TITLE: Derived category of varieties and derived category of quiver algebras
QUESTION [9 upvotes]: 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.

REPLY [11 votes]: 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:

strong exceptional collection,
exceptional collection,
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

derived representations of a directed quiver (with relations),
derived representations of a directed DG-quiver (with relations),
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:-))