Nilpotent cohomological Hall algebras of surfaces

This paper develops a framework for systematically studying cohomological "Hecke operators" associated with modifications of coherent sheaves on a smooth surface $X$ along a fixed proper curve $Z \subset X$ (possibly singular and reducible), using the theory of cohomological Hall algebras. More precisely, we construct a moduli stack of coherent sheaves $\mathbf{Coh}(\widehat{X}_Z)$ on $X$ with set-theoretic support $Z$ and we prove that its reduced is an Artin stack locally of finite type. This provides a vast generalization of the global nilpotent cone. Subsequently, we develop the needed background to define the (motivic, $T$-equivariant) cohomological Hall algebra $\mathbf{HA}^{T}_{X,Z}$ of the moduli stack of coherent sheaves on $X$ with set-theoretic support on $Z$, in the setting of a general motivic formalism $\mathbf{D}$ in the sense of Khan. The algebra $\mathbf{HA}^{\mathbf{D}, A}_{X,Z}$ is functorial with respect to closed immersions $Z' \subset Z$ and transformations of the motivic formalism $\mathbf{D}$, and only depends on the formal neighborhood $\widehat{X}_Z$ of $Z$ in $X$. In the companion paper arXiv:2603.03386, we use the nilpotent COHA $\mathbf{HA}^{T}_{X,Z}$ to answer a question previously raised in arXiv:2004.13685 about the precise relationship between the COHA of a minimal resolution of a Kleinian singularity and the corresponding preprojective COHA.