First Chern form agrees with the topological line class

Let $\gamma \to {\mathbb{C}\mathbb{P}}^m$ be tautological and choose a Hermitian connection on it using the compatible-connection supplier. [F2, A1] First take $m=1$, with affine coordinates $z=z_1\mathrm{/}{z}_0$ and $w=z_0\mathrm{/}{z}_1$. The standard frames $s_U=(1,z)$ and $s_V=(w,1)$ satisfy $s_U=z{s}_V$ on the overlap. If $\mathrm{\nabla }{s}_U={\omega }_U{s}_U$ and $\mathrm{\nabla }{s}_V={\omega }_V{s}_V$, the connection Leibniz rule gives ${\omega }_U-{\omega }_V=\frac{dz}{z}\mathrm{.}$ In their displayed charts, the closed unit disks $D_U=\{[1:z]:\mathrm{\mid }z\mathrm{\mid }\le 1\},D_V=\{[w:1]:\mathrm{\mid }w\mathrm{\mid }\le 1\}$ cover ${\mathbb{C}\mathbb{P}}^1$ and induce opposite orientations on their common boundary. The equator parametrization $z\mapsto [1:z]$ preserves orientation. The coordinate maps from the open unit disks in $z$ and $w$ preserve orientation, have disjoint images, and their closed images cover ${\mathbb{C}\mathbb{P}}^1$, so [F14] gives the integral as the sum of the two disk integrals. Using $\mathrm{\Omega }{\mathrm{\mid }}_U=d{\omega }_U$, $\mathrm{\Omega }{\mathrm{\mid }}_V=d{\omega }_V$, and [F8], ${\int }_{{\mathbb{C}\mathbb{P}}^1}\mathrm{\Omega }={\int }_{\mathrm{\partial }{D}_U}({\omega }_U-{\omega }_V)={\int }_{S^1}\frac{dz}{z}=2\pi i\mathrm{.}$ Thus the degree-two Chern form ${\eta }_{\gamma }=-\mathrm{\Omega }\mathrm{/}(2\pi i)$ has period $-1$. This calculation is for any chosen connection; in particular it applies to the Hermitian connection above.

The de Rham evaluation on the fundamental class is determined by integration [A1, F4, F18]. Identify ${\mathbb{C}\mathbb{P}}^1$ with the unit sphere with its complex orientation. Take a tetrahedron containing the origin in its interior and radially project its oriented boundary to the sphere, orienting the faces by the boundary orientation. The four face maps form a smooth singular cycle: each face map extends smoothly near its standard simplex because its affine plane misses the origin, and the shared edge chains cancel. The radial map carries the oriented tetrahedral triangulation to the sphere, so the resulting cycle has the local orientation restrictions of the fundamental class by [F17]. By [F16], it also represents the corresponding class in smooth singular homology. Each face interior maps orientation-preservingly and diffeomorphically onto one of four disjoint spherical triangles; their closures cover the sphere. Thus the finite-parametrization formula in [F14] identifies the sum of the face integrals with ${\int }_{{\mathbb{C}\mathbb{P}}^1}{\eta }_{\gamma }$. The restriction from continuous to smooth cohomology sends $J[{\eta }_{\gamma }]$ to the class of the integration cochain [F18], so evaluation on this smooth cycle is exactly the integral just computed, namely $-1$. The independent Thom-class calculation is local. On the tautological line $\gamma$, orthogonally project the fixed vector $(1,0)$ onto each complex line. This gives a smooth section with its only zero at $p=[0:1]$. In the chart $w=z_0\mathrm{/}{z}_1$ centered at $p$, use the frame $(w,1)$; the section has fiber coordinate $\bar{w}\mathrm{/}(1+\mathrm{\mid }w{\mathrm{\mid }}^2)$. Its derivative at zero is complex conjugation, with real determinant $-1$. Homotopy through scalar multiples of the section identifies its absolute Thom pullback with the zero-section Euler class. In a small disk about $p$, excision and fiberwise Thom normalization [F10] identify the relative pullback with the local orientation class multiplied by that determinant sign. The fundamental class restricts to the positive local orientation by [F17]. Hence $⟨e({\gamma }_{\mathbb{R}}),[{\mathbb{C}\mathbb{P}}^1]⟩=-1$. Its coefficient image has the same real evaluation by [F13]. Evaluation is an isomorphism in degree two by [F7] and [F6], so the two real singular classes agree on ${\mathbb{C}\mathbb{P}}^1$.

The Schubert cell structure and field UCT control the degree-two comparison [A1, F6, F7]. For $m\ge 1$, the Schubert cell structure in [F6] has one cell in each even dimension and none in odd dimensions. Its cellular chain complex over $\mathbb{R}$ therefore has $C_1=0$, $C_2=\mathbb{R}$, and zero boundary into or out of degree two. The inclusion ${\mathbb{C}\mathbb{P}}^1\hookrightarrow {\mathbb{C}\mathbb{P}}^m$ includes the unique two-cell, so it induces an isomorphism on $H_2$. Cellular homology and [F7] make restriction on $H^2(-;\mathbb{R})$ an isomorphism. By naturality of $J$ and of the Euler class, step 1.2 then gives $J_{{\mathbb{C}\mathbb{P}}^m}[{\eta }_{\gamma }]={\rho }_{{\mathbb{C}\mathbb{P}}^m}(e({\gamma }_{\mathbb{R}}))\mathrm{.}$ For $m=0$, both degree-two groups vanish, so the same equality holds. The sign here comes from the computed periods, not merely from the fact that the tautological class is a generator.

Every real homology class has a smooth cycle representative by [A1, F16]. Fix a smooth real singular two-cycle $z$ in $M$. [F16] Let $K$ be the union of the images of its finitely many singular simplices. Each standard simplex is compact, so [F12] makes $K$ compact. If $K=\mathrm{\varnothing }$, then $z=0$ and this cycle pairs to zero; henceforth assume $K\mathrm{\ne }\mathrm{\varnothing }$. By [F3], $L$ is numerable, so its Thom-defined Euler class lies in the stated scope. Let the index set consist of all local nonvanishing smooth sections ${\lambda }_a$ of $L^{\ast }$, with domain $U_a$; the domains cover $M$ by [F11]. Use [F9] to take a smooth partition $({\varphi }_a)$ subordinate to this indexed cover. The open sets $V_a=\{x:{\varphi }_a(x)>0\}$ cover $K$. Compactness gives finitely many indices $a_1,\dots ,a_N$ whose $V_{a_i}$ cover $K$, so $N\ge 1$. Define ${\sigma }_i={\varphi }_{a_i}{\lambda }_{a_i}$ on $U_{a_i}$ and zero outside. This extension is smooth because $\mathrm{supp}{\varphi }_{a_i}\subset U_{a_i}$. On the open neighborhood $V={\bigcup }_i{V}_{a_i}$ of $K$, at least one ${\sigma }_i$ is nonzero at every point.

The local complex bundle frames give the fiberwise evaluation map [F11]. The evaluation map $\mathrm{\Phi }:L{\mathrm{\mid }}_V⟶V\times {\mathbb{C}}^N,v⟼(\pi (v),({\sigma }_1(v),\dots ,{\sigma }_N(v)))$ is smooth and complex-linear on each fiber. Since some ${\sigma }_i(x)$ is nonzero for every $x\in V$, its fiber map is injective. Its image is a smooth line subbundle: on the open set where the $i$th coordinate is nonzero, the projective coordinate ratios are smooth. Hence it defines a smooth map $f:V\to {\mathbb{C}\mathbb{P}}^{N-1}$ and an isomorphism $L{\mathrm{\mid }}_V\cong f^{\ast }\gamma$. The isomorphism is complex-linear and therefore preserves the complex orientation. This construction uses a finite subcover of $K$; no global finite-dimensional classifying map on $M$ is asserted.

Euler naturality applies to the orientation-preserving line-bundle isomorphism. [F5] By [F5] and the orientation-preserving isomorphism of step 1.5, $e(L_{\mathbb{R}}){\mathrm{\mid }}_V=f^{\ast }e({\gamma }_{\mathbb{R}})\mathrm{.}$ Choose a Hermitian connection on $\gamma$ over ${\mathbb{C}\mathbb{P}}^{N-1}$, and transport its pullback to $L{\mathrm{\mid }}_V$. It may use a different Hermitian metric from the supplied one on $L$, but both are complex-linear connections on the same complex line bundle; their Hermitian metrics need not agree. By [F2], their first Chern forms have the same complex de Rham class, and naturality identifies the pulled-back form class with $f^{\ast }[{\eta }_{\gamma }]$. By step 1.3 this is the complexification of ${\rho }_V(e(L_{\mathbb{R}}){\mathrm{\mid }}_V)$ under the de Rham comparison. The inclusion ${\mathrm{\Omega }}^{\bullet }(V;\mathbb{R})\hookrightarrow {\mathrm{\Omega }}^{\bullet }(V;\mathbb{C})$ is injective on cohomology: a complex primitive of a real exact form has a real part that is a real primitive by [F19]. Therefore the two real classes agree on $V$. Naturality of $J$ gives $J_V[-\frac{{\mathrm{\Omega }}_{\mathrm{\nabla }}}{2\pi i}]={\rho }_V(e(L_{\mathbb{R}}){\mathrm{\mid }}_V)\mathrm{.}$

The UCT evaluation map detects the difference class. [F7] Put $\mathrm{\Delta }=J_M[-\frac{{\mathrm{\Omega }}_{\mathrm{\nabla }}}{2\pi i}]-{\rho }_M(e(L_{\mathbb{R}}))\in H^2(M;\mathbb{R})\mathrm{.}$ If $z=0$, its evaluation is zero. Otherwise step 1.4 gives a neighborhood $V$ containing its image, and step 1.6 makes $\mathrm{\Delta }{\mathrm{\mid }}_V=0$. Naturality in [F4] then makes the UCT evaluation of $\mathrm{\Delta }$ on $[z]$ zero. Every class of $H_2(M;\mathbb{R})$ is represented by a smooth cycle by [F16, A1], so $\mathrm{\Delta }$ evaluates to zero on all of $H_2(M;\mathbb{R})$. Since $\mathbb{R}$ is a field, [F7] makes the evaluation map an isomorphism, hence $\mathrm{\Delta }=0$. This proves the Hermitian assertion globally, including disconnected $M$; the argument only fixes one cycle at a time.

The degree-one transgression applies to any two complex line connections. [F15] Let ${\mathrm{\nabla }}^{\mathrm{\prime }}$ be any complex connection and set $A={\mathrm{\nabla }}^{\mathrm{\prime }}-\mathrm{\nabla }$. The degree-one case of [F15], for $P_1(B)=-\mathrm{tr}(B)\mathrm{/}(2\pi i)$, gives $-\frac{{\mathrm{\Omega }}_{{\mathrm{\nabla }}^{\mathrm{\prime }}}}{2\pi i}+\frac{{\mathrm{\Omega }}_{\mathrm{\nabla }}}{2\pi i}=d(-\frac{A}{2\pi i})\mathrm{.}$ Thus their complex de Rham classes differ by the displayed exact form. Step 1.7 identifies the real class of $-{\mathrm{\Omega }}_{\mathrm{\nabla }}\mathrm{/}(2\pi i)$, and [F19] shows that including real forms into complex forms carries it to the stated complex class. This proves the second assertion and fixes the transgression endpoints in the order $\mathrm{\nabla }$ to ${\mathrm{\nabla }}^{\mathrm{\prime }}$. If $M=\mathrm{\varnothing }$, its singular and de Rham groups are zero by [F13, F19]. A zero curvature form is included in the same transgression equation; the zero cycle was handled in step 1.4. The statement is for a line bundle, and $N=1$ in step 1.5 gives the trivial target ${\mathbb{C}\mathbb{P}}^0$, covered by step 1.3. Degenerate singular simplices remain among the finite chains and have compact standard domains by [F12]. Boundary points use the half-space conventions in [F2, F4, F9, F16]. Full AC is used exactly through the compatible-connection, manifold numerability, Thom/Euler, projective Chern and UCT suppliers; its ${\mathrm{A}\mathrm{C}}_{\omega }$ consequence is used by the partition, smooth-chain and de Rham comparison suppliers. The local cycle argument uses only a finite subcover of $K$, with no global classifying map. There is no if-and-only-if assertion. [A1, F1, F2, F3, F4, F5, F6, F7, F9, F10, F12, F13, F15, F16, F19, cases, step 1.4, step 1.5, step 1.7] $\mathrm{\square }$