Cohomological χ–independence for moduli of one-dimensional sheaves and moduli of Higgs bundles

We prove that the intersection cohomology (together with perverse and Hodge filtrations) for moduli space of one-dimensional semistable sheaves supported in an ample curve class on a toric del Pezzo surface is independent Euler characteristic sheaves. also analogous result Higgs bundles respect to effective divisor $D$ degree $\mathrm{deg}(D)>2g-2$. Our results confirm cohomological $\chi$-independence conjecture by Bousseau $\mathbb{P}^2$, verify Toda's Gopakumar-Vafa invariants certain local curves surfaces. For proof, we combine generalized version Ng\^o's support theorem, dimension estimate stacky Hilbert-Chow morphism, splitting theorem morphism from stack good GIT quotient.