Geodesic
On the Automorphisms of a Rank One Deligne–Hitchin Moduli Space
Symmetry, integrability and geometry: methods and applications, Tome 13 (2017) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $X$ be a compact connected Riemann surface of genus $g \geq 2$, and let ${\mathcal M}_{\rm DH}$ be the rank one Deligne–Hitchin moduli space associated to $X$. It is known that ${\mathcal M}_{\rm DH}$ is the twistor space for the hyper-Kähler structure on the moduli space of rank one holomorphic connections on $X$. We investigate the group $\operatorname{Aut}({\mathcal M}_{\rm DH})$ of all holomorphic automorphisms of ${\mathcal M}_{\rm DH}$. The connected component of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ containing the identity automorphism is computed. There is a natural element of $H^2({\mathcal M}_{\rm DH}, {\mathbb Z})$. We also compute the subgroup of $\operatorname{Aut}({\mathcal M}_{\rm DH})$ that fixes this second cohomology class. Since ${\mathcal M}_{\rm DH}$ admits an ample rational curve, the notion of algebraic dimension extends to it by a theorem of Verbitsky. We prove that ${\mathcal M}_{\rm DH}$ is Moishezon.
Keywords: Hodge moduli space; Deligne–Hitchin moduli space; $\lambda$-connections; Moishezon twistor space. 
@article{SIGMA_2017_13_a71,
     author = {Indranil Biswas and Sebastian Heller},
     title = {On the {Automorphisms} of a {Rank} {One} {Deligne{\textendash}Hitchin} {Moduli} {Space}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2017},
     volume = {13},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a71/}
}
TY  - JOUR
AU  - Indranil Biswas
AU  - Sebastian Heller
TI  - On the Automorphisms of a Rank One Deligne–Hitchin Moduli Space
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2017
VL  - 13
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a71/
LA  - en
ID  - SIGMA_2017_13_a71
ER  - 
%0 Journal Article
%A Indranil Biswas
%A Sebastian Heller
%T On the Automorphisms of a Rank One Deligne–Hitchin Moduli Space
%J Symmetry, integrability and geometry: methods and applications
%D 2017
%V 13
%U http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a71/
%G en
%F SIGMA_2017_13_a71
Indranil Biswas; Sebastian Heller. On the Automorphisms of a Rank One Deligne–Hitchin Moduli Space. Symmetry, integrability and geometry: methods and applications, Tome 13 (2017). http://geodesic.mathdoc.fr/item/SIGMA_2017_13_a71/

[1] Atiyah M. F., Bott R., “The Yang–Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A, 308 (1983), 523–615 | DOI | MR | Zbl

[2] Baraglia D., “Classification of the automorphism and isometry groups of Higgs bundle moduli spaces”, Proc. Lond. Math. Soc., 112 (2016), 827–854, arXiv: 1411.2228 | DOI | MR | Zbl

[3] Baraglia D., Biswas I., Schaposnik L. P., “Automorphisms of $\mathbb{C}^*$ moduli spaces associated to a Riemann surface”, SIGMA, 12 (2016), 007, 14 pp., arXiv: 1508.06587 | DOI | MR | Zbl

[4] Corlette K., “Flat $G$-bundles with canonical metrics”, J. Differential Geom., 28 (1988), 361–382 | DOI | MR | Zbl

[5] Donaldson S. K., “Twisted harmonic maps and the self-duality equations”, Proc. London Math. Soc., 55 (1987), 127–131 | DOI | MR | Zbl

[6] Goldman W. M., “The symplectic nature of fundamental groups of surfaces”, Adv. Math., 54 (1984), 200–225 | DOI | MR | Zbl

[7] Goldman W. M., Xia E. Z., Rank one Higgs bundles and representations of fundamental groups of Riemann surfaces, Mem. Amer. Math. Soc., 193, 2008, viii+69 pp., arXiv: math.DG/0402429 | DOI | MR

[8] Hitchin N. J., “The self-duality equations on a Riemann surface”, Proc. London Math. Soc., 55 (1987), 59–126 | DOI | MR | Zbl

[9] Hitchin N. J., Karlhede A., Lindström U., Roček M., “Hyperkähler metrics and supersymmetry”, Comm. Math. Phys., 108 (1987), 535–589 | DOI | MR | Zbl

[10] Simpson C., “Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization”, J. Amer. Math. Soc., 1 (1988), 867–918 | DOI | MR | Zbl

[11] Simpson C., “The Hodge filtration on nonabelian cohomology”, Algebraic Geometry (Santa Cruz, 1995), Proc. Sympos. Pure Math., 62, Amer. Math. Soc., Providence, RI, 1997, 217–281, arXiv: alg-geom/9604005 | DOI | MR | Zbl

[12] Simpson C., “A weight two phenomenon for the moduli of rank one local systems on open varieties”, From Hodge Theory to Integrability and TQFT tt*-Geometry, Proc. Sympos. Pure Math., 78, Amer. Math. Soc., Providence, RI, 2008, 175–214, arXiv: 0710.2800 | DOI | MR | Zbl

[13] Verbitsky M., “Holography principle for twistor spaces”, Pure Appl. Math. Q, 10 (2014), 325–354, arXiv: 1211.5765 | DOI | MR | Zbl

[14] Weil A., “Zum Beweis des Torellischen Satzes”, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa, 1957, 1957, 33–53 | MR