Moduli space of Higgs coherent systems with stable underlying vector bundles over a curve

Castañeda-González, Edgar I.

doi:10.15517/90r2vk27

Autores/as

Edgar I. Castañeda-González Universidad Autónoma de Zacatecas, Unidad Académica de Matemáticas, Zacatecas, México. Autor/a https://orcid.org/0009-0008-5375-2229

DOI:

https://doi.org/10.15517/90r2vk27

Palabras clave:

Espacios moduli, Haces de Higgs, Sistemas coherentes, Sistemas coherentes de Higgs

Resumen

Sea X una curva proyectiva compleja, suave y conexa, de género al menos 2. Un sistema coherente de Higgs se define como un haz aumentado (E, V), donde E es un haz vectorial holomorfo sobre X y V es un subespacio lineal del espacio de campos de Higgs de E. El propósito de este artículo es doble. En primer lugar, introducimos el problema moduli de los sistemas coherentes de Higgs sobre X. En segundo lugar, construimos el espacio moduli fino para los sistemas coherentes de Higgs cuyo haz vectorial subyacente es estable. La construcción consiste en establecer que el funtor moduli asociado es representado por un esquema proyectivo junto con una familia universal.

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Biografía del autor/a

Edgar I. Castañeda-González, Universidad Autónoma de Zacatecas, Unidad Académica de Matemáticas, Zacatecas, México.

I am a mathematician whose research focuses on constructing moduli spaces of augmented bundles via Geometric Invariant Theory (GIT).

Referencias

M. F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308(1983), no. 1505, 523–615. doi: 10.1098/rsta.1983.0017

N. Bao Châu, Le lemme fondamental pour les algèbres de Lie. Publ. Math. Inst. Hautes Études Sci. 111(2010), 1–169. doi: 10.1007/s10240-010-0026-7

S. B. Bradlow, Coherent Systems: a Brief Survey. Moduli spaces and vector bundles. Vol. 359. London Math. Soc. Lecture Note Ser. With an appendix by H. Lange. Cambridge Univ. Press, Cambridge, 2009, 229–264. doi: 10.1017/CBO9781139107037.009

S. B. Bradlow, O. García-Prada, An application of coherent systems to a Brill-Noether problem. J. Reine Angew. Math. 2002(2002), no. 551, 123–143. doi: 10.1515/crll.2002.079

S. B. Bradlow, O. García-Prada, P. B. Gothen, What is. . . a Higgs bundle? Notices Amer. Math. Soc. 54(2007), no. 8, 980–981.

L. Brambila-Paz, Algebras of endomorphisms of semistable vector bundles of rank 3 over a Riemann surface. J. Algebra 123(1989), no. 2, 414–425. doi: 10.1016/0021-8693(89)90054-9

E. I. Castañeda-González, The moduli space of Higgs coherent systems over a curve via GIT. PhD thesis. Centro de Investigación en Matemáticas, A.C, 2025.

O. García Hernández, Espacios moduli de haces vectoriales y parejas de tipo (n,d,k). Bachelor’s thesis. Instituto Politécnico Nacional, 2017.

U. Görtz, T. Wedhorn., Algebraic geometry I. Schemes—with examples and exercises. Second. Springer Studium Mathematik—Master. Springer Spektrum, Wiesbaden, 2020, vii+625. doi: 10.1007/978-3-658-30733-2

R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977, xvi+496.

R. Hartshorne, Deformation Theory. Vol. 257. Graduate Texts in Mathematics. Springer, New York, 2010, viii+234. doi: 10.1007/978-1-4419-1596-2

T. Hausel, Geometry of the moduli space of Higgs bundles. PhD thesis. University of Cambridge, 1998.

N. J. Hitchin, The self-duality equations on a Riemann surface. Proc. London Math. Soc. s3-55(1987), no. 1, 59–126. doi: 10.1112/plms/s3-55.1.59

D. Huybrechts, M. Lehn, The geometry of moduli spaces of sheaves. Second. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2010, xviii+325. doi: 10.1017/CBO9780511711985

D. Knutson, Algebraic spaces. Vol. 203. Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1971, vi+261. doi: 10.1007/bfb0059753

J. Le Potier, Lectures on Vector Bundles. Vol. 54. Cambridge Studies in Advanced Mathematics. Translated by A. Maciocia. Cambridge University Press, Cambridge, 1997, viii+251.

Q. Liu, Algebraic Geometry and Arithmetic Curves. Vol. 6. Oxford Graduate Texts in Mathematics. Translated from the French by Reinie Erné, Oxford Science Publications. Oxford University Press, Oxford, 2002, xvi+576. doi: 10.5860/choice.40-3456

D. Mumford, Abelian Varieties. Vol. 5. Tata Institute of Fundamental Research Studies in Mathematics. With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition. Published for the Tata Institute of Fundamental Research, Bombay; by Hindustan Book Agency, New Delhi, 2008, xii+263.

P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces. Vol. 51. Tata Institute of Fundamental Research Lectures on Mathematics and Physics. Tata Institute of Fundamental Research, Bombay; Narosa Publishing House, New Delhi, 1978, vi+183.

P. E. Newstead, Higher rank Brill-Noether theory and coherent systems open questions. Proyecciones 41(2022), no. 2, 449–480. doi: 10.22199/issn.0717- 6279-5276

N. Raghavendra, P. A. Vishwanath, Moduli of pairs and generalized theta divisors. Tohoku Math. J. 46(1994), no. 3, 321–340. doi: 10.2748 / tmj/1178225715

S. Ramanan, The moduli spaces of vector bundles over an algebraic curve. Math. Ann. 200(1973), 69–84. doi: 10.1007/BF01578292

L. P. Schaposnik, Higgs bundles—recent applications. Notices Amer. Math. Soc. 67(2020), no. 5, 625–634. doi: 10.1090/noti2074

A. H. W. Schmitt, A general notion of coherent systems. Rev. Mat. Iberoam. 38(2022), no. 6, 1783–1826. doi: 10.4171/RMI/1314

C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety. I. Inst. Hautes Études Sci. Publ. Math. 79(1994), 47–129. doi: 10.1007/BF02698887

J. Swoboda, Moduli spaces of Higgs bundles—old and new. Jahresber. Dtsch. Math.-Ver. 123(2021), no. 2, 65–130. doi: 10.1365/s13291-021-00229-1