Archivos suplementarios
Palabras clave
moduli space
Hitchin map
first fundamental theorem of invariant theory
sistema coherente
espacio de módulos
morfismo de Hitchin
primer teorema fundamental de la teoría de invariantes
Cómo citar
Resumen
En estas notas se presenta un nuevo enfoque para el estudio de las condiciones de semi-estabilidad, así como de los espacios de modulos, de los sistemas coherentes asociados a fibrados vectoriales con estructura adicional. Bajo este enfoque, se abre la posibilidad de definir un morfismo de Hitchin. Se muestra, además, la relación entre algunos ejemplos concretos con problemas clásicos presentes en la teoría geométrica de invariantes.
Citas
M. Bader, Quivers, geometric invariant theory, and moduli of linear dynamical systems, Linear Algebra Appl. 428(2008), no. 11-12, 2405–3034. Doi: 10.1016/j.laa.2007.11.027
M. Bhargava, W. Ho, A. Kumar, Orbit parametrizations for K3 surfaces, Forum Math. Sigma 4(2016), no. e18. Doi: 10.1017/fms.2016.12
S.B. Bradlow, Special metrics and stability for holomorphic bundles with global sections, J. Differential Geom. 33(1991), no. 1, 169–213. Doi: 10.4310/jdg/1214446034
S.B. Bradlow, O. García-Prada, An application of coherent systems to a Brill-Noether problem, J. reine angew. Math. 551(2002), 123–143. Doi: 10.1515/crll.2002.079
I. Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Note Series, vol. 296, Cambridge University Press, Cambridge, 2003. Doi: 10.1017/CBO9780511615436
T.L. Gómez, A. Langer, A.H.W. Schmitt, I. Sols, Moduli spaces for principal bundles in large characteristic, Teichmüller theory and moduli problem, 281–371, Ramanujan Math. Soc. Lect. Notes Ser., vol. 10, Ramanujan Math. Soc., Mysore, 2010. Doi: 10.1016/j.aim.2008.05.015
E. González, P. Solis, Ch.T. Woodward, Stable gauged maps, in: Algebraic geometry (Salt Lake City 2015), 243–275,
Proc. Sympos. Pure Math., vol. 97.1, Amer. Math. Soc., Providence RI, 2018. https://arxiv.org/abs/1606.01384
R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York
Heidelberg, 1977. Doi: 10.1007/978-1-4757-3849-0
M. Hazewinkel, (Fine) moduli (spaces) for linear systems: what are they and what are they good for?, in: C.I.
Byrnes C.F. Martin (Eds.) Geometrical Methods for the Study of Linear Systems, Nato Advanced Study Institutes
Series (Series C - Mathematical and Physical Sciences), vol 62. Springer, Dordrecht, 1980. Doi: 10.1007/978-94-009
-1_3
D. Hilbert, Ueber die Theorie der algebraischen Formen, Math. Ann. 36(1890), 473–534. Doi: 10.1007/BF01208503
D. Hilbert, Ueber die vollen Invariantensysteme, Math. Ann. 42(1893), no. 3, 313–373. Doi: 10.1007/BF01444162
D. Hinrichsen, D. Prätzel-Wolters, A wild quiver in linear systems theory, Linear Algebra Appl. 91(1987), 143–175.
Doi: 10.1016/0024-3795(87)90068-1
N.J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. s3-55(1987), 59–126. Doi: 10.1112/plms/s3-55.1.59
G.H. Hitching, M. Hoff, P.E. Newstead, Nonemptiness and smoothness of twisted Brill–Noether loci, Ann. Mat. Pura Appl. 4(2020), 25 pp. Doi: 10.1007/s10231-020-01009-x
A.D. King, P.E. Newstead, Moduli of Brill-Noether pairs on algebraic curves, Internat. J. Math. 6(1995), no. 5, 733 748. Doi: 10.1142/S0129167X95000316
H. Lange, Higher secant varieties of curves and the theorem of Nagata on ruled surfaces, Manuscripta Math. 47(1984), no. 1-3, 263–269. Doi: 10.1007/BF01174597
H. Lange, M.S. Narasimhan, Maximal subbundles of rank two vector bundles on curves, Math. Ann. 266(1983), no.
, 55–72. Doi: 10.1007/BF01458704
H. Lange, P.E. Newstead, Coherent systems of genus 0, Internat. J. Math.15(2004), no. 4, 409–424. Doi: 10.1142/S0129167X04002326
J. Le Potier, Systèmes Cohérents et Structures de Niveau, Astérisque, vol. 214, 1993. http://www.numdam.org/item/AST_1993__214__1_0/
M. Lübke, A. Teleman, The universal Kobayashi–Hitchin correspondence on Hermitian manifolds, Mem. Amer. Math. Soc., vol. 183, 2006. Doi:10.1090/memo/0863
I. Mundet i Riera, A Hitchin–Kobayashi correspondence for Kähler fibrations, J. reine angew. Math. 528(2000), 41 80. https://arxiv.org/abs/math/9901076
D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, 3rd edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, Springer-Verlag, Berlin, 1994.
Ch. Okonek, A. Schmitt, A. Teleman, Master spaces for stable pairs, Topology 38(1999), no. 1, 117–139. Doi: 10.1016/S0040-9383(98)00006-8
S.Y. Oudot, Persistence Theory: From Quiver Representations to Data Analysis, Mathematical Surveys and Monographs, vol. 209, American Mathematical Society, Providence RI, 2015.
V.L. Popov, E.B. Vinberg, Invariant theory, in: Algebraic Geometry 4, Encycl. Math. Sci., vol. 55, 1994; translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya, vol. 55, 1989, pp. 137–309.
N. Raghavendra, P.A. Vishwanath, Moduli of pairs and generalized theta divisors, Tohoku Math. J. (2) 46(1994), no. 3, 321–340. Doi: 10.2748/TMJ/1178225715
Z. Reichstein, Stability and equivariant maps, Invent. Math. 96(1989), 349–383. Doi: 10.1007/BF01393967
A.H.W. Schmitt, Projective moduli for Hitchin pairs, Internat. J. Math. 9(1998), no. 1, 107–118; erratum, Internat. J. Math. 11(2000), no. 4, 589. Doi: 10.1142/S0129167X98000075
A.H.W. Schmitt, Moduli problems of sheaves associated with oriented trees, Algebr. Represent. Theory 6(2003), no. 1, 1–32. Doi: 10.1023/A:1022322529046
A.H.W. Schmitt, A universal construction for moduli spaces of decorated vector bundles over curves, Transform. Groups 9(2004), no. 2, 167–209. Doi: 10.1007/s00031-004-7010-6
A.H.W. Schmitt, Global boundedness for decorated sheaves, Int. Math. Res. Not. 68(2004), 3637–3671. Doi:
/S1073792804141652
A.H.W. Schmitt, Geometric Invariant Theory and Decorated Principal Bundles, Zurich Lectures in Advanced
Mathematics, vol. 11, European Mathematical Society, Zürich, 2008.
A.H.W. Schmitt, Semistability and instability in products and applications, in: V. Bouchard, C. Doran, S. Méndez Diez & C. Quigley (Eds.) StringMath 2014, Proc. Sympos. Pure Math., vol. 93, Amer. Math. Soc., Providence RI, 2016, pp. 201-214.
A.H.W. Schmitt, A general notion of coherent systems. https://hal.archives-ouvertes.fr/hal-02391836/document
M. Schottenloher, Geometrie und Symmetrie in der Physik, Vieweg+Teubner Verlag, Wiesbaden, 1995. Doi: 10.1007/978-3-322-89928-6
G.W. Schwarz, On classical invariant theory and binary cubics, Ann. Inst. Fourier (Grenoble), 37(1987), no. 3, 191 216. Doi: 10.5802/aif.1104
B. Sturmfels, Algorithms in Invariant Theory, 2nd edition, Texts and Monographs in Symbolic Computation, Springer, Vienna, 2008. Doi: 10.1007/978-3-211-77417-5
M. Teixidor i Bigas, L.W. Tu, Theta divisors for vector bundles, in: R. Donagi (Ed.) Curves, Jacobians, and Abelian
Varieties, Contemporary Mathematics, vol. 136, Amer. Math. Soc., Providence RI, 1992, pp. 327–342. https://bookstore.ams.org/conm-136
M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117(1994), no. 2, 317–353. Doi: 1007/BF01232244