Let $X,Y$ be two connected, non-singular compact Riemann surfaces such that $X$ intersects $Y$ transversely at two distinct points. Let $L$ be a $\mathbb{C}$-local system on $X$. Let $L'$ be the trivial local system on $Y$. Suppose that the fibers of $L$ and $L'$ are isomorphic. Does there exist a local system $L''$ on $X \cup Y$ such that its restriction to $X$ (resp. $Y$) is $L$ (resp. $L'$)?

Intuitively, I would think this is not always true i.e., there should be some restriction on $L$. This is roughly because (due to Riemann-Hilbert correspondence) local systems are in 1-1 correspondence with representations of the fundamental group. Since fundamental group is a topological invariant (in the curve case depending only on the genus) and the arithmetic genus of $X$ is different from that of $X \cup Y$, the space of local systems on $X \cup Y$ should vary from that on $X$. This should give rise to extra conditions on $X$.

However, I do not understand why the kernel of the following morphism should not always be a local system: $$L \oplus L' \to \mathbb{C}^r_{X \cap Y}$$ for $L_{X \cap Y} \cong \mathbb{C}^r_{X \cap Y} \cong L'_{X \cap Y}$ and the morphism is given by a choice of identification of the fibers of $L$ and $L'$ at the points $X \cap Y$. Here $\mathbb{C}^r_{X \cap Y}$ denotes the skyscraper sheaf supported at $X \cap Y$ with fibers $\mathbb{C}^r$.


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