Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians: a comprehensive account
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arXiv
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2025
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| _version_ | 1866910037200338944 |
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| author | Diaconescu, Duiliu-Emanuel Porta, Mauro Sala, Francesco Schiffmann, Olivier Vasserot, Eric |
| author_facet | Diaconescu, Duiliu-Emanuel Porta, Mauro Sala, Francesco Schiffmann, Olivier Vasserot, Eric |
| contents | We begin the systematic study of cohomological Hecke operators of modifications of coherent sheaves on a smooth surface $X$, along a fixed proper curve $Z \subset X$. We develop the necessary geometric foundations in order to define the $T$-equivariant cohomological Hall algebra $\mathbf{HA}^{\mathbf{D}, T}_{X,Z}$ of the moduli stack of coherent sheaves on $X$ with set-theoretic support on $Z$, in the setting of a general motivic formalism $\mathbf{D}$. 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 $X$.
Assume $\mathbf{D}$ gives rise to Borel-Moore homology. When $X$ is a resolution of a Kleinian singularity and $Z$ is the exceptional divisor, we explicitly identify $\mathbf{HA}^T_{X,Z}$ with a completed nonstandard positive half $\mathbb{Y}^+_\infty(\mathfrak{g})$ of the affine Yangian $\mathbb{Y}(\mathfrak{g})$ of the corresponding affine ADE type Lie algebra $\mathfrak{g}$. Let $Z_1,Z_2 \subset X$ be curves with $Z_1\cap Z_2$ being zero-dimensional. We conjecture a PBW type theorem relating the COHAs of the pairs $(X, Z_1\cup Z_2), (X, Z_1)$, and $(X, Z_2)$ and partially prove it in several interesting cases, including all Kleinian resolutions of singularities and elliptic surfaces of types $D$ and $E$.
Our main tools are: (i) a continuity theorem describing the behavior of COHAs of objects in the heart of $t$-structures $τ_n$ when the sequence $(τ_n)_n$ converges in an appropriate sense to a fixed $t$-structure $τ_\infty$; (ii) a theorem relating the action of the braid group $B_Q$ by derived autoequivalences on the preprojective algebra of a quiver $Q$ with the algebraic action of $B_Q$ on the associated COHA $\mathbf{HA}^T_Q$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_19445 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians: a comprehensive account Diaconescu, Duiliu-Emanuel Porta, Mauro Sala, Francesco Schiffmann, Olivier Vasserot, Eric Algebraic Geometry High Energy Physics - Theory Quantum Algebra Representation Theory 14A20 (Primary), 17B37, 55P99 (Secondary) We begin the systematic study of cohomological Hecke operators of modifications of coherent sheaves on a smooth surface $X$, along a fixed proper curve $Z \subset X$. We develop the necessary geometric foundations in order to define the $T$-equivariant cohomological Hall algebra $\mathbf{HA}^{\mathbf{D}, T}_{X,Z}$ of the moduli stack of coherent sheaves on $X$ with set-theoretic support on $Z$, in the setting of a general motivic formalism $\mathbf{D}$. 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 $X$. Assume $\mathbf{D}$ gives rise to Borel-Moore homology. When $X$ is a resolution of a Kleinian singularity and $Z$ is the exceptional divisor, we explicitly identify $\mathbf{HA}^T_{X,Z}$ with a completed nonstandard positive half $\mathbb{Y}^+_\infty(\mathfrak{g})$ of the affine Yangian $\mathbb{Y}(\mathfrak{g})$ of the corresponding affine ADE type Lie algebra $\mathfrak{g}$. Let $Z_1,Z_2 \subset X$ be curves with $Z_1\cap Z_2$ being zero-dimensional. We conjecture a PBW type theorem relating the COHAs of the pairs $(X, Z_1\cup Z_2), (X, Z_1)$, and $(X, Z_2)$ and partially prove it in several interesting cases, including all Kleinian resolutions of singularities and elliptic surfaces of types $D$ and $E$. Our main tools are: (i) a continuity theorem describing the behavior of COHAs of objects in the heart of $t$-structures $τ_n$ when the sequence $(τ_n)_n$ converges in an appropriate sense to a fixed $t$-structure $τ_\infty$; (ii) a theorem relating the action of the braid group $B_Q$ by derived autoequivalences on the preprojective algebra of a quiver $Q$ with the algebraic action of $B_Q$ on the associated COHA $\mathbf{HA}^T_Q$. |
| title | Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians: a comprehensive account |
| topic | Algebraic Geometry High Energy Physics - Theory Quantum Algebra Representation Theory 14A20 (Primary), 17B37, 55P99 (Secondary) |
| url | https://arxiv.org/abs/2502.19445 |