logo AFST
Instantons and framed sheaves on Kähler Deligne–Mumford stacks
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 3, pp. 599-628.

Nous donnons une généralisation champêtre de résultats classiques de théorie de jauge, comme la caractérisation par Donaldson des instantons sur 4 en termes algébro-géométriques, le théorème de Uhlenbeck–Yau et diverses variantes dûes à Bando et ses collaborateurs. Nous appliquons cette machinerie à la classification des instantons sur certains espaces ALE.

We provide stacky generalizations of classical gauge-theoretic results inspired by Donaldson, the Uhlenbeck–Yau theorem and variants due to Bando and his collaborators. Moreover, we show an application of this machinery in the study of ALE spaces.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1579
@article{AFST_2018_6_27_3_599_0,
     author = {Philippe Eyssidieux and Francesco Sala},
     title = {Instantons and framed sheaves on {K\"ahler} {Deligne{\textendash}Mumford} stacks},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {599--628},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {3},
     year = {2018},
     doi = {10.5802/afst.1579},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1579/}
}
TY  - JOUR
AU  - Philippe Eyssidieux
AU  - Francesco Sala
TI  - Instantons and framed sheaves on Kähler Deligne–Mumford stacks
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2018
DA  - 2018///
SP  - 599
EP  - 628
VL  - Ser. 6, 27
IS  - 3
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1579/
UR  - https://doi.org/10.5802/afst.1579
DO  - 10.5802/afst.1579
LA  - en
ID  - AFST_2018_6_27_3_599_0
ER  - 
Philippe Eyssidieux; Francesco Sala. Instantons and framed sheaves on Kähler Deligne–Mumford stacks. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 3, pp. 599-628. doi : 10.5802/afst.1579. https://afst.centre-mersenne.org/articles/10.5802/afst.1579/

[1] Michael F. Atiyah; Nigel J. Hitchin; V.G. Drinfeld; Yu.I. Manin Construction of instantons, Phys. Lett. A, Volume 65 (1978) no. 3, pp. 185-187

[2] Shigetoshi Bando Einstein-Hermitian metrics on non-compact Kähler manifolds, Einstein metrics and Yang-Mills connections (Lecture Notes in Pure and Applied Mathematics) Volume 145, Marcel Dekker, 1993, pp. 27-33

[3] Kai Behrend Cohomology of stacks, School and conference on intersection theory and moduli (ICTP Lecture Notes) Volume 19, Abdus Salam International Centre for Theoretical Physics, 2004, vii+327 pages

[4] Kai Behrend; Brian Conrad; Dan Edidin; Barbara Fantechi; William Fulton; Lothar Göttsche; Andrew Kresch Algebraic stacks (2015) (http://www.math.uzh.ch/ws0607/mat513)

[5] Kai Behrend; Grégory Ginot; Behrang Noohi; Ping Xu String topology for stacks, Astérisque, Volume 343, Société Mathématique de France, 2012, xiv+169 pages

[6] Kai Behrend; Behrang Noohi Uniformization of Deligne-Mumford curves, J. Reine Angew. Math., Volume 599 (2006), pp. 111-153

[7] Ugo Bruzzo; Mattia Pedrini; Francesco Sala; Richard J. Szabo Framed sheaves on root stacks and supersymmetric gauge theories on ALE spaces, Adv. Math., Volume 288 (2016), pp. 1175-1308

[8] Ugo Bruzzo; Francesco Sala; Mattia Pedrini Framed sheaves on projective stacks, Adv. Math., Volume 272 (2015), pp. 20-95

[9] Nicholas P. Buchdahl Instantons on nℂℙ 2 , J. Differ. Geom., Volume 37 (1993) no. 3, p. 669-387

[10] Charles Cadman Using stacks to impose tangency conditions on curves, Am. J. Math., Volume 129 (2007) no. 2, pp. 405-427

[11] Neil Chriss; Victor Ginzburg Representation theory and complex geometry, Birkhäuser, 2010, x+495 pages

[12] David A. Cox; John B. Little; Henry K. Schenck Toric varieties, Graduate Studies in Mathematics, Volume 124, American Mathematical Society, 2011, xxiv+841 pages

[13] Jean-Pierre Demailly Cohomology of q-convex spaces in top degrees, Math. Z., Volume 204 (1990) no. 2, pp. 283-295

[14] Jean-Pierre Demailly Complex analytic and differential geometry (2012) (OpenContent Book available at https://www-fourier.ujf.grenoble.fr/~demailly/manuscripts/agbook.pdf)

[15] Jean-Pierre Demailly; Mihai Paun Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. Math., Volume 159 (2004) no. 3, pp. 1247-1274

[16] Simon K. Donaldson Instantons and geometric invariant theory, Commun. Math. Phys., Volume 93 (1984), pp. 453-460

[17] Emily B. Dryden; Carolyn S. Gordon; Sarah J. Greenwald; David L. Webb Asymptotic expansion of the heat kernel for orbifolds, Mich. Math. J., Volume 56 (2008) no. 1, pp. 205-238

[18] Philippe Eyssidieux; Vincent Guedj; Ahmed Zeriahi Singular Kähler-Einstein metrics, J. Am. Math. Soc., Volume 22 (2009) no. 3, pp. 607-639

[19] Barbara Fantechi; Etienne Mann; Fabio Nironi Smooth toric Deligne-Mumford stacks, J. Reine Angew. Math., Volume 648 (2010), pp. 201-244

[20] Victor Ginzburg Lectures on Nakajima’s quiver varieties, Geometric methods in representation theory. I (Séminaires et Congrès) Volume 24 (2012), pp. 145-219

[21] Gérard Gonzalez-Sprinberg; Jean-Louis Verdier Construction géométrique de la correspondance de McKay, Ann. Sci. Éc. Norm. Supér., Volume 16 (1983), pp. 409-449

[22] Krishnamurthi Guruprasad; André Haefliger Closed geodesics on orbifolds, Topology, Volume 45 (2006) no. 3, pp. 611-641

[23] Jack Hall General Existence Theorems in Moduli Theory (2011) (Ph. D. Thesis)

[24] Seán Keel; Shigefumi Mori Quotients by groupoids, Ann. Math., Volume 145 (1997) no. 1, pp. 193-213

[25] Alastair King Study of conformally self-dual 4-manifolds: instantons and holomorphic bundles on the blown-up plane (1989) (Ph. D. Thesis)

[26] Ryoichi Kobayashi Einstein–Kähler V-metrics on open Satake V-surfaces with isolated quotient singularities, Math. Ann., Volume 272 (1985), pp. 385-398

[27] Andrew Kresch On the geometry of Deligne-Mumford stacks, Algebraic geometry, Seattle 2005 (Proceedings of Symposia in Pure Mathematics) Volume 80, American Mathematical Society, 2009, pp. 259-271

[28] Peter B. Kronheimer The construction of ALE spaces as hyper-Kähler quotients, J. Differ. Geom., Volume 29 (1989) no. 3, pp. 665-683

[29] Peter B. Kronheimer; Hiraku Nakajima Yang-Mills instantons on ALE gravitational instantons, Math. Ann., Volume 288 (1990) no. 2, pp. 263-307

[30] Gérard Laumon; Laurent Moret-Bailly Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete 3., Volume 39, Springer, 2000, xi++208 pages

[31] Camille Laurent-Gengoux; Jean-Louis Tu; Ping Xu Chern-Weil map for principal bundles over groupoids, Math. Z., Volume 255 (2007) no. 3, pp. 451-491

[32] Eugene Lerman Orbifolds as stacks?, Enseign. Math., Volume 56 (2010) no. 3-4, pp. 315-363

[33] Kenji Matsuki; Martin Olsson Kawamata-Viehweg vanishing as Kodaira vanishing for stacks, Math. Res. Lett., Volume 12 (2005) no. 2-3, pp. 207-217

[34] Ieke Moerdijk Orbifolds as groupoids: an introduction, Orbifolds in mathematics and physics (Contemporary Mathematics) Volume 310, American Mathematical Society, 2002, pp. 205-222

[35] Hiraku Nakajima Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. Math., Volume 102 (1990) no. 2, 267 pages

[36] Hiraku Nakajima Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J., Volume 76 (1994) no. 2, pp. 365-416

[37] Hiraku Nakajima Lectures on Hilbert schemes of points on surfaces, University Lecture Series, Volume 18, American Mathematical Society, 1999, xi+132 pages

[38] Hiraku Nakajima Sheaves on ALE spaces and quiver varieties, Mosc. Math. J., Volume 7 (2007) no. 4, pp. 699-722

[39] Fabio Nironi Moduli spaces of semistable sheaves on projective Deligne-Mumford stacks (2009) (https://arxiv.org/abs/0811.1949)

[40] Behrang Noohi Foundations of Topological Stacks I (2005) (https://arxiv.org/abs/math/0503247)

[41] Dorette A. Pronk Etendues and stacks as bicategories of fractions, Compos. Math., Volume 102 (1996) no. 3, pp. 243-303

[42] Julius Ross; Richard Thomas Weighted projective embeddings, stability of orbifolds, and constant scalar curvature Kähler metrics, J. Differ. Geom., Volume 88 (2011) no. 1, pp. 109-159

[43] Olivier Schiffmann Variétés carquois de Nakajima (d’après Nakajima, Lusztig, Varagnolo, Vasserot, Crawley-Boevey, et al.), Bourbaki seminar. Volume 2006/2007 (Astérisque) Volume 317 (2008), p. 395-344

[44] Bertrand Toën Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford, K-Theory, Volume 18 (1999) no. 1, pp. 33-76

[45] Karen Keskulla Uhlenbeck; Shing-Tung Yau On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Commun. Pure Appl. Math., Volume 39 (1986), p. S257-S293

[46] Angelo Vistoli Intersection theory on algebraic stacks and on their moduli spaces, Invent. Math., Volume 97 (1989) no. 3, pp. 613-670

[47] Shing-Tung Yau On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation I, Commun. Pure Appl. Math., Volume 31 (1978), pp. 339-411

Cité par Sources :