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(1) Is the definition of flop given by Wikipedia the industry standard?

(2) Regardless of the answer to (1), when is it expected that a birational transformation gives rise to a derived equivalence?

References to places where precise conjectures are recorded will be very much appreciated!

The reason I'm asking: apparently it is conjectured that different crepant resolutions are derived equivalent. On page 40 of this paper of Bondal-Orlov, they conjecture that flops induce derived equivalences. Apparently "flop" is sometimes used to mean birational transformation preserving canonical classes (without specifying the type of surgery actually being performed). So I'm interested to know whether such transformations are expected to be factorizable into (Wikipedia) flops, or produce derived equivalences for other reasons.

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    $\begingroup$ Interestingly, Atiyah invented his flop in 1958, 5 years before Fosbury invented his, but for some reason nobody used the Atiyah flop in the 1960 Summer Olympics. $\endgroup$
    – GS
    Commented Jun 14, 2010 at 14:30
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    $\begingroup$ For dimensions three, check out Bridgeland's arxiv.org/abs/math/0009053. $\endgroup$
    – Aaron Bergman
    Commented Jun 14, 2010 at 20:50
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    $\begingroup$ Thanks; my understanding is that Bridgeland's theorem that crepant resolutions (in dim'n 3) are derived equivalent relies on the fact they are connected by a sequence of (Kollar's definition, which I think is the same as Wikipedia) flops. Is this now known in (any) higher dimension? Is it a conjecture (folklore or otherwise)? $\endgroup$
    – GS
    Commented Jun 14, 2010 at 21:09
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    $\begingroup$ In higher dimensions, I would guess the best known result should be Theorem 1 from Kawamata's paper "Flops connect minimal models" <href arxiv.org/abs/0704.1013>. By that theorem, any two crepant resolutions (or even terminal crepant partial resolutions) are connected by a sequence of flops over the base (where flop here is in the standard sense of Kollar--Mori's book, or Wikipedia). $\endgroup$
    – user5117
    Commented Jul 25, 2010 at 10:58
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    $\begingroup$ Also, if I remember rightly, according to Miles Reid's survey paper "Twenty-five years of 3-folds --- an old person's view", Atiyah's flop was already known to Zariski. He then goes on to claim that it is really already present in 19th century work on the standard Cremona transformation of P^3, and arrives at an original birthdate for the flop of 1837! $\endgroup$
    – user5117
    Commented Jul 25, 2010 at 11:03

2 Answers 2

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I'm not any kind of expert on this stuff and I'm not sure what the current state of this conjecture is, but Kawamata has conjectures in this paper and this paper regarding when two birational varieties have equivalent derived categories. He also discusses flops in the first paper.

He has partial results, including: if $X$ is general type and $\mathcal{D}^b(X) \cong \mathcal{D}^b(Y)$ as triangulated categories then $X$ and $Y$ are K-equivalent. This generalizes the famous theorem of Bondal-Orlov that the bounded derived category of a Fano variety determines the variety. IIRC, in the proof of his theorem he takes the kernel of the Fourier-Mukai transform that gives the equivalence, shows that the support of the kernel (meaning the union of the supports of the cohomology sheaves of the kernel) has a component $Z$ dominating both varieties and uses $Z$ for the "roof" of the K-equivalence. The assumption that $X$ is general type is used to show that the projections from $Z$ are birational.

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Concerning (2) there is Conjecture 6.24 in the book of Huybrechts, Fourier-Mukai transforms in algebraic geometry. The conjecture predicts that two birational Calabi-Yau varieties are derived equivalent.

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  • $\begingroup$ Thanks! Due to my shocking level of ignorance I had not looked at Huybrechts' book. You actually undersell the conjecture there: it asserts that if there exists a birational correspondence between (smooth, projective over alg. closed field of char. 0---seem to be standing assumptions) varieties that matches up canonical classes then the 2 varieties in question are derived equivalent (though the equivalence need not be induced by the given birational correspondence). Since I'm also interested in the question about flops, I hope you don't mind if I don't accept this helpful answer! $\endgroup$
    – GS
    Commented Jun 17, 2010 at 10:41

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