Curvature and first Chern form of a complex line

The tautological line is the smooth subbundle of ${\mathbb{C}\mathbb{P}}^1\times {\mathbb{C}}^2$ whose fiber at a line $\mathrm{\ell }$ is $\mathrm{\ell }$; on $U=\{[1:z]\}$ and $V=\{[w:1]\}$ its nowhere-zero frames are $s_U([1:z])=(1,z)$ and $s_V([w:1])=(w,1)$. These smooth local trivializations make it a smooth complex line, and the induced Hermitian metric and [F3] provide a Hermitian connection.

In a local unitary frame of any Hermitian line, the metric derivative identity in [F5] gives $\omega +\bar{\omega }=0$. The line curvature structure equation has no quadratic term because a scalar one-form wedges with itself to zero, so [F6] gives ${\mathrm{\Omega }}_{\mathrm{\nabla }}=d\omega$; [F7] then gives the normalized Chern-form formula, which is real-valued by [F8]. The same scalar structure equation applies in any smooth complex frame, even when that frame is not unitary.

On $U\cap V$, $z=1\mathrm{/}w$ and $s_U=z{s}_V$; applying the connection Leibniz rule in [F4] yields ${\omega }_U-{\omega }_V=z^{-1}dz=dz\mathrm{/}z$. Set $D_U=\{[1:z]:\mathrm{\mid }z\mathrm{\mid }\le 1\}$ and $D_V=\{[w:1]:\mathrm{\mid }w\mathrm{\mid }\le 1\}$. These disks cover ${\mathbb{C}\mathbb{P}}^1$; their boundary orientations are opposite, and $z=e^{it}$ positively parametrizes $\mathrm{\partial }{D}_U$. By [F9], ${\int }_{{\mathbb{C}\mathbb{P}}^1}{\mathrm{\Omega }}_{\mathrm{\nabla }}={\int }_{D_U}d{\omega }_U+{\int }_{D_V}d{\omega }_V={\int }_{\mathrm{\partial }{D}_U}({\omega }_U-{\omega }_V)={\int }_{S^1}\frac{dz}{z}=2\pi i\mathrm{.}$ The determinant normalization [F7] therefore gives ${\int }_{{\mathbb{C}\mathbb{P}}^1}{c}_1(\mathrm{\nabla })=-1$. In particular, this curvature cannot vanish identically.

By [F10], the real de Rham class of $c_1(\mathrm{\nabla })$ is the image of $c_1(\gamma )$. The de Rham isomorphism [F11], the smooth-chain comparison [F12], the complex-oriented fundamental class [F13], and the pairing [F14] identify the calculated integral with $⟨\rho (c_1(\gamma )),[{\mathbb{C}\mathbb{P}}^1]⟩$. This proves the stated topological normalization, with the sign fixed by the complex orientation and the library's convention.

If a local unitary frame changes by $s^{\mathrm{\prime }}=gs$ for a smooth $g:U\to U(1)$, the Leibniz rule gives ${\omega }^{\mathrm{\prime }}=\omega +g^{-1}dg$. This added one-form is imaginary and closed: $\mathrm{\mid }g\mathrm{\mid }=1$ gives $d\bar{g}=-g^{-2}dg$, so $\overline{g^{-1}dg}=-g^{-1}dg$, and $d(g^{-1}dg)=-g^{-2}dg\wedge dg=0$. It is locally exact by writing $g=e^{i\theta }$ locally. It need not be globally exact: for $g(z)=z$ on $S^1$, ${\int }_{S^1}{g}^{-1}dg=2\pi i$, whereas Stokes [F9] makes the integral of an exact one-form on $S^1$ zero, applying Stokes to the real and imaginary parts. Thus curvature and Chern form are unchanged under every unitary frame change, without asserting a global primitive. The disk computation uses explicit charts and adds no choices once $\mathrm{\nabla }$ is supplied; full AC enters through the projective-line, compatible-connection, and first-Chern comparison suppliers. [A1, F3, F4, F6, F7, F9, algebra] $\mathrm{\square }$