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Higgs Bundles and Hermitian Symmetric Spaces

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Higgs Bundles and Hermitian Symmetric Spaces Higgs bundles and Hermitian symmetric spaces Roberto Rubio N´u~nez ICMAT CSIC-UAM-UC3M-UCM Tesis presentada para la obtenci´ondel t´ıtulode Doctor en Matem´aticas por la Universidad Aut´onomade Madrid. Director: Oscar´ Garc´ıa-Prada. 24 de febrero de 2012 2 A mis padres y a mi hermano 4 Contents Acknowledgementsi List of Symbols iii Introducci´on:resumen y conclusiones vii 1 Introduction: summary and main results1 2 Hermitian symmetric spaces 11 2.1 Basics on symmetric spaces and root theory.............. 11 2.2 Cayley transform............................. 19 2.3 Restricted root theory.......................... 24 2.3.1 Classical results.......................... 24 2.3.2 More results on restricted roots................. 26 2.3.3 Sums of roots........................... 28 2.4 The Toledo character........................... 30 2.4.1 Definition of the Toledo character................ 30 2.4.2 Jordan algebra structure..................... 34 2.5 Parabolic subgroups........................... 40 2.5.1 Basics on R-parabolic subgroups................. 40 2.5.2 Relevant parabolic subgroups.................. 42 2.5.3 Subtubes and Levi factors.................... 48 2.6 Normalizer of the maximal tube subdomain.............. 51 3 Higgs bundles 56 3.1 Basics on twisted Higgs bundles and stability.............. 56 3.2 HK correspondence and moduli spaces................. 59 3.3 G-Higgs bundles when G is of Hermitian type............. 60 3.4 α-Milnor-Wood inequality........................ 62 3.5 The Toledo invariant and the topological class............. 66 4 Maximal Toledo invariant 68 4.1 Milnor-Wood inequality......................... 68 4.2 Tube-type groups and Cayley correspondence............. 70 4.3 Non-tube groups and stabilization of a maximal tube......... 77 5 Surface group representations 83 5.1 Correspondence with Higgs bundles................... 83 5.2 Equivalent definitions of the Toledo invariant.............. 87 5.3 Milnor-Wood inequality and rigidity of maximal representations... 89 5.4 Future directions............................. 91 A Examples 93 B Additional remarks 104 C Tables 107 Bibliography 112 Acknowledgements This thesis would not have been possible without the aid and support of countless people and many institutions during the past seven years. I must first express my gratitude towards my supervisor Oscar´ Garc´ıa-Pradafor suggesting me this problem and for his support, guidance and sponsoring during all this time. I would also like to express my deepest gratitude to Olivier Biquard, for hosting me during several visits to the University of Paris VI, and for his time and his commitment with the research project. He has indeed acted as an unofficial co-supervisor. I would also like to thank the people and institutions that have hosted other valu- able research stays: Ignasi Mundet at the University of Barcelona, Peter Gothen at the University of Porto and, in particular, Nigel Hitchin at the University of Oxford. I would wish to convey thanks to the institutions that have hosted this project, the Instituto de Ciencias Matem´aticasICMAT CSIC-UAM-UC3M-UCM and the Univer- sity Aut´onoma of Madrid, as well as to their members and staff. Many other mathematicians have helped me with useful conversations, as Luis Alvarez-C´onsul,Steve´ Bradlow, Marco Castrill´on,Vicente M´u~noz,and in particu- lar Tom´asG´omez. I wish to give my thanks to some fellow graduate students and postdocs for their (not only mathematical) support and friendship. My gratitude to Paloma Bengoechea, Ang´elicaBenito, Javier Fres´an,Luis Emilio Garc´ıa, Car- los Gonz´alez-Guill´en,Luis Hern´andez-Corbato,Jes´usP. Moreno, Jos´eand Alberto Navarro, Mar´ıaPe, Ana Primo, Steve Rayan, Juanjo Ru´e,Laura Shaposnik, Car- los Vinuesa, Alfonso Zamora, especially to the 1001 people: Marta Aparicio, Emilio Franco, Fernando Jim´enez,Marina Logares, C´edricMart´ınez-Campos, Ana Pe´on,and very especially to Alvaro´ Ant´onand Mario Garc´ıa-Fern´andez. Tracking back in time, I do not forget my years as an undergraduate in the Uni- versity of Valencia. Thanks to professors Juan Climent, Rafael Crespo, Olga Gil, Antonio Mart´ınez-Naveira, Pepe Mart´ınez-Verduch, Gabriel Navarro, and in partic- ular, to Rafael Sivera and Francisca Mascar´o. Thanks also to my teachers Carmen i Ballester and Angel´ Solera. With no doubt, a person who deserves a special mention in this paragraph is Antonio Ledesma, for his encouragement. I could not have keep on working without the support of my family and many friends. I am deeply and forever indebted to my parents and my brother for their love and support throughout my entire life. I would like to thank all the people with whom I lived during four years in the Residencia de Estudiantes of Madrid, both scholars and staff. I still feel it as my second home. Allow me to make a mention of a few: Alberto B., Andr´es,Diego, Iv´an,Jos´e,Jos´eLuis, Juan M., the maintenance people, Marcelo, Miguel, Nere, Nerea, Pablo, Pamela, Quela, Ra´ul,Sergio, Tagore, and especially Abraham, Joserra, Juanjo, Mar´ıaA. and Abelardo, Marian, Nacho, Pedro, Rafa J., To~no,V´ıctorand Teresa. Thanks also to my classmates in Valencia: Anabel, Bea, Diego, Dioni, Elena, Isa, Jannet, Joserra, Juanje, Mario, Marta, Pablo, Pascual, Paula, Pili, Ra´ul,Rafa, Rebeca, Rosaura and Su´arez. When I was out of Valencia, Madrid or Paris, two people made me feel as if I were at home: Cristo in Barcelona and Juan Carlos in Oxford, thank you. This long list would not be complete though, without Agata,´ Arnau, Judith, Manu, Patricio, Pili, V´ıctor,Ximo, the people from the Colegio de Espa~nain Paris, the people from the Spanish Mathematical Olympiad, and my friends from Titaguas, especially Alberto and Sara. I apologize if I have missed anyone. In that case, let me know and I will put you in the next thesis. I wish to acknowledge financial help from a predoctoral I3P-JAE grant from CSIC from 2006 to 2010, a scholarship of the Ayuntamiento de Madrid in the Residencia de Estudiantes of Madrid from 2006 to 2011, and the research projects Moduli spaces and geometric structures (MEC, MTM2007-67623), Topology and Hodge-theory of Higgs bundles moduli space (CSIC and FCT, 2007PT0011), Geometry of moduli spaces (CSIC and CAM, 200550M170) and Interactions of Low-Dimensional Topology and Geometry with Mathematical Physics (European Science Foundation). ii List of Symbols a Maximal subalgebra contained in m, hence abelian . 16 a± Image of a by '±, subalgebra of m± ...............................18 1 1 a Multiplicity of the roots ± 2 γj ± 2 γk (j 6= k)inΣ ................. 25 B Killing form . 12 Bθ Positive definite quadratic form . 12 Bτ Hermitian form defined y Bτ (X; Y ) = B(X; τY ).................. 22 1 b Multiplicity of the roots ± 2 γj inΣ ............................... 25 β Component of the Higgs field . 60 + C0 Subset of roots fα 2 ∆C j π(α) = 0g .............................. 24 + 1 Ci Subset of roots fα 2 ∆C j π(α) = − 2 γig .......................... 24 + 1 Cij Subset of roots fα 2 ∆C j π(α) = 2 (γj − γi)g ..................... 24 C∗∗ Subset of compact restricted roots . 44 C C CG Subgroup of H defined by CHC (gT ).............................. 52 0 CG Semisimple part of CG ............................................ 52 C C C Centralizer C of HT in H ........................................51 π C c Cayley transform, c = exp 4 iyΓ 2 U ⊂ G ...................... 19 C 2 P χ Toledo character in h , given by + α ...................... 31 T N α2∆Q P χΓ0 Character defined by γ2Γ0 γ ..................................... 45 0 χ Character defined by χT − χΓ0 .................................... 45 ∗ ∗ χ Character of HT describing semi-invariance of det . 36 cD Cayley transform of the bounded domain D ⊂ m+ ................ 23 D Realization of M as a bounded domain in m+ ..................... 15 d Toledo invariant . 61 ∆C Set of compact roots . 17 ∆Q Set of non-compact roots . 17 ± ∆Q Set of positive (negative) non-compact roots . 17 ± ∆C Set of positive (negative) compact roots . .17 det Determinant of the Jordan algebra . 36 iii Det Usual determinant of the linear group GL(V )..................... 36 C eα Generator of gα such that τeα = e−α and [eα; e−α] = hα ........... 17 C e−α Generator of g−α such that τeα = e−α and [eα; e−α] = hα .......... 17 eΓ Sum of eγ for γ 2 Γ...............................................19 GC Simply connected group with Lie algebra gC ...................... 15 G0 Subgroup of GC corresponding to the Lie subalgebra g ⊂ gC ...... 15 Ge Universal cover of the Lie group G ................................ 13 GT subgroup of G corresponding to the subalgebra gT ⊂ g ............ 20 0 GLs Real Lie group defined by (Ls \ H) exp(ms \ m).................. 58 C C C gα Root space corresponding to α 2 ∆(g ; t )........................ 17 gC[T ] Subalgebra of gC given by a set of roots T ........................ 44 gλ Root space corresponding to λ 2 Σ(g; a).......................... 16 g∗ Dual of the Lie algebra g ......................................... 12 C C C C gΓ0 Subalgebra of g given by hΓ0 + mΓ0 ...............................48 C gΓ0 Subalgebra of g given by g \ gΓ0 .................................. 48 γ Component of the Higgs field . 60 Γ System of st-orthogonal roots . 18 Γ0 Subset of Γ . 43 0 0 ΓC Finite subgroup of CG defined by CG \ Z(CG).................... 53 C C ΓH Subgroup of HT defined by HT \ Z(CG).......................... 53 H0 Subgroup of GC corresponding to the Lie subalgebra h ⊂ gC ...... 15 H0;C Subgroup of GC corresponding to the Lie subalgebra hC ⊂ gC ..... 15 HT subgroup of H corresponxding to the subalgebra hT ⊂ h .......... 20 0 H Isotropy subgroup of ieΓ in H .....................................21 0 H0 Identity component of H ..........................................21 He Universal cover of the Lie group H ................................13 h Lie algebra of H .................................................. 11 C hα Element of t satisfying α(hα)=2 ...............................
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