I am looking for more detailed expositions that treat "zero sets of sections of vector bundles". I am aware of the fact that in modern algebraic geometry, "zero-schemes" of global sections are well defined and studied within the context of schemes.
But for now, I am more interested in a classical approach. I.e., let $\pi:E\rightarrow X$ be a vector bundle. Let $s_0$ be its zero section and $s$ some arbitrary section. The zero set of $s$ can be defined as $$Z(s) = \{ x\in X|\text{ } s(x) = 0_x\in \pi^{-1}(x)\} = X\times_{s = s_0} X.$$
My general question is: What can be said about the geometry/topology of $Z(s)$?
I am aware of the following situations:
1.) If $E$ and $X$ are $C^\infty$-manifolds and $E$ is a real $n$ - plane bundle, then if $s$ is transverse to the zero section, then $Z(s)$ is a submanifold by standard transversality arguments
2.) If $E$ and $X$ are algebraic varieties, $Z(s)$ can be constructed by gluing together local zero sets of $s$ in a chart of $E$.
I want to know what happens for complex vector bundles over manifolds $E$, $X$. Also, what about the homology/cohomology of $Z(S)$? How does it relate to characteristic classes of $E$? I have read that the homology class $[Z(s)]$ is dual to the top Chern class of $E$, but couldn't find any reference.
This is a very unprecise question, so I don't expect exact answers. But maybe someone can hint towards literature, where these questions are treated, since I couldn't find a lot in standard books by Milnor&Stasheff, Hatcher...(I guess I don't know what to look for)