Tame Harmonic Bundles on Punctured Riemann Surfaces

Reisert, Pascal (31. März 2011): Tame Harmonic Bundles on Punctured Riemann Surfaces. Masterarbeit, Fakultät für Physik, Ludwig-Maximilians-Universität München.

Abstract

A punctured Riemann surface is a compact Riemann surface with finitely many points removed. We will discuss an equivalence by [Sim90] between tame harmonic bundles, regular filtered stable Higgs bundles resp. D-modules and regular filtered local systems over these surfaces.

Abstract

Eine punktierte Riemannsche Fläche ist eine kompakte Riemannsche Fläche ohne einer endlichen Anzahl ausgezeichneter Punkte. Wir zeigen eine Äquivalenz aus [Sim90] zwischen zahmen harmonischen Bündeln, regulär gefilterten Higgs Bündeln bzw. D-Modulen und reguläre gefilterten lokalen Systemen über einer punktierten Riemannschen Fläche.

• Dokumententyp: LMU München: Studienabschlussarbeit
• Keywords: Higgs bundle; D-module; parabolic bundle; Kobayashi-Hitchin correspondence; stability; Hermitian-Einstein equations
• Fakultät: Mathematik, Informatik und Statistik ; Physik
• Themengebiete: 500 Naturwissenschaften und Mathematik > 510 Mathematik ; 500 Naturwissenschaften und Mathematik > 530 Physik
• Sprache: Englisch

Literaturliste

[AM08] Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, American Mathematical Society, 2008.

[Ara12] Donu Arapura, Algebraic Geometry over the Complex Numbers, Springer, New York, 2012.

[Atk08] Richard Atkins, When is a Connection a Levi-Civita Connection (2008).

[Bas09] Stéphane Baseilhac, Chern Simon Theory in Dimension Three, Université de Grenoble, 2009.

[Bau09] Helga Baum, Eichfeldtheorie, Springer, Berlin, Heidelberg, 2009.

[Bed91] Eric Bedford, Several complex variables and complex geometry, American Mathematical Society, 1991.

[Bha06] Rajendra Bhatia, Positive Definite Matrices, Princeton University Press, 2006.

[Biq97] O. Biquard, Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse), Ann. Sci. École Norm. Sup. 30 (1997), 41-96.

[Boo75] William Boothby, An introduction to differentiable manifolds and Riemannian geometry,Academic Press, New York, 1975.

[Bou07] Bourbaki, Variétés différentielles et analytiques, Springer, Berlin, Heidelberg, 2007.

[Che46] Claude Chevalley, Theory of Lie Groups, Princeton University Press, 1946.

[CG75] Maurizio Cornalba and Phillip Griffiths, Analytic Cycles and Vector Bundles on Non- Compact Algebraic Varieties, Inventiones math. 28 (1975), 1-106.

[Cor88] Kevin Corlette, Flat G-bundles with canonical metrics, J. Differential Geom. 28 (1988), no. 3, 361-382.

[Cor93] Kevin Corlette, Nonabelian Hodge theory, Proc. Symp. Pure Math. 54 (1993), no. 2, 125-140.

[CPR93] Gustavo Corach, Horacio Porta, Lázaro Recht, The geometry of the space of selfadjoint invertible elements in a C*-algebra, Integral Equations and Operator Theory 16 (1993), no. 3, 333-359.

[Dah77] E. J. Dahlberg, On exceptional sets at the boundary for subharmonic functions, Ark. Math. 15 (1977), 305-312.

[Dar94] R.W.R. Darling, Differential Forms and Connections, Cambridge University Press, 1994.

[Del70] Pierre Deligne, Equations différentielles à Points Singuliers Réguliers, Vol. 163, 1970.

[DH87] Pierre Deligne and D. Husemöller, Survey of Drinfeld modules, Vol. 67, 1987.

[Don85] Simon K. Donaldson, Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985), no. 3, 1-26.

[Don87] Simon K. Donaldson, Infinite determinants, stable bundles, and curvature, Duke Math. J. 54 (1987), 231-247.

[Del70] Pierre Deligne, Equations différentielles à points singulieres régulieres, Lecture notes in math. 163, Springer, 1970.

[For81] Otto Forster, Lectures on Riemann Surfaces, Springer, Berlin, 1981.

[Fre07] Edward Frenkel, Langlands Correspondence for Loop Groups (2007).

[GAGA] Jean-Pierre Serre Serre, Géometrie Algébrique et Géométrie Analytique, Géometrie Algébrique et Géométrie Analytique 6 (1956), 1-42.

[Gar91] Stephan J. Gardiner, Maximum Principle for Subharmonic Functions, Math. Scand. 68 (1991), 210-220.

[GH78] Phillip Griffiths and Joseph Harris, Principles of Algebraic Geometry, Wiley- Interscience, 1978.

[GI63] O. Goldman and N. Iwahori, The space of p-adic norms, Acta Math. 109 (1963), 137-177.

[GS92] Mikhail Gromov and Richard Schön, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. Math. I.H.E.S. 76 (1992), 165- 246.

[GY93] Robert E. Greene and Shing-Tung Yau, Differential Geometry, Proceedings of Symposia in Pure Mathematics, vol. 54, American Mathematical Society, 1993.

[Ham75] Richard S. Hamilton, Harmonic maps of manifolds with boundary, Springer, New York, 1975.

[Hat02] Allen Hatcher, Algebraic topology, Cambridge University Press, 2002.

[Hat03] Alan Hatcher, Vector Bundles and K-Theory, http://www.math.cornell.edu/~hatcher/VBKT/VB.pdf, 2003.

[Hit87] Nigel J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), no. 3, 59-126.

[Huy05] Daniel Huybrechts, Complex Geometry, Springer, Berlin, Heidelberg, 2005.

[Jos05] Jürgen Jost, Riemannian Geometry and Geometric Analysis, 4th ed., Springer, Berlin, Heidelberg, 2005.

[JZ97] Jürgen Jost and Kang Zuo, Harmonic maps of infinite energy and rigidity results for representations of fundamental groups of quasiprojective varieties, J. Differential Geom. 47 (1997), no. 3, 469-503.

[JZ87] Jürgen Jost and Kang Zuo, Harmonic maps into Bruhat-Tits buildings and Factorization p-adically unbounded representations of pi_1 of algrebaic varieties I, Math. Ann. 278 (1987), 481-496.

[Kob80] Shoshichi Kobayashi, First Chern class and holomorphic tensor fields, Vol. 77, 1980.

[Kob87] Shoshichi Kobayashi, Differential Geometry of Complex Vector Bundles, Mathematical Society of Japan, 1987.

[Lee00] John M. Lee, Introduction to Topological Manifolds, Springer, New York, 2000.

[LiLo00] Elliott H. Lieb and Michael Loss, Analysis, 2nd ed., American Mathematical Society, 2000.

[Lüb82] M. Lübke, Chernklassen von Hermitian-Einstein-Vektorbündeln, Vol. 260, 1982.

[Mir95] Rick Miranda, Algebraic Curves and Riemann Surfaces, American Mathematical Society, 1995.

[Moc02a] Takuro Mochizuki, Asymptotic Behaviour of tame nilpotent Harmonic Bundles with Trivial Parabolic Structure, J. Differential Geometry 62 (2002), 351-559.

[Moc02b] Takuro Mochizuki, Asymptotic behaviour of tame nilpotent harmonic bundles with trivial parabolic structure, arXiv:math/0212232v1 (2002).

[Moc06a] Takuro Mochizuki, Kobayashi-Hitchin Correspondence For Tame Harmonic Bundles and an Application, arXiv:math/0411300v3 (2006).

[Moc06b] Takuro Mochizuki, Kobayashi-Hitchin correspondence for tame harmonic bundles II, arXiv:math/0602266 (2006).

[Moc07a] Takuro Mochizuki, Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor D-modules, Vol. 1, American Mathematical Society, 2007.

[Moc07b] Takuro Mochizuki, Asymptotic Behaviour of Tame Harmonic Bundles and an Application to Pure Twistor D-modules, Vol. 2, American Mathematical Society, 2007.

[Moc10] Takuro Mochizuki, Wild Harmonic Bundles and Wild Pure Twistor D-Modules, arXiv:0803.1344v4 (2010).

[MR84] V. B. Mehta and A. Ramanathan, Restriction of stable sheaves and representations of the fundamental group, Invent. Math. 77 (1984), 163-172.

[MS80] V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), no. 3, 205-239.

[FA08] Peter Müller, Functional Analysis I (Lecture), Ludwig-Maximilians University, Munich, 08/09.

[FA09] Peter Müller, Functional Analysis II (Lecture), Ludwig-Maximilians University, Munich, 2009.

[PDE09] Peter Müller, Partial Differential Equations I (Lecture), Ludwig-Maximilians University, Munich, 09/10.

[Mun04] Gheorghe Munteanu, Complex spaces in Finsler, Lagrange, and Hamilton geometries, Kluwer Academic Publishers, 2004.

[NS65] M. S. Narasimhan and C. S. Seshadri, Stable and unitary bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-564.

[Rei09] Pascal Reisert, Explizite Darstellung der multiplikativen Verknüpfung in den Verlinde Algebren über dem Darstellungsring von SU(2), Bachelor Thesis, Ludwig-Maximilians University, 2009.

[Rei10] Pascal Reisert, Flat Principal G-Bundles and Local Systems, Theoreticum, Ludwig-Maximilians University, 2010.

[Sab05] Claude Sabbah, Polarizable twistor D-modules, math.AG/0503038 (2005).

[Sab07] Claude Sabbah, Isomonodromic Deformations and Frobenius Manifolds, Springer, Berlin, 2007.

[Sch05] Pierre Schapira, An introduction to algebraic topology, Course at Paris VI University, 2005/2006. [Sch06] Pierre Schapira, Categories, sites, sheaves and stacks, Course at Paris VI University, 2006.

[ScSc95] Peter Scheinost and Martin Schottenloher, Metaplectic quantization of the moduli spaces of flat and parabolic bundles, Vol. 466, 1995.

[Sch73] W. Schmid, Variation of Hodge structure: the singularities of the period mapping, Invent. Math. 22 (1973), 211-319.

[Sim88] Carlos T. Simpson, Constructing Variations of Hodge Structure using Yang-Mills Theory and Applications to Uniformization, Journal of the American Mathematical Society 1 (1988), no. 4.

[Sim90] Carlos T. Simpson, Harmonic Bundles on Noncompact Curves, Journal of the American Mathematical Society 3 (1990), no. 3.

[Sim92] Carlos T. Simpson, Higgs bundles and local systems, Publications Mathématiques de l'I.H.É.S. 75 (1992), 5-95.

[Sim94a] Carlos T. Simpson, Moduli of representations of the fundamental group of a smooth projective group variety I, Publ. Math. l'I.H.É.S. 79 (1994), 47-129.

[Sim94b] Carlos T. Simpson, Moduli of representations of the fundamental group of a smooth projective group variety II, Publ. Math. l'I.H.É.S. 80 (1994), 5-79.

[Sim96] Carlos T. Simpson, The Hodge filtration on nonabelian cohomology, arXiv:alg-geom/9604005v1 (1996).

[Sim97] Carlos T. Simpson, Mixed twistor structures, arXiv:alg-geom/9705006v1 (1997).

[Ste91] Willi-Hans Steeb, Kronecker product of matrices and applications, BI-Wiss.Verlag, 1991.

[Sza09] Tamás Szamuely, Galois Groups and Fundamental Groups, Cambridge University Press, 2009.

[UY86] Karen K. Uhlenbeck and Shing-Tung Yau, On the existence of Hermitian- Yang-Mills connections in stable vector bundles, Comm. Pure and Appl. Math 39-S (1986), 257-293.

[Zuo99] Kang Zuo, Representations of the Fundamental Group of Algebraic Varieties, Springer, Berlin, 1999.