Finite group actions on moduli spaces of vector bundles
Florent Schaffhauser1
1 Departamento de Matemáticas Universidad de Los Andes Bogotá (Colombia)
Séminaire de théorie spectrale et géométrie, Volume 34 (2016-2017), pp. 33-63.

We study actions of finite groups on moduli spaces of stable holomorphic vector bundles and relate the fixed-point sets of those actions to representation varieties of orbifold fundamental groups.

DOI: 10.5802/tsg.354
Classification: 14H60, 14H30
Keywords: Vector bundles on curves and their moduli, Fundamental groups
Florent Schaffhauser 1

1 Departamento de Matemáticas Universidad de Los Andes Bogotá (Colombia) 
@article{TSG_2016-2017__34__33_0,
     author = {Florent Schaffhauser},
     title = {Finite group actions on moduli spaces of vector bundles},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {33--63},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {34},
     year = {2016-2017},
     doi = {10.5802/tsg.354},
     zbl = {1360.30037},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.354/}
}
TY  - JOUR
AU  - Florent Schaffhauser
TI  - Finite group actions on moduli spaces of vector bundles
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2016-2017
SP  - 33
EP  - 63
VL  - 34
PB  - Institut Fourier
PP  - Grenoble
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.354/
DO  - 10.5802/tsg.354
LA  - en
ID  - TSG_2016-2017__34__33_0
ER  - 
%0 Journal Article
%A Florent Schaffhauser
%T Finite group actions on moduli spaces of vector bundles
%J Séminaire de théorie spectrale et géométrie
%D 2016-2017
%P 33-63
%V 34
%I Institut Fourier
%C Grenoble
%U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.354/
%R 10.5802/tsg.354
%G en
%F TSG_2016-2017__34__33_0
Florent Schaffhauser. Finite group actions on moduli spaces of vector bundles. Séminaire de théorie spectrale et géométrie, Volume 34 (2016-2017), pp. 33-63. doi : 10.5802/tsg.354. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.354/

[1] Jørgen E. Andersen; Jakob Grove Automorphism fixed points in the moduli space of semi-stable bundles, Q. J. Math, Volume 57 (2006) no. 1, pp. 1-35 | DOI | MR | Zbl

[2] Michael F. Atiyah K-theory and reality, Q. J. Math., Oxf. II. Ser., Volume 17 (1966), pp. 367-386 | DOI | MR | Zbl

[3] Michael F. Atiyah; Raoul Bott The Yang–Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond., A, Volume 308 (1983) no. 1505, pp. 523-615 | DOI | MR | Zbl

[4] Vikraman Balaji; Conjeeveram S. Seshadri Moduli of parahoric 𝒢-torsors on a compact Riemann surface, J. Algebr. Geom., Volume 24 (2015) no. 1, pp. 1-49 | DOI | MR | Zbl

[5] Indranil Biswas; Johannes Huisman; Jacques Hurtubise The moduli space of stable vector bundles over a real algebraic curve, Math. Ann., Volume 347 (2010) no. 1, pp. 201-233 | DOI | MR | Zbl

[6] Philip P. Boalch Riemann–Hilbert for tame complex parahoric connections, Transform. Groups, Volume 16 (2011) no. 1, pp. 27-50 | DOI | MR | Zbl

[7] Georgios D. Daskalopoulos The topology of the space of stable bundles on a compact Riemann surface, J. Differ. Geom., Volume 36 (1992) no. 3, pp. 699-746 | MR | Zbl

[8] Georgios D. Daskalopoulos; Karen K. Uhlenbeck An application of transversality to the topology of the moduli space of stable bundles, Topology, Volume 34 (1995) no. 1, pp. 203-215 | DOI | MR | Zbl

[9] Simon K. Donaldson A new proof of a theorem of Narasimhan and Seshadri, J. Differ. Geom., Volume 18 (1983) no. 2, pp. 269-277 | MR | Zbl

[10] Mikio Furuta; Brian Steer Seifert fibred homology 3-spheres and the Yang–Mills equations on Riemann surfaces with marked points, Adv. Math., Volume 96 (1992) no. 1, pp. 38-102 | DOI | MR | Zbl

[11] Günter Harder; Mudumbai S. Narasimhan On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann., Volume 212 (1975), pp. 215-248 | DOI | MR | Zbl

[12] Victoria Hoskins; Florent Schaffhauser Rational points of quiver moduli spaces, 2017 (https://arxiv.org/abs/1704.08624) | Zbl

[13] Victoria Hoskins; Florent Schaffhauser Group actions on quiver varieties and applications, Internat. J. Math., Volume 30 (2019) no. 02, 1950007, 46 pages | DOI | MR | Zbl

[14] Chiu-Chu Liu; Florent Schaffhauser The Yang–Mills equations over Klein surfaces, J. Topol., Volume 6 (2013) no. 3, pp. 569-643 | DOI | MR | Zbl

[15] Vikram B. Mehta; Conjeeveram S. Seshadri Moduli of vector bundles on curves with parabolic structures, Math. Ann., Volume 248 (1980) no. 3, pp. 205-239 | DOI | MR | Zbl

[16] David Mumford Projective invariants of projective structures and applications, Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, 1963, pp. 526-530 | MR | Zbl

[17] Mudumbai S. Narasimhan; Conjeeveram S. Seshadri Holomorphic vector bundles on a compact Riemann surface, Math. Ann., Volume 155 (1964), pp. 69-80 | DOI | MR | Zbl

[18] Mudumbai S. Narasimhan; Conjeeveram S. Seshadri Stable and unitary vector bundles on a compact Riemann surface, Ann. Math., Volume 82 (1965), pp. 540-567 | DOI | MR | Zbl

[19] Johan Råde On the Yang–Mills heat equation in two and three dimensions, J. Reine Angew. Math., Volume 431 (1992), pp. 123-163 | DOI | MR | Zbl

[20] Sundararaman Ramanan Orthogonal and spin bundles over hyperelliptic curves, Proc. Indian Acad. Sci., Math. Sci., Volume 90 (1981) no. 2, pp. 151-166 | DOI | MR | Zbl

[21] A. Ramanathan Stable principal bundles on a compact Riemann surface, Math. Ann., Volume 213 (1975), pp. 129-152 | DOI | MR | Zbl

[22] A. Ramanathan; Swaminathan Subramanian Einstein–Hermitian connections on principal bundles and stability, J. Reine Angew. Math., Volume 390 (1988), pp. 21-31 | MR | Zbl

[23] Florent Schaffhauser Real points of coarse moduli schemes of vector bundles on a real algebraic curve, J. Symplectic Geom., Volume 10 (2012) no. 4, pp. 503-534 | DOI | MR | Zbl

[24] Florent Schaffhauser On the Narasimhan–Seshadri correspondence for Real and Quaternionic vector bundles, J. Differ. Geom., Volume 105 (2017) no. 1, pp. 119-162 | MR | Zbl

[25] Conjeeveram S. Seshadri Space of unitary vector bundles on a compact Riemann surface, Ann. Math., Volume 85 (1967), pp. 303-336 | DOI | MR | Zbl

[26] Isadore M. Singer The geometric interpretation of a special connection, Pac. J. Math., Volume 9 (1959), pp. 585-590 | DOI | MR | Zbl

Cited by Sources: