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    "# First Chern Form on $\\mathbb{CP}^1$\n",
    "\n",
    "A complex line bundle is characterized by the curvature of a unitary\n",
    "connection.  On an affine chart of $\\mathbb{CP}^1$, use the complex\n",
    "coordinate\n",
    "\n",
    "$$u=r e^{2\\pi i\\theta},\\qquad \\bar u=r e^{-2\\pi i\\theta}.$$\n",
    "\n",
    "We compute a Fubini--Study connection, its curvature, and the normalized\n",
    "first Chern form.\n"
   ]
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    "## Polar chart and connection one-form\n",
    "\n",
    "The chart variables are $(r,\\theta)$ with $r\\ge0$ and\n",
    "$0\\le\\theta<1$.  The local connection is\n",
    "\n",
    "$$\\omega=\\frac{\\bar u\\,du}{1+u\\bar u}.$$\n"
   ]
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    "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"
   ]
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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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   "source": [
    "## Curvature two-form\n",
    "\n",
    "For a line bundle the connection is Abelian, so $\\omega\\wedge\\omega=0$\n",
    "and $\\Omega=d\\omega$.  We display its antisymmetric component matrix\n",
    "explicitly.\n"
   ]
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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"
   ]
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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"
   ]
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   "source": [
    "## First Chern form\n",
    "\n",
    "With the orientation and connection convention used here,\n",
    "\n",
    "$$c_1=\\frac{\\Omega}{-2\\pi i}.$$\n"
   ]
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   "cell_type": "code",
   "execution_count": 5,
   "id": "6ec7ca40",
   "metadata": {},
   "outputs": [],
   "source": [
    "def c1Form : DiffForm MathValue := Ω / ((-2) * π * i)\n"
   ]
  },
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   "execution_count": 6,
   "id": "afc82e15",
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       "$\\begin{pmatrix} 0 & \\frac{-r}{r^{4} + 2 r^{2} + 1} \\\\ \\frac{r}{r^{4} + 2 r^{2} + 1} & 0 \\\\ \\end{pmatrix}$"
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   "source": [
    "c1Form\n"
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   "source": [
    "The full antisymmetric form contributes twice its upper component, so\n",
    "its oriented density is\n",
    "\n",
    "$$-\\frac{2r}{(1+r^2)^2}\\,dr\\wedge d\\theta.$$\n",
    "\n",
    "Integrating over $r\\in[0,\\infty)$ and $\\theta\\in[0,1)$ gives\n",
    "\n",
    "$$\\int_0^\\infty\\frac{-2r}{(1+r^2)^2}\\,dr=-1.$$\n",
    "\n",
    "Thus this convention describes the degree $-1$ line bundle.  Reversing\n",
    "the orientation or using the dual connection reverses the Chern number.\n"
   ]
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