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I am currently trying to learn some complex geometry, using mainly the book by Huybrechts. There is one thing confusing me, however: For example on Wikipedia people are talking about the canonical bundle of a smooth divisor (in the linked article called $\omega_D$) and I am not sure what that is.

I do understand how to define the canonical bundle of a complex manifold. Therefore, I also understand how to define the canonical bundle of, say, a smooth hypersurface. I also know that every hypersurface Y defines a divisor $D = \sum_i [Y_i]$, where $Y_i$ are the irreducible components.

Given a smooth divisor $D = \sum_i a_i [Y_i]$ one could assign the hypersurface $Y = \bigcup_i Y_i$ to it and define $\omega_D = \omega_Y$. But (like Huybrechts also writes in his book) this construction is clearly not very natural.

It just doesn't seem to be right to me. How is the bundle $\omega_D$ defined then? (What is, e.g., the base space of this bundle?)

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This is actually a comment, but too big to say it in comments. That's why I'm writing it as an answer.

Theorem: (R. Hartshorne, Algebraic Geometry, Ch. II, Th. 8.17) Let $X$ be a non-singular variety of a field $k=\bar k.$ Let $Y$ be a closed irreducible subscheme of $X$ with the ideal sheaf $\mathcal I.$ Then $Y$ is non-singular if and only if (1) $\Omega_{Y/k}$ is locally free, and (2) we have the following exact sequence $$ 0 \to \mathcal I/\mathcal I^2 \to \Omega_{X/k} \otimes \mathcal O_Y \to \Omega_{Y/k} \to 0.$$ In this case, the sheaf $\mathcal I/\mathcal I^2$ is a locally free sheaf of rank $r= codim(Y,X)$ on $Y.$

Now let $Y$ be a non-singular subvariety of a non-singular variety $X$ over $k=\bar k.$ Then the sheaf $\mathcal N_{Y/X}:= \mathcal Hom_{\mathcal O_Y}(\mathcal I/\mathcal I^2, \mathcal O_Y)$ is called the normal sheaf of $Y$ in $X.$ It is locally free of rank $r= codim(Y,X).$

Theorem: (R. Hartshorne, Algebraic Geometry, Ch. II, Prop. 8.20) Let $Y$ be a non-singular subvariety of codimension $r$ in a non-singular variety $X$ over $k=\bar k.$ Then $\omega_Y \cong \omega_X \otimes \wedge^r \mathcal N_{Y/X}.$ If $r=1,$ consider $Y$ as a divisor, and let $\mathcal L$ be the associated invertible sheaf on $X.$ Then $\omega_Y \cong \omega_X \otimes \mathcal L \otimes \mathcal O_Y.$

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  • $\begingroup$ Thanks for the answer! To be completely honest I don't understand all of this and I certainly need to learn more algebraic geometry. Do I understand it correctly that you're saying: If $Y$ is a non-singular subvariety of codimension one, then we can consider $Y$ as a divisor as well and the formula in Wiki makes sense? But not every divisor is a subvariety, right? $\endgroup$ – Noiralef Mar 19 '15 at 14:01
  • $\begingroup$ Every divisor is a formal sum of hypersurfaces, so yes, it defines a subvariety, albeit not generally an irreducible one. For these adjunctions formulas in the context of complex geometry, you might want to check Griffiths and Harris $\endgroup$ – Artur Araujo Mar 22 '15 at 21:28

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