In his paper "Mukai flops and derived categories", Namikawa reduces a general Mukai flop of a smooth projective $2n$-dimensional variety $Z$ along a subvariety $W\cong \mathbb P^n$ with $N_{W/Z}\cong \Omega^1_{\mathbb P^n}$ to another smooth projective variety $Z^+$ with subvariety $W^+$, satisfying the above properties and obtained by blowing up $Z$ along $W$ and then blowing down the exceptional divisor in the other direction, to the simpler case where $X,X^+$ are $\mathbb P^n$ bundles over $\mathbb P^n$ (given by $\mathbb P(\Omega^1_{\mathbb P^n}\oplus \mathcal O_{\mathbb P^n})$) with the projective space subvarieties $Y,Y^+$ coming from the sections of these bundles corresponding to the quotient line bundle $\mathcal O_{\mathbb P^n}$ where the "plus" versions are obtained again by blowing up and blowing down in the other direction.

He proves the simpler case directly, and then reduces the above case to this one by noting that the formal completion of $X$ along $Y$, $X_Y$, is naturally isomorphic to $Z_W$, the formal completion of $Z$ along $W$, and likewise for the "plus" versions.

I was wondering if these identifications only relied on the fact that the normal bundles and underlying variety completed along were the same in each case. If so, can someone point me to a general reference for this technique? If not, what is the proof of these identifications, since no mention of a proof is given there?


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Even the first-order infinitesimal neighborhood of a subvariety typically depends on more than the normal bundle, cf. Exercise III.4.10, p. 225 of Hartshorne's "Algebraic Geometry". However, for some special choices of subvariety and normal bundle, the formal neighborhood is automatically formally isomorphic to the formal neighborhood of the zero section in the normal bundle.

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    $\begingroup$ 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. $\endgroup$
    – HNuer
    Aug 6, 2013 at 14:58
  • $\begingroup$ 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? $\endgroup$
    – HNuer
    Aug 6, 2013 at 15:01
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    $\begingroup$ 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. $\endgroup$ Aug 6, 2013 at 16:27

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