Fixed point for the specified Borel on a projective variety

Build a normal one-dimensional filtration of the specified $B=T⋉U$. Order the positive roots ${\beta }_1,\dots ,{\beta }_m$ by decreasing height, breaking ties arbitrarily, and put $S_j=\{{\beta }_1,\dots ,{\beta }_j\}$, $U_j={\prod }_{\beta \in S_j}{U}_{\beta }$ with $U_0=1$. If $\beta \in S_j$, $\gamma \in {\mathrm{\Phi }}^{+}$ and $\beta +\gamma$ is a root, then $\mathrm{ht}(\beta +\gamma )>\mathrm{ht}(\beta )$, so $\beta +\gamma \in S_j$. The height-raising BCH commutator law and polynomial root coordinates of [F1] therefore make each $U_j$ a closed connected subgroup normalized by $U$; $T$ normalizes it because it scales every root coordinate, so $U_j◃B$. The multiplication $U_{j-1}\times U_{{\beta }_j}\to U_j$ is a polynomial isomorphism by the same triangular root-coordinate recursion, and $U_{{\beta }_j}\cong {\mathbb{G}}_a$; thus $U_j$ is generated by $U_{j-1}$ and one copy of ${\mathbb{G}}_a$. Choose a coordinate decomposition $T\cong {\mathbb{G}}_m^l$ and let $T_k\cong {\mathbb{G}}_m^k$ be the first $k$ factors, $0\le k\le l$. The preimages $B_{m+k}:=U⋊T_k$ of $T_k$ under $B\to T$ are closed and normal in $B$, and each is generated by $B_{m+k-1}$ and the next coordinate copy of ${\mathbb{G}}_m$. With $B_j:=U_j$ for $0\le j\le m$, this gives $1=B_0◃B_1◃\cdots ◃B_{m+l}=B$, with each extension generated by its predecessor and one algebraic root or torus subgroup isomorphic to ${\mathbb{G}}_a$ or ${\mathbb{G}}_m$.

Base case of ${\mathbb{G}}_a$. Let $Y$ be projective with a ${\mathbb{G}}_a$ action, let $Z\subseteq Y$ be a nonempty stable closed subvariety, choose $y\in Z$, and write $F:{\mathbb{A}}_t^1\to Y$, $t\mapsto t\cdot y$. Projectivity makes $Y$ proper by [F2], so the valuative criterion [F3] extends the generic map $\mathrm{Spec}\mathbb{C}(t)\to Y$ uniquely to the discrete valuation ring $\mathbb{C}[s{]}_{(s)}$ at $\mathrm{\infty }$, where $s=t^{-1}$. This local-ring map extends to an actual Zariski neighbourhood: choose an affine open $V\subseteq Y$ containing the image of the closed point; the map from the local spectrum factors through $V$, and the images of finitely many generators of $\mathcal{O}(V)$ are fractions in $\mathbb{C}[s{]}_{(s)}$ with denominators nonzero at $s=0$. Invert their product $h(s)$, with $h(0)\mathrm{\ne }0$, to obtain a morphism $D(h)\subseteq {\mathbb{A}}_s^1\to V\subseteq Y$ agreeing with the valuation-ring lift. On the integral overlap $D(h)\cap {\mathbb{A}}_t^1$ it and $F$ agree at the generic point; because $Y$ is separated, their equalizer is closed, and because the overlap is reduced and irreducible, a closed equalizer containing its generic point is the whole overlap as a scheme. Thus they glue over the open cover ${\mathbb{P}}^1={\mathbb{A}}_t^1\cup D(h)$ to a morphism $\mathrm{\Phi }:{\mathbb{P}}^1\to Y$. For $a\in \mathbb{C}$, translation ${\tau }_a(t)=t+a$ extends to an automorphism of ${\mathbb{P}}^1$ fixing $\mathrm{\infty }$, so $a\cdot \mathrm{\Phi }(t)$ and $\mathrm{\Phi }(t+a)$ agree on ${\mathbb{A}}_t^1$ and therefore on ${\mathbb{P}}^1$ by [F4]. Evaluating at $\mathrm{\infty }$ gives $a\cdot \mathrm{\Phi }(\mathrm{\infty })=\mathrm{\Phi }(\mathrm{\infty })$. The point $\mathrm{\Phi }(\mathrm{\infty })$ lies in $Z$ because $Z$ is closed and contains the dense-open image $\mathrm{\Phi }({\mathbb{A}}_t^1)$, so it is the required ${\mathbb{G}}_a$-fixed point.

Base case of ${\mathbb{G}}_m$. Let $Y$ be projective with a ${\mathbb{G}}_m$ action, let $Z\subseteq Y$ be a nonempty stable closed subvariety, choose $y\in Z$, and put $\phi :{\mathbb{G}}_m\to Y$, $t\mapsto t\cdot y$. Apply [F3] to the generic map at the two missing points $0,\mathrm{\infty }$ of ${\mathbb{P}}^1$. At $0$ use the local ring $\mathbb{C}[t{]}_{(t)}$, and at $\mathrm{\infty }$ use $\mathbb{C}[s{]}_{(s)}$ with $s=t^{-1}$; each lift extends to an affine open neighbourhood $D(h_0)$ or $D(h_{\mathrm{\infty }})$ by the finite-generator denominator argument of step 1.2. The three maps on ${\mathbb{G}}_m$, $D(h_0)$ and $D(h_{\mathrm{\infty }})$ agree on each integral pairwise overlap: their equalizer is closed because $Y$ is separated, contains the generic point, and therefore equals the reduced irreducible overlap as a scheme. Hence they glue to $\mathrm{\Phi }:{\mathbb{P}}^1\to Y$. For $a\in {\mathbb{G}}_m$, multiplication $t\mapsto at$ extends to an automorphism of ${\mathbb{P}}^1$ fixing $0$, and the maps $a\cdot \mathrm{\Phi }(t)$ and $\mathrm{\Phi }(at)$ agree on the dense open ${\mathbb{G}}_m$, hence everywhere by [F4]. Thus $\mathrm{\Phi }(0)$ is ${\mathbb{G}}_m$-fixed; it lies in $Z$ because $Z$ is closed and contains the dense-open image $\mathrm{\Phi }({\mathbb{G}}_m)$.

Induct along the filtration of step 1.1. For any nonempty projective $B$-variety $Y$, set $Y_0=Y$ and let $Y_j$ be the reduced closed subscheme of points fixed by $B_j$. It is closed: for each $b\in B_j(\mathbb{C})$ the equalizer of the automorphism $y\mapsto b\cdot y$ with ${\mathrm{id}}_Y$ is closed because $Y$ is separated, and the intersection of these closed subsets is closed; taking the reduced induced structure gives $Y_j$. Since $B_j◃B$, the action of $B$ preserves $Y_j$ setwise, and the restricted action factors through the reduced closed subscheme $Y_j$: the source $B\times Y_j$ is reduced over the perfect field $\mathbb{C}$, so a morphism whose closed-point image lies in $Y_j$ annihilates its radical ideal. Each $Y_j$ is projective as a closed subscheme of $Y$. Suppose $Y_{j-1}$ is nonempty. The next one-dimensional subgroup $H_j\cong {\mathbb{G}}_a$ or ${\mathbb{G}}_m$ from step 1.1 acts on the nonempty projective variety $Y_{j-1}$, since $B_{j-1}$ is normal in $B$; step 1.2 or step 2.1 gives an $H_j$-fixed point there. As $B_j$ is generated by $B_{j-1}$ and $H_j$, that point lies in $Y_j$, so $Y_j$ is nonempty. Induction from $Y_0$ to $Y_{m+l}=Y^B$ yields a $B$-fixed point. This uses normality of the chosen root-height and torus subgroups, without asserting that an arbitrary kernel of a vector-group character is normal.

The closed subvariety $Z$ is projective over $\mathbb{C}$ (a closed subvariety of a projective scheme in the same projective embedding) and nonempty and $B$-stable, so it is a nonempty projective $B$-variety; applying step 3.1 with $Y=Z$ yields $Z^B\mathrm{\ne }\mathrm{\varnothing }$, that is, a $B$-fixed point of $Z$. The Axiom of Choice is assumed in the statement and declared as the dependency The Axiom of Choice; it is inherited by the suppliers [F1], [F2], [F3] and [F4], each of which assumes it, and no additional choice is made in the argument beyond the choice of the point $y$ inside the nonempty variety.