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The Adjunction Formula is given to be

$K_V = (K_X \otimes [V])_V$

Where $K_V$ is the canonical class on $V\subset X$ and $K_X$ of $X$. And $[V]$ denotes the line bundle associated to $V$.

Now say, Instead of the canonical bundle on $V$, that I'm interested in some general Line bundle $[D]$. Is there some equivalent formula like

$[D]_V = ([D]_X \otimes [V])_V$?

Or how do you actually restrict a line bundle on $X$ to $V$? How do their chern classes relate?

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What kind of a formula do you want?

If we write the formula you posit for $[D]$ doesn't make sense: if we restrict $[D]$ to $V$, we get $[D]_V$; that's a tautology.

The reason that the adjunction formula has a more complicated shape is precisely because the restriction of the canonical bundle of $X$ to $V$ is not the canonical bundle of $V$.

The Chern class of $[D]_V$ is precisely the intersection of (a generic representative of the linear equivalence class of) the Chern class of $[D]$ with $V$. (If $s$ is a generic section of $D$, assuming it admits one, then the Chern class of $[D]$ is precisely the zero locus of $s$. If we restrict $[D]$ to $V$, then $s_{|V}$ will be a section of $[D]_V$, and its zero locus will be the intersection of the zero locus of $s$ with $V$.)

Based on your question, it seems that you are confused about something at a more basic level. Perhaps asking a more specific question would help.

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    Ahhh nevermind, I now see the wickedness of my ways :) Thanks!2012-06-27