Scattering diagrams, stability conditions, and coherent sheaves on ℙ²

We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on P 2 \mathbb {P}^2 . This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on P 2 \mathbb {P}^2 , or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local P 2 \mathbb {P}^2 .

As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on P 2 \mathbb {P}^2 is Hodge-Tate, and we give the first non-trivial numerical checks of the general χ \chi -independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local P 2 \mathbb {P}^2 .