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Let $E\to X$ be a holomorphic vector bundle over a complex manifold $X$ and denote by $\mathbb{P}(E)$ its projectivization. There is then a notion of a tautological line bundle $\mathcal{O}_E(1)\to \mathbb{P}(E)$ (cf. Kobayashi section 3.5).

I do not understand why Demailly (in Complex Analytic and Differential Geometry page 281) considers $\mathcal{O}_E(1)$ to be the bundle $\pi^*E/S\to \mathbb{P}(E^*)$ where $S\subset \pi^*E$ is the tautological hyperplane bundle inside $E$.

I know these two bundles are related, but why does Demailly choose the less intuitive $\pi^*E/S$?

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  • $\begingroup$ The obvious answer is the correct one: there is a "tautological" short exact sequence $$ 0\to S\to \pi^* E \to \mathcal{O}_E(1) \to 0. $$ $\endgroup$ Jan 9, 2022 at 12:52
  • $\begingroup$ @PiotrAchinger you could consider $$0\to \mathcal{O}_E(1)\to \pi^*E\to S\to 0$$ for the first option $\endgroup$
    – BinAcker
    Jan 9, 2022 at 12:54
  • $\begingroup$ Ah, so is this about the convention whether $\mathbf{P}(E)$ parametrizes one-dimensional subbundles or quotients of $E$? Then yes, indeed unfortunately both conventions are in use. The "quotients" version, while less geometric, is slighlty more natural (essentially since pullback of coherent sheaves is right exact). $\endgroup$ Jan 9, 2022 at 12:59
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    $\begingroup$ Then $\mathbf{P}_\text{subbundles}(E^\vee) = \mathbf{P}_\text{quotients}(E)$, and under this isomorphism "my" sequence is the dual of yours. $\endgroup$ Jan 9, 2022 at 13:01
  • $\begingroup$ @PiotrAchinger I guess that the projectivization of vector bundles should be subbundles, not quotients. It does not seem to contradict your comment on coherent sheaves, because sheaves are about "lineat functions" on the bundle, so a dual is involved. The same thing happens when one talks about the line bundle associated to a Cartier divisor, where one does not take the ideal (considered as functions, "invertible sheaf") but its dual (considered as a geometric object, " line bundle"). $\endgroup$
    – Z. M
    Jan 16, 2022 at 1:02

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