# Hodge Theory, Complex Geometry, and Representation Theory

Share this page *Edited by *
*Robert S. Doran; Greg Friedman; Scott Nollet*

This volume contains the proceedings of an NSF/Conference Board of the
Mathematical Sciences (CBMS) regional conference on Hodge theory,
complex geometry, and representation theory, held on June 18, 2012, at
the Texas Christian University in Fort Worth, TX. Phillip Griffiths,
of the Institute for Advanced Study, gave 10 lectures describing
now-classical work concerning how the structure of Shimura varieties
as quotients of Mumford-Tate domains by arithmetic groups had been
used to understand the relationship between Galois representations and
automorphic forms. He then discussed recent breakthroughs of Carayol
that provide the possibility of extending these results beyond the
classical case. His lectures will appear as an independent volume in
the CBMS series published by the AMS.

This volume, which is dedicated to Phillip Griffiths, contains
carefully written expository and research articles. Expository papers
include discussions of Noether-Lefschetz theory, algebraicity of Hodge
loci, and the representation theory of \(SL_{2}(\mathbb{R})\).
Research articles concern the Hodge conjecture, Harish-Chandra modules, mirror
symmetry, Hodge representations of \(Q\)-algebraic groups, and
compactifications, distributions, and quotients of period domains. It
is expected that the book will be of interest primarily to research
mathematicians, physicists, and upper-level graduate students.

# Table of Contents

## Hodge Theory, Complex Geometry, and Representation Theory

- Preface ix10 free
- The smooth center of the cohomology of a singular variety 118 free
- 1. Definition and properties of the smooth center of the cohomology 219
- 2. 𝐻_{𝑠𝑚}¹(𝐶) of a singular curve 320
- 3. A simple strange surface 623
- 4. Revisiting the example of Barbieri-Viale and Srinivas 825
- 5. Examples of varieties with normal crossings 1027
- 6. More on varieties with normal crossings 1431
- References 1835

- Developments in Noether-Lefschetz theory 2138
- Compact quotients of non-classical domains are not Kähler 5168
- Algebraicity of Hodge loci for variations of Hodge structure 5976
- On the differential equations satisfied by certain Harish-Chandra modules 85102
- 1. Sectionformat {Introduction}{1} 85102
- 2. Sectionformat {Notations and preliminaries}{1} 92109
- 3. Sectionformat {The three filtrations}{1} 100117
- 4. Sectionformat {The spectral sequences associated to the filtrations $F^�ullet _a$ and $protect lprime F^�ullet _a$}{1} 101118
- 5. Sectionformat {The spectral sequence associated to the filtration $F^�ullet _b$}{1} 104121
- 6. Sectionformat {The spectral sequence associated to the filtration $F^�ullet _c$}{1} 108125
- 7. Sectionformat {Involutivity and the characteristic module of the PDE associated to $widehat V^mu $}{1} 114131
- 8. Sectionformat {The characteristic variety $Xi _A$ and characteristic sheaf $mathscr {M}_{mu ,A}$\ associated to~$V^mu $}{1} 120137
- Appendix A. Sectionformat {The symbol spectral sequence and higher characteristic varieties}{1} 129146
- Hy @SectionAnchorHref {conm12177.section*.161}Sectionformat {References}{@m } 139156

- Kato-Usui partial compactifications over the toroidal compactifications of Siegel spaces 143160
- On the equivalence problem for bracket-generating distributions 157174
- Notes on the representation theory of 𝑆𝐿₂(ℝ) 173190
- Introduction 173190
- 1. Irreducible representations of 𝑠𝑙₂ 174191
- 2. Parabolic induction 176193
- 3. Eisenstein series 181198
- 4. Modular forms 182199
- 5. Cuspidal automorphic forms 185202
- 6. Cohomology 188205
- 7. Appendix I: 𝐿²(𝐺) and the {𝐷_{𝑚}^{±}} 191208
- 8. Appendix II: Poincaré series 194211
- References 198215

- Cup products in automorphic cohomology: The case of 𝑆𝑝₄ 199216
- Hodge type conjectures and the Bloch-Kato theorem 235252
- 1. Introduction 235252
- 2. Notation 237254
- 3. Cycle class maps 238255
- 4. Key example I 240257
- 5. Beilinson-Hodge conjectures 241258
- 6. Galois actions 242259
- 7. Integral formulations and the Milnor regulator 249266
- 8. Rigidity revisited 253270
- 9. Bloch-Kato theorem and its consequences 254271
- References 256273

- Principal Hodge representations 259276
- A study of mirror symmetry through log mixed Hodge theory 285302