Géométrie algébrique
The equivariant Atiyah class
Comptes Rendus. Mathématique, Tome 359 (2021) no. 3, pp. 257-282.

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 MB des complexes parfaits simples sur une famille lisse projective YB 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 MB of simple perfect complexes on a G-invariant smooth projective family YB 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 :
Accepté le :
Accepté après révision le :
Publié le :
DOI : 10.5802/crmath.166
Classification : 14F05, 14D20, 14N10
Ricolfi, Andrea T. 1

1 Scuola Internazionale Superiore di Studi Avanzati (SISSA), Via Bonomea 265, 34136 Trieste, Italy
@article{CRMATH_2021__359_3_257_0,
     author = {Ricolfi, Andrea T.},
     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},
     url = {http://archive.numdam.org/articles/10.5802/crmath.166/}
}
TY  - JOUR
AU  - Ricolfi, Andrea T.
TI  - The equivariant Atiyah class
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 257
EP  - 282
VL  - 359
IS  - 3
PB  - Académie des sciences, Paris
UR  - http://archive.numdam.org/articles/10.5802/crmath.166/
DO  - 10.5802/crmath.166
LA  - en
ID  - CRMATH_2021__359_3_257_0
ER  - 
%0 Journal Article
%A Ricolfi, Andrea T.
%T The equivariant Atiyah class
%J Comptes Rendus. Mathématique
%D 2021
%P 257-282
%V 359
%N 3
%I Académie des sciences, Paris
%U http://archive.numdam.org/articles/10.5802/crmath.166/
%R 10.5802/crmath.166
%G en
%F CRMATH_2021__359_3_257_0
Ricolfi, Andrea T. The equivariant Atiyah class. Comptes Rendus. Mathématique, Tome 359 (2021) no. 3, pp. 257-282. doi : 10.5802/crmath.166. http://archive.numdam.org/articles/10.5802/crmath.166/

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

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

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

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

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

[6] Bernstein, Joseph; Lunts, Valery 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. (Berthelot, Pierre; Grothendieck, Alexander; Illusie, Luc, eds.), Lecture Notes in Mathematics, 225, Springer, 1971 | Zbl

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

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

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

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

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

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

[14] Gholampour, Amin; Kool, Martijn; Young, Benjamin 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] Graber, Tom; Pandharipande, Rahul Localization of virtual classes, Invent. Math., Volume 135 (1999) no. 2, pp. 487-518 | DOI | MR | Zbl

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

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

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

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

[22] Huybrechts, Daniel; Thomas, Richard P. 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] Huybrechts, Daniel; Thomas, Richard P. 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] Illusie, Luc Complexe cotangent et déformations. I, Lecture Notes in Mathematics, 239, Springer, 1971, xv+355 pages | MR | Zbl

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

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

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

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

[32] Neeman, Amnon 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] Neeman, Amnon An improvement of the base-change theorem and the functor f ! (2017) (https://arxiv.org/abs/1406.7599)

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

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

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

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

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

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

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

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

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

[46] Thomason, Robert W. 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] Thomason, Robert W.; Trobaugh, Thomas 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

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

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

Cité par Sources :