on the two families of fibres. In particular ${\mathcal{L}}_{{\omega }_1}\otimes {\mathcal{L}}_{{\omega }_2}\cong {\mathcal{L}}_{{\omega }_1+{\omega }_2}={\mathcal{L}}_{\rho }$ has fibre-degree pair $(1,1)$, and ${\omega }_{G\mathrm{/}B}\cong {\mathcal{L}}_{-2\rho }$ has degree $⟨-2\rho ,{\alpha }_j^{\vee }⟩=-2$ on both families, matching the behaviour $\mathcal{O}(-2)$ of a canonical bundle on the fibres of a ${\mathbb{P}}^1$-bundle.

Wrap-up and the Axiom of Choice. The two projections of the statement are $f_{{\alpha }_1}$ and $f_{{\alpha }_2}$ of step 1.3, with fibers ${\mathbb{P}}^1$ by step 1.4 and fibre-degree pairs $(1,0)$, $(0,1)$ by step 2.1; the consistency check with ${\omega }_{G\mathrm{/}B}$ is step 2.1. The Axiom of Choice [A1] is assumed in the statement and inherited from the named suppliers [F1]-[F6]; no choice is made in the example itself, whose only selections are the finitely many standard data ${\alpha }_i,{\omega }_i$ and the standard flag.