Vector bundle as an extension of line bundles This is a question about vector bundles on a smooth non-proper (non-projective) algebraic surface $X$ over $\mathbb{C}$. Are there any known examples of a non-proper surface $X$ and a rank two vector bundle $V$ on $X$ which is not an extension of line bundles, i.e. does not fit into a short exact sequence  $0 \to L \to V \to M \to 0$ where $L$ and $M$ are line bundles on $X$?
Notice that the usual topological idea of using the second Chern class does not work here.  I am asking about obstructions to restricting the structure group to the upper triangular subgroup of $GL_{2}(\mathbb{C})$.
 A: Only a partial answer, starting with the disclaimer that it does not apply when the base is the total space of a vector bundle on a projective curve. 
''If $X$ is a Stein manifold of dimension $2$, then each holomorphic vector bundle $V \to X$ of rank $r >1$ has a trivial one-dimensional subbundle''. In particular, it fits into a short exact sequence.
Proof: $X$ contains a $2$-dimensional CW-complex $K \subset X$ as a deformation retract. A generic (smooth) section of $V$ has a zero set of real dimension $4-2r < 2$, and moreover the zero set does not meet $K$ for dimensional reasons and by transversality. So $V|_K$ has a trivial complex line bundle as subbundle.
By homotopy invariance of vector bundles, this shows that $V$ has a trivial smooth subbundle. Now study the holomorphic fibre bundle $Mon(\mathbb{C};V)\to X$ (a point over $x $ is a complex linear monomorphism $\mathbb{C} \to V_x$). The fibre is the complex homogeneous space $Mon(\mathbb{C};\mathbb{C}^r)$, which is the quotient of $GL_r (\mathbb{C})$ by the stabilizer subgroup $G$ of the action of $GL_r (\mathbb{C})$ on $\mathbb{C}^{r} \setminus 0$.
The first paragraph says that there is a global smooth section $X \to Mon(\mathbb{C};V)$. Finding a holomorphic subbundle is the same as finding a holomorphic section.
Grauerts theorem (''Analytische Faserungen \"uber holomorph-vollst\"andigen R\"aumen'') says that for bundles of the above type over Stein manifolds, each smooth section is homotopic to a holomorphic section.
A: I am not sure what surface you have in mind, but it will depend on the surface. For example, if $X=\mathrm{Spec}\,A$ is affine and Pic is trivial, but $A^2(X)\neq 0$, then you can always represent a non-zero class in $A^2(X)$ by a zero cycle defined by an ideal $I$ which is a local complete intersection. By Serre construction, one has an exact sequence, $0\to A\to P\to I\to 0$, where $P$ is a rank 2 vector bundle and its second Chern class is non-zero by construction. If $P$ was filtered by line bundles, since Pic is trivial, the vector bundle is trivial and thus second Chern class must be zero. Such examples are easy, by taking a general hypersurface of large degree in affine three space. (Technically, take a general one in projective three space of large degree and remove a hyperplane section). 
