Let $S$ be a compact submanifold of $X$ smooth manifold. I know that $T_X|_S=T_S\oplus N_{S/X}$ where $N_{S/X}$ is the normal bundle. I have read that the euler class $e(N_{S/X})$ corresponds (via integration over S, i suppose) to the self intersection number $S\cdot S$. I've thought about it, but i don't know how to prove it, also i can't find the proof in any book. Do you know something about it?
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$\begingroup$ To start: what is your definition of $e(N_{S/X})$? $\endgroup$– JHMCommented Mar 8, 2013 at 2:14
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1$\begingroup$ IMO you are depriving yourself of some important suffering. $\endgroup$– JHMCommented Mar 8, 2013 at 2:20
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$\begingroup$ Your statement that $T_X|_S = T_S \oplus N_{S/X}$ is not true. There is an exact sequence relating these things, but it generally does not split. $\endgroup$– Jack HuizengaCommented Mar 8, 2013 at 2:49
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12$\begingroup$ @Jack: words like "compact", "manifold", and "integration" suggest to me that Konrad is working in the differentiable category, where he may split to his heart's content. $\endgroup$– Allen KnutsonCommented Mar 8, 2013 at 2:59
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$\begingroup$ @Konrad: How do you define the self-intersection "number" unless $\dim(X)=2\dim(S)$? $\endgroup$– Mark GrantCommented Mar 8, 2013 at 3:07
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I've just taught this in my graduate class. Check these notes on intersection theory.
The result you want is contained in Thm. 4.7. Again, you need $2\dim S=\dim X$.
answered Mar 8, 2013 at 13:13
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1$\begingroup$ Your notes look very nice, thanks for sharing. $\endgroup$– DaveCommented Mar 8, 2013 at 15:17
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$\begingroup$ yes, i forgot the hypotesis $2dim S=dim X$, in particular $S$ is a complex curve in $X$. $S\cdot S=\int_X \eta_S\wedge \eta_S$. If $s_0:S\rightarrow N$ is the zero section of the normal bundle, to use thm 4.7 i guess you are impling that $\int_X \eta_S \wedge \eta_S =\int_N \eta_{s_0(S)}\wedge \eta_{s_0(S)}$.. but why is that? $\endgroup$– KonradCommented Mar 12, 2013 at 15:04