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  "body": "\n## Statement\n\nLet $k$ be an algebraically closed field.\nLet $G$ be a connected [[solvable|0FXJ7YFMMX]] [[linear algebraic group|GA81VVQHEP]] over $k$.\nLet $X$ be a nonempty complete [[variety|HV6YBS0YDH]] over $k$ with an [[action|RW85C34GQW]] of $G$.\nThen $G$ has a [[fixed point|0RRWNMYRXC]] in $X$.\n\n## Remarks\n\nThe proof goes by induction on $\\dim G$.\nThe derived group $G'$ is connected, normal and of smaller dimension, so its fixed-point set $Y = X^{G'}$ is nonempty, and it is closed, hence complete.\nThe group $G$ acts on $Y$, and an orbit of minimal dimension $G \\cdot y$ is closed, so it is complete.\nThe stabilizer $G_y$ contains $G'$, so it is normal and $G / G_y$ is an affine group.\nThe orbit map gives a bijective morphism from $G / G_y$ onto the complete orbit $G \\cdot y$, and one checks that $G / G_y$ is then complete as well.\nA connected variety that is both affine and complete is a point, so the orbit is a point and $y$ is fixed.\n\nExample: a connected solvable closed subgroup $H$ of $\\mathrm{GL}_n$ acts on the variety of full flags in $k^n$, which is complete.\nThe theorem gives a flag fixed by $H$, and this proves the Lie–Kolchin theorem: $H$ is conjugate to a subgroup of the group $B$ of invertible upper triangular matrices.\n\nCompleteness is needed: $\\mathbb{G}_m$ acts on $\\mathbb{A}^1 \\setminus \\set{0}$ by multiplication without a fixed point.\nSolvability is needed: $\\mathrm{SL}_2$ acts on $\\mathbb{P}^1$ transitively, so without a fixed point.\nConnectedness is needed: the group $\\Z / 2$ swaps the two points of a two-point variety.\n\n## Explanations\n\n### L1 human\n\nA connected solvable group acting on a complete variety always leaves some point where it is.\nComplete varieties, like projective space, have no \"escape to infinity\", so the group cannot push every point away.\nThe theorem is the algebraic analogue of fixed-point theorems for compact spaces.\n\n### L2 human\n\nThe theorem is the basic tool for Borel subgroups.\nApplied to a Borel subgroup $B$ acting on $G / P$, it shows that $B$ is conjugate into every parabolic subgroup $P$.\nIt gives the conjugacy of all [[Borel subgroups|5JSYA1JNCC]], and with it the conjugacy of maximal tori.\nIt also gives the Lie–Kolchin theorem on triangularizing connected solvable matrix groups.\n",
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  "markdown": "https://tamnd.github.io/math2/node/MG20HMMYKY.md",
  "msc": [
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