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I have heard that derived category of coherent sheaves $\mathrm{Coh}(X)$ on any Fano varieties $X$ may be realized as derived category $\mathrm{Coh}(\mathrm{Rep}(Q,W))$ of representation of quiver $Q$ with relations $W$. I wonder for what kind of varieties their derived category of sheaves are realized by quiver. For example, is it possible for a quintic threefold $X$?

Thank you very much.

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2 Answers 2

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Any variety with a strong exceptional collection that generates the derived category of coherent sheaves will work. This goes back in some form to Bondal and is a consequence of Rickard's derived Morita equivalence. (This is sufficient, but not necessary. For non-compact varieties, for example, you can get derived equivalences with the path algebra of quivers with loops that are not given by the endomorphism of a strong exceptional collection, but are closely related. See Bridgeland's 0502050 for an example, and more in my paper with Nick Proudfoot 0512166.)

In general, given a generator of the derived category (or any triangulated category or stable infinity-category I suppose), the (generalized) theory of derived Morita equivalence says that there is an equivalence of categories between the derived category of modules over the endomorphisms of this object and the original category. If the generator only has endomorphisms of degree zero (ie, $Ext^i(E,E)$ is zero for non-zero i), the endomorphisms form an algebra, and you get an equivalence between the derived categories of this algebra and the original category. The exceptional collection makes it easy to interpret this algebra as the path algebra of a quiver with relations.

If the generator does have endomorphisms of non-zero degree, we instead have to think of the endomorphisms as a dg- or A-infinity algebra (or some other spectrum thingie if you're not working over a field of characteristic zero I think). Since the quintic is a compact Calabi-Yau, Serre duality means that any object that has endomorphisms of degree zero also has endomorphisms of degree 3, so we can't represent its derived category of coherent sheaves as the derived category of the path algebra of a quiver with relations.

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Thank you for the wonderful answer, Aaron! It gives me a good overview of the field and motivates me to study the subject. –  Pooya Nov 26 '12 at 4:02
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I now wonder where you use "compactness". Let $X$ be a conifold singularity and $Y$ be its crepant resolution with exceptional curve $C$. Y may be identified with the normal bundle $\pi:\mathcal{O}_C(-1)^2\rightarrow C$. Define $E=\pi^*\mathcal{O}_C\oplus \pi^*\mathcal{O}_C(1)$ and $A=\mathrm{End}(E)$. Then there exists a derived equivalence of Calabi-Yau 3 categories $D(Y)\cong D(Mod(A))$. So in this case Calabi-Yau 3 category is realized as quiver. –  Pooya Nov 26 '12 at 9:58
    
Those are the noncompact examples I was referring to in the parenthetical remark in the first paragraph. The last paragraph is evaded there because Serre duality doesn't hold for noncompact varieties. –  Aaron Bergman Nov 26 '12 at 12:44
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The question is the mother of all confusions. For the starters, a quintic threefold is not Fano but rather Calabi-Yau, and there you are doomed as Aaron has explained.

In general, a triangulated category may have three types of nice gadgets:

  1. strong exceptional collection,
  2. exceptional collection,
  3. tilting object.

Number 1 gives both 2 and 3. 2 or 3 do not give each other or 1. Each of these gadgets gives you a realization of your category as

  1. derived representations of a directed quiver (with relations),
  2. derived representations of a directed DG-quiver (with relations),
  3. derived representations of a quiver (with relations)

correspondingly. None of these gadgets exist on Calabi-Yau-s as explained by Aaron. All of these gadgets are expected to exist on Fano-s but the reality is much tougher. While we know these gadgets on Fano-s in many partial cases, we do not know them, in general. IMHO, it is a matter of time when a smooth compact Fano without none of them is discovered (or maybe even already known!).

Having said all this, you can reasonably expect that any smooth projective variety would have its bounded derived category of coherent sheaves realized as derived representations of a DG-quiver, but this gets technical and you should consult a real expert, not just a wabbitt:-))

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Thank you for the explanation. –  Pooya Nov 26 '12 at 9:48
    
Could you give examples for 2 or 3 do not give each other or 1? I'm mostly interested in 2 does not give 1. –  Julian Kuelshammer Mar 14 at 13:09
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