Complex Algebraic Foliations, Book by A. Lins Neto and Bruno Scárdua, Chapter 3, Page 82.
Let $M$ be a Complex Manifold of complex dimension two (i.e., $M$ is a Complex Surface).
Let $S$ be a Compact Connected Riemann Surface (the complex dimension of $S$ is one) such that $S$ is Embedded in the complex manifold $M$.
We aim to define the Number of Self-Intersections of the Riemann Surface $S$.
I have some questions:
- What do they exactly mean by saying "Let $\widetilde{S}$ be a $C^{\infty}$ deformation of $S$"?
- Does "$\widetilde{S}$ cuts $S$ transversely" mean that for every point $p \in S \cap \widetilde{S}$ we have $T_p(S) + T_p(\widetilde{S}) = T_p(M)$?
- what is the "isotopy of $S$"?
- Why is $S \cap \widetilde{S}$ finite?
- what do they mean by saying "a basis $\{u_1,u_2,v_1,v_2\}$ is positive in $T_p(M)$?
