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Let $X$, $Y$ and $Z$ be smooth manifolds and $$ X\xrightarrow{~f~}Y\xrightarrow{~g~}Z $$ be two composable embeddings. One can consider the normal bundles $N_{X/Y}\to X$ and $N_{Y/Z} \to Y$ and the normal bundle $N_{X/Z} \to X$ of the composition.

We know:

$N_{X/Z} \cong N_{X/Y} \oplus f^*N_{Y/Z}$ in this smooth setting, but the proof (as far as I know) uses splittings. To what extent does this work in the holomorphic category? Specifically, does this work for diagonal maps $\Delta: X \rightarrow X \times X$ and their iterative compositions? ($\Delta: X \times X \xrightarrow{id \times \Delta} X \times X \times X$ etc.)

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  • $\begingroup$ In the holomorphic setting there is only (in general non-split) short exact sequence of normal bundles. But such a question would be more appropriate on MSE. $\endgroup$
    – Sasha
    Commented Sep 3, 2021 at 4:21

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Blow up a point on a surface. The normal bundle of the projective line there will have a negative degree summand.

Embed the surface in a projective space. The normal bundle of the projective line there won't have a negative degree summand.

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  • $\begingroup$ Can you say a bit more about how this answers the question? $\endgroup$
    – John Rached
    Commented Sep 3, 2021 at 13:55

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