First Chern class of tensor, dual, and conjugate lines

The tensor formula is the additivity clause of the group isomorphism of [F1]: linearity of $c_1$ under the group structure of ${\mathrm{Pic}}_{\mathrm{t}\mathrm{o}\mathrm{p}}$ is exactly $c_1(L\otimes M)=c_1(L)+c_1(M)$.

The dual formula: the evaluation pairing of [F2] shows $L\otimes L^{\ast }\cong {\epsilon }^1$, so by step 1.1 and $c_1({\epsilon }^1)=0$ one has $c_1(L^{\ast })=-c_1(L)$.

The conjugate formula: write a Hermitian metric from [F3] as $h$, conjugate-linear in its first argument and linear in its second (transpose the arguments if using the opposite convention). It provides, for each $x$, the conjugate-linear isomorphism $L_x\to L_x^{\ast }$, $v\mapsto h(v,-)$, which is complex-linear on the conjugate line; in a local frame $e$, the functional sends $we$ to $\bar{z}wh(e,e)$ for $v=ze$, so it is continuous with nonzero coefficient $h(e,e)>0$. Hence the maps define an isomorphism $\bar{L}\cong L^{\ast }$. Hence $c_1(\bar{L})=c_1(L^{\ast })=-c_1(L)$ by step 2.1.

Specialization to underlying real bundles: for a complex line with $c_1(L)=e(L_{\mathbb{R}})$ by [F4], the dual identity reads $e((L^{\ast }{)}_{\mathbb{R}})=-e(L_{\mathbb{R}})$, consistent with the orientation-reversal sign.

Boundary cases. For the trivial line $L={\epsilon }^1$ the formulas read $c_1(M)=0+c_1(M)$ and $0=0$; for a point base both sides vanish. The empty base is excluded by the path-connected hypothesis; the group ${\mathrm{Pic}}_{\mathrm{t}\mathrm{o}\mathrm{p}}(X)$ is abelian and $H^2(X;\mathbb{Z})$ is nonzero as a group in general but may be zero, in which case all three identities still hold. AC is used only through [A1] in the classification and metric suppliers.