Moduli of Bridgeland semistable objects on P

Let X be a smooth projective surface, Coh(X) the abelian category of coherent sheaves on X, and D(X) the bounded derived category of coherent sheaves on X. We study the Bridgeland stability conditions on D(X) and see that for some stability conditions on D(X) the moduli spaces of (semi)stable objects in D(X) coincide with the moduli spaces of (semi)stable coherent sheaves, while for some other stability conditions the moduli spaces of (semi)stable objects inD(X) coincide with the moduli spaces of (semi)stable modules over a finite dimensional C-algebra in the case of X with a full strong exceptional collection. In particular, we construct the moduli spaces of (semi)stable coherent sheaves on P2 as the moduli spaces of (semi)stable quiver representations. This gives another proof of Le Potier’s result [P1] and some variants.