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Let $M$ be a smooth manifold of dimension $m$ and $\pi:E\rightarrow M$ a vector bundle of rank $e$. Given a section $s$ of the bundle $\pi:E\rightarrow M$, we expect that the zero locus $Z(s)$ of $s$ is a submanifold $N\subset M$ of dimension $m-e$. This is true if $s$ can be perturbed into a general position so that $s(M)$ and the zero section intersect transversally.

Perturbation is not always possible (for example in holomorphic category category). In this case we need "excess intersection theory"; if the section $s$ lies in a subbundle $F\subset E$ and is a transversal section of $F$, the correct $(m-e)$-cycle we should take is the Euler class of the quotient bundle $E/F$, which is homologous to $Z(s)$ if transverse perturbation of $s$ exists.

My problem is that I don't really know good explicit examples with which I can compute things. Could anyone give me an example, which shows how useful excess intersection theory is?

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Doubled mathoverflow.net/questions/107292/… –  Ehsan M. Kermani Sep 16 '12 at 3:39

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