I have seen, but don't fully understand, the following statement and sketch proof:
Statement: A codimension $2$ submanifold $C \subset \mathbb{C}^2$ such that $C$ has positive intersection index with every complex line is complex analytic (i.e. every tangent space is complex).
Sketch Proof: Given a point $p \in C$ we can write $C$ as a graph over it's tangent space $T_pC$ in some neighbourhood of $p$. Now if the tangent space is not complex then one can "easily" find a complex line which has negative intersection index with $C$ at $p$. This is a contradiction and hence $T_pM$ is complex.
My problems are as follows:
1) What exactly is meant by the tangent space not being complex here? 2) How does one find this line precisely?
For the first point if we are considering $C$ as a subset in $\mathbb{C}^2$ then I guess we are talking about the tangent space as a plane in $\mathbb{C}^2$ which would necessarily be complex?
For the second point I guess that if we take a non-complex plane a complex plane both in $\mathbb{C}^2$ then their orientations will be oposite and hence the intersection index will be negative?
Any help will be greatly appreciated!