Timeline for Are formal completions along a subvariety only dependent on the normal bundle?
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| Aug 6, 2013 at 16:27 | comment | added | Jason Starr | The formal neighborhood of a smooth subvariety of a smooth variety is isomorphic to the formal neighborhood of the zero section of the normal bundle <B>if and only if</B>, for every integer $r$, there is a splitting of the algebra surjection $\mathcal{O}_X/I^r \to \mathcal{O}_X/I$ that is an algebra surjection (local on stalks). You can try to prove this by induction. Such a splitting for $r$ gives rise to an infinitesimal extension of $\mathcal{O}_Y$ by $I^r/I^{r+1}$. So if $Ext^1(\Omega_Y,I^r/I^{r+1})$ vanishes, then the splitting lifts to $r+1$. In your case, use Bott vanishing. | |
| Aug 6, 2013 at 15:01 | comment | added | HNuer | I was also wondering if there were other examples that come up in practice. My method above can be very quickly used to give a proof of derived equivalence of classical flops since the normal bundle of $\mathbb P^n$ is then $\mathcal O_{\mathbb P^n}(-1)^{n+1}$ and the analogous vanishing as above is clear. Are there other examples? | |
| Aug 6, 2013 at 14:58 | comment | added | HNuer | That's what I thought from trying to work this out, but do you know of any important examples and proofs of why this works for them? With the above example, I can show that $\mathcal O_Z/I^n$ has a unique module structure for $n\leq 3$, where $I$ is the ideal sheaf of $\mathbb P^n$ in $Z$. So up to the second-order infinitesimal neighborhood both completions are the same. But this is a brute-force method and to continue it I'd need to know that Ext$^1(Sym^k(T_{\mathbb P^n}),Sym^l(T_{\mathbb P^n}))=0$ for $k>l$ or $l>k$ (I forget which) which I don't know is true at the moment. | |
| S Aug 6, 2013 at 14:34 | history | answered | Jason Starr | CC BY-SA 3.0 | |
| S Aug 6, 2013 at 14:34 | history | made wiki | Post Made Community Wiki by Jason Starr |