[proofplan] We construct the integer Čech cocycle directly from logarithms of the transition functions and take its Čech cohomology class as the transition-function definition of $c_1(L)$. The cocycle condition follows by applying the Čech differential to the logarithmic $1$-cochain and using $\delta^2=0$. We then check that changing logarithms or local frames changes the integer cocycle by an integral Čech coboundary, and the explicit cofinal good-cover hypothesis lets two chosen good covers be compared after passage to a common good refinement in the refinement model of Čech cohomology. Finally, tensor products multiply transition functions and duals invert them, so the corresponding logarithmic cocycles add and negate. [/proofplan] [step:Declare the transition cocycle definition of the first Chern class] We use the stated good-cover model for [Čech cohomology](/page/Cech%20Cohomology): a class in $\check H^2(X;\mathbb Z)$ is represented by an integer Čech $2$-cocycle on a good cover, with two representatives identified when their pullbacks to a common good refinement differ by an integral Čech coboundary. The cofinal good-cover hypothesis is used at this point to ensure that chosen good trivializing covers can be compared by common good refinements and that this refinement comparison model computes the desired Čech group. For a Hermitian complex line bundle $L\to X$, define $c_1(L)\in \check H^2(X;\mathbb Z)$ to be the class represented by the integer transition cocycle constructed below from a good trivializing cover, unitary local frames, and logarithmic lifts of the transition functions. The remaining steps prove that this definition is independent of all those choices and then compute its behavior under tensor products and duals. [guided] The theorem is about the first Chern class, so we must specify exactly which model of $c_1$ is being used. We use the transition-function model: choose a good trivializing cover, write the line bundle by $S^1$-valued transition functions, lift those transition functions locally to real logarithms, and take the integer obstruction on triple overlaps. The resulting Čech class is, by definition in this model, $c_1(L)$. The good-cover system is not used in the algebraic computation on one overlap. It enters through the background Čech framework: the covers under consideration admit common good refinements, and classes computed on different good covers are compared by pulling both cocycles to such a refinement. Thus, once we prove invariance under logarithms, frames, and refinement pullback, the construction gives a well-defined element of $\check H^2(X;\mathbb Z)$. [/guided] [/step] [step:Show that the logarithmic triple sums are integer-valued] Fix unitary local frames for the Hermitian line bundle $L\to X$, so the transition functions take values in $S^1\subset\mathbb C^\times$. For every nonempty overlap $U_i\cap U_j$, the good-cover hypothesis gives that $U_i\cap U_j$ is contractible. Hence the covering map $\mathbb R\to S^1$, $t\mapsto \exp(2\pi i t)$, admits a continuous lift along the transition function $g_{ij}:U_i\cap U_j\to S^1$ by the standard [lifting criterion](/theorems/1897) for [covering maps](/page/Covering%20Map). Choose logarithmic lifts antisymmetrically: for every nonempty overlap $U_i\cap U_j$, the map \begin{align*} a_{ij}:U_i\cap U_j &\to \mathbb R \end{align*} satisfies \begin{align*} g_{ij}=\exp(2\pi i a_{ij}), \qquad a_{ji}=-a_{ij}. \end{align*} This antisymmetric choice is part of the construction: choose $a_{ij}$ for one orientation of each unordered pair and define the opposite orientation by $a_{ji}:=-a_{ij}$, which is valid because $g_{ji}=g_{ij}^{-1}$. For each nonempty triple intersection $U_i\cap U_j\cap U_k$, define \begin{align*} n_{ijk}:U_i\cap U_j\cap U_k &\to \mathbb Z \end{align*} by \begin{align*} n_{ijk}(x):=a_{ij}(x)+a_{jk}(x)+a_{ki}(x). \end{align*} The transition functions satisfy the line-bundle cocycle identity \begin{align*} g_{ij}(x)g_{jk}(x)g_{ki}(x)=1 \end{align*} for every $x\in U_i\cap U_j\cap U_k$. Therefore \begin{align*} 1 &=g_{ij}(x)g_{jk}(x)g_{ki}(x)\\ &=\exp(2\pi i a_{ij}(x))\exp(2\pi i a_{jk}(x))\exp(2\pi i a_{ki}(x))\\ &=\exp(2\pi i n_{ijk}(x)). \end{align*} Thus $n_{ijk}(x)\in\mathbb Z$ for every $x\in U_i\cap U_j\cap U_k$. Since each $a_{ij}$ is continuous, $n_{ijk}$ is continuous as a map to $\mathbb R$. Since $\mathbb Z\subset\mathbb R$ has the discrete [subspace topology](/page/Subspace%20Topology), $n_{ijk}$ is locally constant as an integer-valued function. Hence $n=(n_{ijk})$ is an element of $C^2(\mathcal U;\mathbb Z)$. [/step] [step:Verify that the integer cochain is a Čech cocycle] Let $\delta$ denote the Čech coboundary operator. Regard $a=(a_{ij})$ as a real-valued Čech $1$-cochain on $\mathcal U$. By definition, \begin{align*} (\delta a)_{ijk}=a_{jk}-a_{ik}+a_{ij}. \end{align*} Since the logarithmic lifts were chosen antisymmetrically, $a_{ki}=-a_{ik}$ on $U_i\cap U_j\cap U_k$. Therefore the originally defined triple sum satisfies \begin{align*} n_{ijk}=a_{ij}+a_{jk}+a_{ki}=a_{ij}+a_{jk}-a_{ik}=(\delta a)_{ijk}. \end{align*} Now apply $\delta$ once more. Since the Čech coboundary satisfies $\delta^2=0$, \begin{align*} \delta n=\delta(\delta a)=0. \end{align*} Thus the integer cochain $n$ defined above is a Čech $2$-cocycle with coefficients in $\mathbb Z$. [guided] The purpose of this step is to check that the integer data obtained on triple overlaps satisfy the compatibility condition on quadruple overlaps. Let $\delta$ be the Čech coboundary operator. For a real-valued $1$-cochain $a=(a_{ij})$, the coboundary is \begin{align*} (\delta a)_{ijk}=a_{jk}-a_{ik}+a_{ij}. \end{align*} The expression from the construction is \begin{align*} n_{ijk}=a_{ij}+a_{jk}+a_{ki}. \end{align*} The important normalization is $a_{ki}=-a_{ik}$. This was not imposed after the fact; it was built into the choice of logarithms. Therefore the triple sum is exactly the Čech coboundary of the real-valued $1$-cochain $a$: \begin{align*} n_{ijk}=a_{ij}+a_{jk}-a_{ik}=(\delta a)_{ijk}. \end{align*} The Čech differential always satisfies $\delta^2=0$. Applying this identity to the real-valued cochain $a$ gives \begin{align*} \delta n=\delta(\delta a)=0. \end{align*} Since the previous step proved that every $n_{ijk}$ is integer-valued and locally constant, this proves that $n$ is a Čech $2$-cocycle with coefficients in $\mathbb Z$. [/guided] [/step] [step:Changing logarithms changes the cocycle by an integral coboundary] Let \begin{align*} a'_{ij}:U_i\cap U_j &\to \mathbb R \end{align*} be another antisymmetric choice of logarithmic lifts for the same transition functions $g_{ij}$, so $a'_{ji}=-a'_{ij}$. For each $i,j$, define \begin{align*} m_{ij}:U_i\cap U_j &\to \mathbb Z \end{align*} by \begin{align*} m_{ij}:=a'_{ij}-a_{ij}. \end{align*} Since $\exp(2\pi i a'_{ij})=\exp(2\pi i a_{ij})$, each $m_{ij}$ is integer-valued and locally constant. The antisymmetry of $a$ and $a'$ gives $m_{ji}=-m_{ij}$. Let $n'$ be the integer $2$-cocycle obtained from $a'$. Then \begin{align*} n'_{ijk} &=a'_{ij}+a'_{jk}+a'_{ki}\\ &=(a_{ij}+m_{ij})+(a_{jk}+m_{jk})+(a_{ki}+m_{ki})\\ &=n_{ijk}+m_{ij}+m_{jk}-m_{ik}\\ &=n_{ijk}+(\delta m)_{ijk}. \end{align*} Thus \begin{align*} n'=n+\delta m. \end{align*} Therefore $n$ and $n'$ define the same class in $\check H^2(\mathcal U;\mathbb Z)$. [/step] [step:Changing local frames changes the cocycle by an integral coboundary] Choose another unitary local frame $e'_i$ of $L|_{U_i}$ for each $i\in I$. Since both $e_i$ and $e'_i$ are unitary frames of the same Hermitian complex line bundle over $U_i$, there is a continuous function \begin{align*} b_i:U_i &\to S^1 \end{align*} such that \begin{align*} e'_i=b_i e_i. \end{align*} Because $\mathcal U$ is a good cover, each $U_i$ is contractible, so the covering map $\mathbb R\to S^1$, $t\mapsto \exp(2\pi i t)$, admits a continuous lift along $b_i$. Choose continuous logarithmic lifts \begin{align*} c_i:U_i &\to \mathbb R \end{align*} with \begin{align*} b_i=\exp(2\pi i c_i). \end{align*} Let $g'_{ij}:U_i\cap U_j\to\mathbb C^\times$ be the transition functions for the frames $e'_i$. From $e'_j=g'_{ij}e'_i$ and $e_j=g_{ij}e_i$, we obtain \begin{align*} g'_{ij}=b_j g_{ij} b_i^{-1}. \end{align*} Therefore a compatible logarithmic lift is \begin{align*} a'_{ij}:=a_{ij}+c_j-c_i. \end{align*} The corresponding triple sum satisfies \begin{align*} n'_{ijk} &=a'_{ij}+a'_{jk}+a'_{ki}\\ &=(a_{ij}+c_j-c_i)+(a_{jk}+c_k-c_j)+(a_{ki}+c_i-c_k)\\ &=n_{ijk}. \end{align*} If a different logarithmic lift for $g'_{ij}$ is chosen, the previous step shows that the resulting cocycle differs from $n'$ by an integral coboundary. Hence changing frames does not change the cohomology class. [/step] [step:Compare arbitrary good covers by passing to a common refinement] First let $\mathcal V=\{V_\alpha\}_{\alpha\in A}$ be a good refinement of $\mathcal U$, and choose a refinement map \begin{align*} \rho:A &\to I \end{align*} such that \begin{align*} V_\alpha\subset U_{\rho(\alpha)} \end{align*} for every $\alpha\in A$. Restrict the frame $e_{\rho(\alpha)}$ to $V_\alpha$. The transition function on $V_\alpha\cap V_\beta$ is \begin{align*} g^{\mathcal V}_{\alpha\beta} = g_{\rho(\alpha)\rho(\beta)}\big|_{V_\alpha\cap V_\beta}, \end{align*} and the logarithmic lift is \begin{align*} a^{\mathcal V}_{\alpha\beta} = a_{\rho(\alpha)\rho(\beta)}\big|_{V_\alpha\cap V_\beta}. \end{align*} Thus on $V_\alpha\cap V_\beta\cap V_\gamma$, \begin{align*} n^{\mathcal V}_{\alpha\beta\gamma} = n_{\rho(\alpha)\rho(\beta)\rho(\gamma)} \big|_{V_\alpha\cap V_\beta\cap V_\gamma}. \end{align*} This is exactly the Čech pullback of the cocycle $n$ along the refinement map $\rho$. Now let $\mathcal U$ and $\mathcal W$ be two arbitrary good trivializing covers, with independently chosen unitary frames and logarithmic lifts. Choose a common good refinement $\mathcal V$ of both covers, with refinement maps $\rho_{\mathcal U}:A\to I$ and $\rho_{\mathcal W}:A\to J$. Pull the two constructions back to $\mathcal V$. The preceding computation identifies each pulled-back cocycle with the cocycle obtained from the restricted frames and restricted logarithms. On the fixed cover $\mathcal V$, the earlier logarithm-independence and frame-independence steps show that these two pulled-back cocycles differ by an integral Čech coboundary. Therefore the two original cocycles define the same element of $\check H^2(X;\mathbb Z)$ in the refinement model for [Čech cohomology](/page/Cech%20Cohomology). [/step] [step:Add transition logarithms for tensor products] Let $L_1\to X$ and $L_2\to X$ be Hermitian complex line bundles. Choose a common good trivializing cover $\mathcal U=\{U_i\}_{i\in I}$, local frames $e_{L_1,i}$ for $L_1|_{U_i}$ and $e_{L_2,i}$ for $L_2|_{U_i}$, and transition functions \begin{align*} g_{L_1,ij},g_{L_2,ij}:U_i\cap U_j &\to \mathbb C^\times. \end{align*} Choose logarithmic lifts \begin{align*} a_{L_1,ij},a_{L_2,ij}:U_i\cap U_j &\to \mathbb R \end{align*} with \begin{align*} g_{L_\ell,ij}=\exp(2\pi i a_{L_\ell,ij}) \end{align*} for $\ell\in\{1,2\}$. The [tensor product](/page/Tensor%20Product) frame \begin{align*} e_{L_1,i}\otimes e_{L_2,i} \end{align*} is a local frame for $(L_1\otimes L_2)|_{U_i}$, and its transition function is \begin{align*} g_{L_1\otimes L_2,ij}=g_{L_1,ij}g_{L_2,ij}. \end{align*} A logarithmic lift is therefore \begin{align*} a_{L_1\otimes L_2,ij}:=a_{L_1,ij}+a_{L_2,ij}. \end{align*} The resulting integer cocycle satisfies \begin{align*} n_{L_1\otimes L_2,ijk} &=a_{L_1\otimes L_2,ij}+a_{L_1\otimes L_2,jk}+a_{L_1\otimes L_2,ki}\\ &=\bigl(a_{L_1,ij}+a_{L_1,jk}+a_{L_1,ki}\bigr) +\bigl(a_{L_2,ij}+a_{L_2,jk}+a_{L_2,ki}\bigr)\\ &=n_{L_1,ijk}+n_{L_2,ijk}. \end{align*} Passing to cohomology gives \begin{align*} c_1(L_1\otimes L_2)=c_1(L_1)+c_1(L_2). \end{align*} [/step] [step:Negate transition logarithms for the dual bundle] Let $L^\vee\to X$ be the dual line bundle. For a local frame $e_i$ of $L|_{U_i}$, let \begin{align*} \epsilon_i:L|_{U_i} &\to U_i\times \mathbb C \end{align*} denote the dual local frame characterized fiberwise by \begin{align*} \epsilon_i(e_i)=1. \end{align*} If $e_j=g_{ij}e_i$, then the dual frames satisfy \begin{align*} \epsilon_j=g_{ij}^{-1}\epsilon_i. \end{align*} Thus the transition function for $L^\vee$ is \begin{align*} g_{L^\vee,ij}=g_{ij}^{-1}. \end{align*} If $a_{ij}$ is a logarithmic lift of $g_{ij}$, then \begin{align*} a_{L^\vee,ij}:=-a_{ij} \end{align*} is a logarithmic lift of $g_{L^\vee,ij}$. Hence the corresponding integer cocycle is \begin{align*} n_{L^\vee,ijk} &=a_{L^\vee,ij}+a_{L^\vee,jk}+a_{L^\vee,ki}\\ &=-(a_{ij}+a_{jk}+a_{ki})\\ &=-n_{ijk}. \end{align*} Therefore \begin{align*} c_1(L^\vee)=-c_1(L). \end{align*} This completes the proof of the transition function formula and its functorial identities. [/step]