I am new to StackExchange, and self-taught in the field of characteristic classes, vector bundles, etc... so apologies in advance if my question is somewhat trivial or ill-posed.
I am trying to do a concrete computation using general facts I've gathered from notes and books on the web. The setup is as follows. Let $M$ be the complex manifold $M = \mathbb{C} \times \mathbb{CP}^1$ and the $N \subset M$ the submanifold $N = \mathbb{CP}^1$ . I would like to compute the Chern character of $K_M|N$, the canonical bundle over $M$ restricted to $N$.
From what I gathered, it seems the adjunction formula could be helpful (see e.g. Corollary 2.25 here), \begin{equation} K_N = K_M|N \otimes L_N^{-1}|N \, . \end{equation} This should somehow allow me to express $ch(K_M|N)$ in terms of $ch(K_N) = 1 + c_1(K_N) = 1 - c_1(\mathbb{CP}^1)$, using that $ch(E\otimes F) = ch(E) ch(F)$.
The question is about the meaning and contribution of the second factor in the adjunction formula. Can someone give me a clearer idea of what this inverse line bundle over $N$ is, what does it mean for it to be itself restricted to $N$, and how to compute its Chern character in terms of characteristic classes of $N$ for the case $N = \mathbb{CP}^1$?
I feel that, due to the triviality of $M$ and $N$ above, the answer should be somewhat simple. Any help would be greatly appreciated!