# What is a flop (and when are they conjectured to give derived equivalences)?

(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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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. –  GS Jun 14 '10 at 14:30
For dimensions three, check out Bridgeland's arxiv.org/abs/math/0009053. –  Aaron Bergman Jun 14 '10 at 20:50
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)? –  GS Jun 14 '10 at 21:09
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). –  Artie Prendergast-Smith Jul 25 '10 at 10:58
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! –  Artie Prendergast-Smith Jul 25 '10 at 11:03

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.