Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$

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 $\mathbb{P}^2$. This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on $\mathbb{P}^2$, or equivalently the refined Donaldson-Thomas invariants for compactly supported sheaves on local $\mathbb{P}^2$. As applications, we prove that the intersection cohomology of moduli spaces of Gieseker semistable sheaves on $\mathbb{P}^2$ is Hodge-Tate, and we give the first non-trivial numerical checks of the general $\chi$-independence conjecture for refined Donaldson-Thomas invariants of one-dimensional sheaves on local $\mathbb{P}^2$.