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my question is as follows.

Let $\chi$ a compact Calabi-Yau 3-fold and $A,B \subset \chi$ two 2-complex dimensional manifolds such that their intersection $C := A \cap B$ is a 1-complex dimensional manifold. Now let $L_A$ and $L_B$ holomorphic line bundles over $A$ and $B$ respectively. One can then consider their pullback onto $C$. I will denote these holomorphic line bundles over $C$ as $\left. L_A \right|_C$ and $\left. L_B \right|_C$. In particular we can construct the following holomorphic line bundle over $C$ $$ \tilde{L} = \left. L_A \right|_C \otimes \left. L_B \right|^\vee \otimes \sqrt{K}$$ where $^\vee$ denotes the use of the dual bundle and $K$ denotes the holomorphic cotangent bundle (i.e. the canonical bundle). I am now interested in the Cech cohomology groups $$ \check{H}^i \left( C, \tilde{L} \right)$$ where in an abuse of notation $\tilde{L}$ represents the sheaf of holomorphic sections of $\tilde{L}$. In particular I am interested in their dimensionality.

Given knowledge about $\chi$, $A$, $B$, $L_A$ and $L_B$, what can I learn about $ \text{dim}_{\mathbb{C}}\left( \check{H}^i \left(C, \tilde{L} \right) \right)$?

As I am still new to algebraic geometry, it would be very nice if you could provide some reference where I can read more about this topic.

Thank you guys very much in advance for your answers!

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