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   "source": [
    "# $\\mathbb{CP}^1$ 上の第1チャーン形式\n",
    "\n",
    "複素直線束は、ユニタリ接続の曲率によって特徴付けられます。\n",
    "$\\mathbb{CP}^1$ のアフィンチャート上で、複素座標\n",
    "\n",
    "$$u=r e^{2\\pi i\\theta},\\qquad \\bar u=r e^{-2\\pi i\\theta}.$$\n",
    "\n",
    "を使います。フビニ＝スタディ接続とその曲率、そして正規化された第1チャーン\n",
    "形式を計算します。\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a2458e82",
   "metadata": {},
   "source": [
    "## 極座標チャートと接続1形式\n",
    "\n",
    "チャートの変数を $(r,\\theta)$ とし、$r\\ge0$、$0\\le\\theta<1$ とします。\n",
    "局所的な接続は\n",
    "\n",
    "$$\\omega=\\frac{\\bar u\\,du}{1+u\\bar u}.$$\n"
   ]
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  {
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   "execution_count": 1,
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   "source": [
    "declare symbol r, θ : MathValue\n",
    "\n",
    "def params : Vector MathValue := [| r, θ |]\n",
    "def u : MathValue := r * e ^ (2 * π * i * θ)\n",
    "def ū : MathValue := r * e ^ ((-2) * π * i * θ)\n",
    "\n",
    "def d (X : MathValue) : DiffForm MathValue :=\n",
    "  !(flip ∂/∂) params X\n",
    "\n",
    "def ω : DiffForm MathValue := ū * d u / '(1 + u * ū)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "05dfa9a8",
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       "$\\begin{pmatrix} \\frac{r}{r^{2} + 1} \\\\ \\frac{2 i r^{2} π}{r^{2} + 1}\\\\ \\end{pmatrix}$"
      ]
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   "source": [
    "ω\n"
   ]
  },
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   "cell_type": "markdown",
   "id": "8d60dd09",
   "metadata": {},
   "source": [
    "## 曲率2形式\n",
    "\n",
    "直線束の接続は可換なので、$\\omega\\wedge\\omega=0$ であり、\n",
    "$\\Omega=d\\omega$ となります。その反対称成分行列を明示的に表示します。\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "3f7d884f",
   "metadata": {},
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   "source": [
    "def ωr : MathValue := ω_1\n",
    "def ωθ : MathValue := ω_2\n",
    "def Ωrθ : MathValue :=\n",
    "  (∂/∂ ωθ r - ∂/∂ ωr θ) / 2\n",
    "def Ω : DiffForm MathValue :=\n",
    "  [| [| 0, Ωrθ |], [| -Ωrθ, 0 |] |]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "d4a44bcb",
   "metadata": {},
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       "$\\begin{pmatrix} 0 & \\frac{2 i r π}{r^{4} + 2 r^{2} + 1} \\\\ \\frac{-2 i r π}{r^{4} + 2 r^{2} + 1} & 0 \\\\ \\end{pmatrix}$"
      ]
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   "source": [
    "Ω\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5358cf3c",
   "metadata": {},
   "source": [
    "## 第1チャーン形式\n",
    "\n",
    "ここで採用する向きと接続の規約では、\n",
    "\n",
    "$$c_1=\\frac{\\Omega}{-2\\pi i}.$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "6ec7ca40",
   "metadata": {},
   "outputs": [],
   "source": [
    "def c1Form : DiffForm MathValue := Ω / ((-2) * π * i)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "afc82e15",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$\\begin{pmatrix} 0 & \\frac{-r}{r^{4} + 2 r^{2} + 1} \\\\ \\frac{r}{r^{4} + 2 r^{2} + 1} & 0 \\\\ \\end{pmatrix}$"
      ]
     },
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   "source": [
    "c1Form\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "37561697",
   "metadata": {},
   "source": [
    "完全な反対称形式には上三角側の成分が2回寄与するため、その向き付き密度は\n",
    "\n",
    "$$-\\frac{2r}{(1+r^2)^2}\\,dr\\wedge d\\theta.$$\n",
    "\n",
    "となります。$r\\in[0,\\infty)$、$\\theta\\in[0,1)$ の範囲で積分すると\n",
    "\n",
    "$$\\int_0^\\infty\\frac{-2r}{(1+r^2)^2}\\,dr=-1.$$\n",
    "\n",
    "となります。したがって、この規約は次数 $-1$ の直線束を記述しています。\n",
    "向きを反転するか、双対接続を使うと、チャーン数の符号が反転します。\n"
   ]
  }
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