Geodesic
On Higgs Bundles on Nodal Curves
Symmetry, integrability and geometry: methods and applications, Tome 15 (2019) Cet article a éte moissonné depuis la source Math-Net.Ru

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This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
Keywords: Higgs bundles; nodal curves; generalised parabolic structures. 
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     author = {Marina Logares},
     title = {On {Higgs} {Bundles} on {Nodal} {Curves}},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a23/}
}
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Marina Logares. On Higgs Bundles on Nodal Curves. Symmetry, integrability and geometry: methods and applications, Tome 15 (2019). http://geodesic.mathdoc.fr/item/SIGMA_2019_15_a23/

[1] Balaji V., Barik P., Nagaraj D. S., “A degeneration of moduli of Hitchin pairs”, Int. Math. Res. Not., 2016 (2016), 6581–6625, arXiv: 1308.4490 | DOI | MR | Zbl

[2] Bhosle U. N., “Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves”, Ark. Mat., 30 (1992), 187–215 | DOI | MR | Zbl

[3] Bhosle U. N., “Representations of the fundamental group and vector bundles”, Math. Ann., 302 (1995), 601–608 | DOI | MR | Zbl

[4] Bhosle U. N., “Generalized parabolic bundles and applications. II”, Proc. Indian Acad. Sci. Math. Sci., 106 (1996), 403–420 | DOI | MR | Zbl

[5] Bhosle U. N., “Principal $G$-bundles on nodal curves”, Proc. Indian Acad. Sci. Math. Sci., 111 (2001), 271–291 | DOI | MR | Zbl

[6] Bhosle U. N., “Generalized parabolic Hitchin pairs”, J. Lond. Math. Soc., 89 (2014), 1–23 | DOI | MR | Zbl

[7] Bhosle U. N., Biswas I., Hurtubise J., “Grassmannian-framed bundles and generalized parabolic structures”, Internat. J. Math., 24 (2013), 1350090, 49 pp., arXiv: 1202.4239 | DOI | MR | Zbl

[8] Bhosle U. N., Logares M., Newstead P. E., Vector bundles on singular curves, in preparation

[9] Bhosle U. N., Parameswaran A. J., “Holonomy group scheme of an integral curve”, Math. Nachr., 287 (2014), 1937–1953 | DOI | MR | Zbl

[10] Bhosle U. N., Parameswaran A. J., Singh S. K., “Hitchin pairs on an integral curve”, Bull. Sci. Math., 138 (2014), 41–62 | DOI | MR | Zbl

[11] Cook P. R., Local and global aspects of the module theory of singular curves, Ph.D. Thesis, University of Liverpool, 1993

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

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

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

[15] Mazzeo R., Swoboda J., Weiss H., Witt F., “Ends of the moduli space of Higgs bundles”, Duke Math. J., 165 (2016), 2227–2271 | DOI | MR | Zbl

[16] Mehta V. B., Seshadri C. S., “Moduli of vector bundles on curves with parabolic structures”, Math. Ann., 248 (1980), 205–239 | DOI | MR | Zbl

[17] Narasimhan M. S., Seshadri C. S., “Stable and unitary vector bundles on a compact Riemann surface”, Ann. of Math., 82 (1965), 540–567 | DOI | MR | Zbl

[18] Oda T., Seshadri C. S., “Compactifications of the generalized Jacobian variety”, Trans. Amer. Math. Soc., 253 (1979), 1–90 | DOI | MR | Zbl

[19] Seshadri C. S., “Moduli of $\pi $-vector bundles over an algebraic curve”, Questions on Algebraic Varieties, CIME, III Ciclo (Varenna, 1969), Edizioni Cremonese, Rome, 1970, 139–260 | MR

[20] Seshadri C. S., Fibrés vectoriels sur les courbes algébriques, Astérisque, 96, Société Mathématique de France, Paris, 1982 | MR

[21] Simpson C. T., “Harmonic bundles on noncompact curves”, J. Amer. Math. Soc., 3 (1990), 713–770 | DOI | MR | Zbl

[22] Simpson C. T., “Higgs bundles and local systems”, Inst. Hautes Études Sci. Publ. Math., 75 (1992), 5–95 | DOI | MR | Zbl

[23] Simpson C. T., “Moduli of representations of the fundamental group of a smooth projective variety. I”, Inst. Hautes Études Sci. Publ. Math., 79 (1994), 47–129 | DOI | MR | Zbl

[24] Simpson C. T., “Moduli of representations of the fundamental group of a smooth projective variety. II”, Inst. Hautes Études Sci. Publ. Math., 80 (1994), 5–79 | DOI | MR

[25] Swoboda J., “Moduli spaces of Higgs bundles on degenerating Riemann surfaces”, Adv. Math., 322 (2017), 637–681, arXiv: 1507.04382 | DOI | MR | Zbl

[26] Weil A., “Généralisation des fonctions abéliennes”, J. Math. Pure Appl., 17 (1938), 47–87