Nilpotent cohomological Hall algebras of surfaces

Abstract

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 $\mathbf{Coh}({\hat{X}}_Z)$ 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 ${\mathbf{HA}}_{X,Z}^T$ 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 ${\mathbf{HA}}_{X,Z}^{\mathbf{D},A}$ is functorial with respect to closed immersions $Z^{\prime }\subset Z$ and transformations of the motivic formalism $\mathbf{D}$, and only depends on the formal neighborhood ${\hat{X}}_Z$ of $Z$ in $X$. In the companion paper arXiv:2603.03386, we use the nilpotent COHA ${\mathbf{HA}}_{X,Z}^T$ 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.

v3: This is a revised version of Part I of v2 of this manuscript. It contains two new results: the theory of nilpotent sheaves on formal schemes is now developed via a relative version of ind-coherent sheaves, and the nilpotent 0-dimensional COHA of a smooth surface is computed explicitly; 65 pp. v2: Title changed to reflect that the manuscript will be posted in separate parts; 335 pp. v1: 335 pp