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More than once I've come across a statement in a paper about derived categories in which it says something to the effect of "in order to prove that $\Phi:D^b(X)\rightarrow D^b(Y)$ is fully-faithful we may restrict ourselves locally analytically and assume we are in the following situation ..." I was wondering why this is valid. Here are two examples that spring to mind in the work of Kawamata:

1) In "D-equivalence and K-equivalence", on p. 12 he mentions that to show that a given integral transform is fully-faithful, one only need to show that it respects Homs between spanning classes. Then he says "therefore, in order to prove our conjecture, we may consider locally over an analytic neighborhood of a point of $W$ and replace the given flop by any other flop which is analytically isomorphic to the original one." This was in the context of a classical flop of a subvariety $E\subset X$, $X$ a smooth $2m+1$-dimensional projective variety, $E\cong \mathbb P^m,N_{E/X}\cong \mathcal O_{\mathbb P^m}(-1)^{m+1}$ to another pair $F\subset Y$ satisfying the same conditions and obtained by blowing up $X$ along $E$ and then blowing down the exceptional divisor in the other direction. $W$ is the common contraction of $E$ in $X$ and $F$ in $Y$, respectively.

Why is it valid to just look locally analytically to prove this, and how does it follow from the well-known theorem about spanning classes?

2) Another example is found in his paper "Log Crepant Birational Maps and Derived Categories." Here in trying to prove an integral functor is fully-faithful, he notes that "since the set of all the point sheaves span the derived categories, our assertion can be prove analytic locally." Here the context is toroidal birational maps between toroidal varieties, and this statement is used to reduce one to the case of a toric morphism between toric varieties. I don't understand why this reduction via looking analytic locally works, especially since if one reads the proof there, powers of an ample line bundle are the spanning classes used in the proof, not point sheaves.

Any experts in derived categories care to explain?

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  • $\begingroup$ random thought: maybe it's not really a "passing to an analytic nbhrd" phenomenon but really it's "passing to a formal completion"? $\endgroup$ – bananastack Apr 1 '15 at 5:07
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I'll try to answer 1), but I won't give you a formal proof, more like an idea of a proof.

First it seems you worry about analytic/algebraic issues. In fact, you can define $D^b(X)$ in the analytic or algebraic context. If you have derived equivalence in the analytic context and your functor sends algebraic objects to algebraic objects (for instance it is a composition of derived push-forwards/pull-backs of projective morphisms), then you have derived equivalence in the algebraic context (the other way should be true with some projective hypotheses, by some GAGA-type results).

Next you worry about localization. So you have your flop:

\begin{array}{ccccc} & & Z & & \\ & \stackrel{q}\swarrow & & \stackrel{p}\searrow & \\ X & & & & X^{+} \\ & \stackrel{r}\searrow & & \stackrel{t}\swarrow & \\ & & W & & \end{array}

In your situation, $r$ and $t$ are isomorphisms outside $0 \in W$ and $q$ and $p$ are isomorphisms outside $r^{-1}(0)$ and $t^{-1}(0)$. So if you want to prove derived equivalence using the functor:

$$ \mathrm{R} p_* \mathrm{L} q^*$$

you don't care what happens outside of $0 \in W$. Indeed, this functor is the identity outside of $r^{-1}(0) \subset X$ and $t^{-1}(0) \subset X^{+}$ (here the argument is not formal, but you understand it). Hence you can restrict to a neighborhood of $0 \in W$. This could be an alegbraic or an analytic neighborhood of $0 \in W$, it does not matter for the moment.

Why Kawamata chooses an analytic neighborhood of $0 \in W$ is because he knows a more or less standard result, which is not trivial at all. This result tells you that locally ANALYTICALLY (and is NOT true algebraicly), the flops he studies is isomorphic to a standard form of flops (namely it is the contraction of the zero section on the tangent bundle of something). Hence, by the above remark, he only has to deal with the standard form of the flop and the computations with derived categories are then easier.

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