Let $X$ be a compact Kähler manifold and $L$ be a holomorphic line bundle on $X$ with a Hermitian metric $h$. I am struggling to understand how one induces a canonical dual metric $h^*$ on $L^*$. Now let's say $\phi^*$ be a nontrivial holomorphic section on $L^*$, so it induces a corresponding non-trivial anti holomorphic section on $L$, lets call it $\phi,$ is $\|\phi\|_h^2=\|\phi^*\|_{h^*}^2?$
Edit: So as far as I understand, we get a map \begin{align*} \Gamma(X;L)\rightarrow \Gamma(X;L^*)\\ \phi\mapsto \phi^*; \phi^*(\alpha)=h(\alpha,\phi)\\ \end{align*} This map is complex anti-linear and we define $\langle\phi^*,\psi^*\rangle_{h^*}:=\langle\psi,\phi\rangle_{h}.$ Does this map take a holomorphic section to an anti-holomorphic section or we need to make the map complex-linear?