Another example, actually a lot more important, is to take a Hartogs figure, such at [ X:=D\times {0}\cup \partial D \times D \subset C^2 ] which your are welcome to think of as $R^4$.

Take a neighborhood $U$ of $X$. Then the first cohomoloy of a small neighborhood $U$ with values in the sheaf of holomorphic functions is not zero.

I could spell t out if you are interested.

John Hubbard

Another example, actually a lot more important, is to take a Hartogs figure, such at $$ X:=D\times \{0\}\cup \partial D \times D \subset \mathbb C^2 $$ which your are welcome to think of as $\mathbb R^4$.

Take a neighborhood $U$ of $X$. Then the first cohomoloy of a small neighborhood $U$ with values in the sheaf of holomorphic functions is not zero.

I could spell t out if you are interested.

John Hubbard