The equivariant Atiyah class

Andrea T. Ricolfi

doi:10.5802/crmath.166

Géométrie algébrique

The equivariant Atiyah class

Andrea T. Ricolfi ¹

¹ Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy

Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 257-282.

Résumé
Abstract

Soit $X$ un schéma complexe sur lequel agit un groupe algébrique affine $G$ . Nous démontrons que la classe d’Atiyah d’un complexe parfait $G$ -équivariant au dessus de $X$ , construite par Huybrechts et Thomas, est $G$ -équivariante dans un sense précis. Comme application, nous démontrons que, si $G$ est réductif, la théorie d’obstruction sur l’espace de modules relatif fin $M \to B$ des complexes parfaits simples sur une famille lisse projective $Y \to B$ est $G$ -équivariante. Les résultats contenus ici vont suggérer comment vérifier l’équivariance de la théorie d’obstruction naturelle sur un nombre d’espaces de modules munis de l’action d’un tore, notamment ceux qui sont construits en théorie de Donaldson–Thomas et en théorie de Vafa–Witten.

Let $X$ be a complex scheme acted on by an affine algebraic group $G$ . We prove that the Atiyah class of a $G$ -equivariant perfect complex on $X$ , as constructed by Huybrechts and Thomas, is $G$ -equivariant in a precise sense. As an application, we show that, if $G$ is reductive, the obstruction theory on the fine relative moduli space $M \to B$ of simple perfect complexes on a $G$ -invariant smooth projective family $Y \to B$ is $G$ -equivariant. The results contained here are meant to suggest how to check the equivariance of the natural obstruction theories on a wide variety of moduli spaces equipped with a torus action, arising for instance in Donaldson–Thomas theory and Vafa–Witten theory.

Reçu le : 2020-09-04
Accepté le : 2020-12-09
Accepté après révision le : 2021-01-26
Publié le : 2021-04-20

DOI : 10.5802/crmath.166

Classification : 14F05, 14D20, 14N10

Affiliations des auteurs :

Andrea T. Ricolfi ¹

¹ Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Andrea T. Ricolfi},
     title = {The equivariant {Atiyah} class},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {257--282},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {3},
     year = {2021},
     doi = {10.5802/crmath.166},
     language = {en},
}

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Andrea T. Ricolfi. The equivariant Atiyah class. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 257-282. doi : 10.5802/crmath.166. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.166/

Bibliographie
Cité par

[1] Michael Francis Atiyah Complex analytic connections in fibre bundles, Trans. Am. Math. Soc., Volume 85 (1957), pp. 181-207 | DOI | MR | Zbl

[2] Matthew Ballard; David Favero; Ludmil Katzarkov Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math., Volume 746 (2019), pp. 235-303 | DOI | MR | Zbl

[3] Sjoerd Beentjes; Andrea T. Ricolfi Virtual counts on Quot schemes and the higher rank local DT/PT correspondence (2018) (to appear in Math. Res. Lett.)

[4] Kai Behrend; Barbara Fantechi The intrinsic normal cone, Invent. Math., Volume 128 (1997) no. 1, pp. 45-88 | DOI | MR | Zbl

[5] Kai Behrend; Barbara Fantechi Symmetric obstruction theories and Hilbert schemes of points on threefolds, Algebra Number Theory, Volume 2 (2008), pp. 313-345 | DOI | MR | Zbl

[6] Joseph Bernstein; Valery Lunts Equivariant sheaves and functors, Lecture Notes in Mathematics, 1578, Springer, 1994, iv+139 pages | DOI | MR | Zbl

[7] Séminaire de géométrie algébrique du Bois Marie 1966/67, SGA 6. Théorie des intersections et théorème de Riemann-Roch. (Pierre Berthelot; Alexander Grothendieck; Luc Illusie, eds.), Lecture Notes in Mathematics, 225, Springer, 1971 | Zbl

[8] Alberto Cazzaniga; Dimbinaina Ralaivaosaona; Andrea T. Ricolfi Higher rank motivic Donaldson–Thomas invariants of $𝔸^{3}$ via wall-crossing, and asymptotics (2020) (https://arxiv.org/abs/2004.07020)

[9] Ben Davison; Andrea T. Ricolfi The local motivic DT/PT correspondence (2019) (https://arxiv.org/abs/1905.12458)

[10] Igor Dolgachev Lectures on invariant theory, London Mathematical Society Lecture Note Series, 296, Cambridge University Press, 2003, xvi+220 pages | DOI | MR | Zbl

[11] Barbara Fantechi; Lothar Göttsche Riemann-Roch theorems and elliptic genus for virtually smooth schemes, Geom. Topol., Volume 14 (2010) no. 1, pp. 83-115 | DOI | MR | Zbl

[12] Nadir Fasola; Sergej Monavari; Andrea T. Ricolfi Higher rank K-theoretic Donaldson–Thomas theory of points, Forum Math. Sigma, Volume 9 (2021), e15 | DOI | MR | Zbl

[13] Letterio Gatto; Andrea T. Ricolfi Jet bundles on Gorenstein curves and applications, J. Singul., Volume 21 (2020), pp. 50-83 | MR | Zbl

[14] Amin Gholampour; Martijn Kool; Benjamin Young Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts, Int. Math. Res. Not., Volume 2018 (2018) no. 10, pp. 2981-3069 | DOI | MR | Zbl

[15] Tom Graber; Rahul Pandharipande Localization of virtual classes, Invent. Math., Volume 135 (1999) no. 2, pp. 487-518 | DOI | MR | Zbl

[16] Alexander Grothendieck Éléments de géométrie algébrique. I. Le langage des schémas, Publ. Math., Inst. Hautes Étud. Sci. (1960) no. 4, p. 228 | Numdam | MR | Zbl

[17] Jack Hall; Amnon Neeman; David Rydh One positive and two negative results for derived categories of algebraic stacks, J. Inst. Math. Jussieu, Volume 18 (2019) no. 5, pp. 1087-1111 | DOI | MR | Zbl

[18] Jack Hall; David Rydh Algebraic groups and compact generation of their derived categories of representations, Indiana Univ. Math. J., Volume 64 (2015) no. 6, pp. 1903-1923 | DOI | MR | Zbl

[19] Jack Hall; David Rydh Perfect complexes on algebraic stacks, Compos. Math., Volume 153 (2017) no. 11, pp. 2318-2367 | DOI | MR | Zbl

[20] Robin Hartshorne Algebraic geometry, Graduate Texts in Mathematics, 52, Springer, 1977, xvi+496 pages | Zbl

[21] Daniel Huybrechts; Manfred Lehn The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, Cambridge University Press, 2010, xviii+325 pages | DOI | Zbl

[22] Daniel Huybrechts; Richard P. Thomas Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes, Math. Ann., Volume 346 (2010) no. 3, pp. 545-569 | DOI | MR | Zbl

[23] Daniel Huybrechts; Richard P. Thomas Erratum to: Deformation-obstruction theory for complexes via Atiyah and Kodaira-Spencer classes [MR2578562], Math. Ann., Volume 358 (2014) no. 1-2, pp. 561-563 | DOI | Zbl

[24] Luc Illusie Complexe cotangent et déformations. I, Lecture Notes in Mathematics, 239, Springer, 1971, xv+355 pages | MR | Zbl

[25] Luc Illusie Complexe cotangent et déformations. II, Lecture Notes in Mathematics, 283, Springer, 1972, vii+304 pages | MR | Zbl

[26] Srikanth B. Iyengar; Joseph Lipman; Amnon Neeman Relation between two twisted inverse image pseudofunctors in duality theory, Compos. Math., Volume 151 (2015) no. 4, pp. 735-764 | DOI | MR | Zbl

[27] Martijn Kool Fixed point loci of moduli spaces of sheaves on toric varieties, Adv. Math., Volume 227 (2011) no. 4, pp. 1700-1755 | DOI | MR | Zbl

[28] Gérard Laumon; Laurent Moret-Bailly Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 39, Springer, 2000, xii+208 pages | Zbl

[29] Jun Li; Gang Tian Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties, J. Am. Math. Soc., Volume 11 (1998) no. 1, pp. 119-174 | DOI | MR | Zbl

[30] Joseph Lipman Notes on derived functors and Grothendieck duality, Foundations of Grothendieck duality for diagrams of schemes (Lecture Notes in Mathematics), Volume 1960, Springer, 2009, pp. 1-259 | DOI | MR

[31] David Mumford; John Fogarty; Frances C. Kirwan Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 34, Springer, 1994, xiv+292 pages | DOI | MR

[32] Amnon Neeman The Grothendieck duality theorem via Bousfield’s techniques and Brown representability, J. Am. Math. Soc., Volume 9 (1996) no. 1, pp. 205-236 | DOI | MR | Zbl

[33] Amnon Neeman An improvement of the base-change theorem and the functor $f^{!}$ (2017) (https://arxiv.org/abs/1406.7599)

[34] Martin Olsson Sheaves on Artin stacks, J. Reine Angew. Math., Volume 603 (2007), pp. 55-112 | DOI | MR | Zbl

[35] Martin Olsson Algebraic spaces and stacks, Colloquium Publications, 62, American Mathematical Society, 2016, xi+298 pages | DOI | MR | Zbl

[36] Dragos Oprea; Rahul Pandharipande Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics, 2019 (https://arxiv.org/abs/1903.08787) | Zbl

[37] Andrea T. Ricolfi The DT/PT correspondence for smooth curves, Math. Z., Volume 290 (2018) no. 1-2, pp. 699-710 | DOI | MR | Zbl

[38] Andrea T. Ricolfi Local contributions to Donaldson–Thomas invariants, Int. Math. Res. Not., Volume 2018 (2018) no. 19, pp. 5995-6025 | DOI | MR | Zbl

[39] Andrea T. Ricolfi On the motive of the Quot scheme of finite quotients of a locally free sheaf, J. Math. Pures Appl, Volume 144 (2020), pp. 50-68 | DOI | MR | Zbl

[40] Andrea T. Ricolfi Virtual classes and virtual motives of Quot schemes on threefolds, Adv. Math., Volume 369 (2020), p. 107182 | DOI | MR | Zbl

[41] Maxwell Rosenlicht Toroidal algebraic groups, Proc. Am. Math. Soc., Volume 12 (1961), pp. 984-988 | DOI | MR | Zbl

[42] Christian Serpé Resolution of unbounded complexes in Grothendieck categories, J. Pure Appl. Algebra, Volume 177 (2003) no. 1, pp. 103-112 | DOI | MR | Zbl

[43] Nicolas Spaltenstein Resolutions of unbounded complexes, Compos. Math., Volume 65 (1988) no. 2, pp. 121-154 | Numdam | MR | Zbl

[44] The Stacks Project Authors Stacks Project, 2016 (http://stacks.math.columbia.edu)

[45] Hideyasu Sumihiro Equivariant completion II, J. Math. Kyoto Univ., Volume 15 (1975) no. 3, pp. 573-605 | DOI | MR | Zbl

[46] Robert W. Thomason Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math., Volume 65 (1987) no. 1, pp. 16-34 | DOI | MR | Zbl

[47] Robert W. Thomason; Thomas Trobaugh Higher algebraic $K$ -theory of schemes and of derived categories, The Grothendieck Festschrift, Vol. III (Progress in Mathematics), Volume 88, Birkhäuser, 1990, pp. 247-435 | DOI | MR | Zbl

[48] Burt Totaro The resolution property for schemes and stacks, J. Reine Angew. Math., Volume 577 (2004), pp. 1-22 | DOI | MR | Zbl

[49] Michela Varagnolo; Eric Vasserot Double affine Hecke algebras and affine flag manifolds, I, Affine Flag Manifolds and Principal Bundles (2010), pp. 233-289 | DOI | Zbl

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