# Talbot 2011: Non-Abelian Hodge Theory

## Mentored by Carlos Simpson

###
May 1-7, 2011

Salt Lake City, Utah

## References

Below are some preliminary references.The references are divided into sections, including the first historical section which contains basic material which should be covered as background. Then I have chosen a few directions for advanced topics, which are among the current research directions.

The list of references is currently in a very preliminary state. These are not exhaustive, indeed there are whole segments which don't appear (see for example the subjects mentioned at the end). On the other hand, it is clear that we will not be able to consider all of the references here either, as there are already too many. And I am certainly not an expert on many of these subjects. So the reference lists are meant more as an indication for places to start. If we can arrive at a good understanding of even a small number of these papers that will be great.

Google scholar and Math Sci should be used freely to go forward (find references citing a given paper) and backwards (papers refered to by a given paper).

### Quick Surveys

Here are a few resources which can provide a quick introduction to the subject.-
A fairly readable account is in the appendix (by Oscar Garcia-Prada) to the third edition of Wells'
**"Differential Analysis on Complex Manifolds"**. Note that you don't have to know much about ordinary Hodge theory to get started - the subject has quite a different flavour at first. - There is this survey by Carlos Simpson.
- Hitchin's paper The Self Duality Equations on A Riemann Surface is also a good place to start.

### Historical stuff

Some lectures are introductory on character varieties, Higgs bundles, semistability, nonabelian harmonic theory relating them, construction of moduli spaces, the hyperkahler structure and its twistor space, deformation theory, etc.- Define the affine scheme of representations, and its GIT quotient by the conjugation action. Points represent Jordan-Holder equivalence classes of representations. Deformation theory.
- Definition of Higgs bundles, interpretation as sheaves on $T^{\ast}X$, the spectral and cameral coverings. Semistability. The Dolbeault cohomology complex, and deformation theory.
- Given a representation, take an equivariant harmonic map; the Bochner identities yield a Higgs bundle. Starting with a stable Higgs bundle, Donaldson-Uhlenbeck-Yau's techniques give an Hermitian-Yang-Mills metric and, if $c_i=0$, a flat connection.
- There are different techniques for constructing the moduli space of Higgs bundles. The characteristic polynomial of the Higgs field gives Hitchin's fibration to a vector space, the hamiltonian for an integrable system. A similar construction gives the moduli space for vector bundles with integrable connection.
- The Riemann-Hilbert correspondence between vector bundles with integrable connection, and representations, may be seen as an analytic isomorphism between moduli spaces. This goes back to Serre's example for rank $1$ systems on an elliptic curve. A basic question is the asymptotic behavior of this correspondence at infinity.
- The correspondence between Higgs bundles and representations induces an homeomorphism between the moduli spaces. We get two distinct complex structures which fit into a quaternionic structure. Together with the symplectic structure, these give a hyperkahler structure on the open set of smooth points.
- Following Deligne's interpretation, the twistor space can be constructed by glueing together two copies of the moduli space of $\lambda$-connections, via the Riemann-Hilbert correspondence.
- The earliest usage of the term ``nonabelian Hodge theory'' seems to be \cite{Andersson-86}, it would be good to try to see, in retrospect, how that fits in with the theory.

- [Do-83]{Donaldson-83} S. Donaldson. A new proof of a theorem of Narasimhan and Seshadri. J. Diff. Geom. 18 (1983), 269-277.
- [Lu-Ma-85]{Lubotsky-Magid-85} A. Lubotsky, A. Magid. Varieties of representations of finitely generated groups.} {\sc Mem. Amer. Math. Soc. 58 (1985).
- [An-86]{Andersson-86} S. Andersson. Nonabelian Hodge theory via heat flow. Differential Geometry (Pe\~n\'iscola 1985)}, Lecture Notes in Math. 1209 (1986), 8-36.
- [Hi-87a]{Hitchin-87a} N. Hitchin. Stable bundles and integrable systems. Duke Math. J.} 54 (1987), 91-114.
- [Hi-87b]{Hitchin-87b} N. Hitchin. The self-duality equations on a Riemann surface. Proc. London Math. Soc. (3) 55 (1987), 59-126. Appendix by S. Donaldson.
- [Co-88]{Corlette-88} K. Corlette. Flat $G$-bundles with canonical metrics. J. Diff. Geom. 28 (1988), 361-382.
- [Si-88]{Simpson-88} C. Simpson. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc. 1 (1988), 867-918.
- [Si-92]{Simpson-92} C. Simpson. Higgs bundles and local systems. Publ. Math. I.H.E.S. 75 (1992), 5-95.
- [Gr-Sc-92]{Gromov-Schoen-92} M. Gromov, R. Schoen. Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one. I.H.E.S. Publ. Math. 76 (1992), 165-246.
- [Si-95]{Simpson-95} C. Simpson. Moduli of representations of the fundamental group of a smooth projective variety, II. I.H.E.S. Publ. Math. 80 (1995), 5-79.
- [Hi-Ka-Li-Ro-87]{Hitchin-et.al-87} N. Hitchin, A. Karlhede, U. Lindstrom, M. Ro\u{c}ek. {\em Hyperk{a}hler metrics and supersymmetry.} Comm. Math. Phys. {\bf 108 (1987), 535-559.
- [Fu-91]{Fujiki-91} A. Fujiki. Hyperkahler structure on the moduli space of flat bundles. Prospects in Complex Geometry, L.N.M. 1468 (1991), 1-83.
- [Si-97]{Simpson-97} C. Simpson. The Hodge filtration on nonabelian cohomology. Algebraic geometry---Santa Cruz 1995, , Proc. Sympos. Pure Math., 62, Part 2, A.M.S. (1997), 217-281.

### Topology of character varieties

This is a vibrant topic of current research.- One of the main goals would be to understand the important recent paper \cite{deCataldo-Hausel-Migliorini-10}.
- It would be good to start with Hitchin's original calculation for rank $2$.
- Hitchin's method has been pushed to rank $3$ in \cite{Garcia-Prada-Gothen-Munoz-08}.
- A number of works have concerned real groups, where many new phenomena occur such as the Toledo invariant, the Milnor-Woods inequality, etc.
- One major topic which fits well here is the symplectic structure on the moduli space.

- [Hi-92]{Hitchin-92} N. Hitchin. Lie groups and Teichmuller space. Topology 31 (1992), 449-473.
- [Go-To-10]{Goldman-Toledo-10} W. Goldman, D. Toledo. Affine cubic surfaces and relative $SL(2)$-character varieties of compact surfaces. Preprint \verb+arXiv:1006.3838+ (2010).
- [dC-Ha-Mi-10]{deCataldo-Hausel-Migliorini-10} M. de Cataldo, T. Hausel, L. Migliorini. Topology of Hitchin systems and Hodge theory of character varieties. Arxiv preprint arXiv:1004.1420 (2010).
- [Fo-Go-06]{Fock-Goncharov-06} V. Fock and A. Goncharov. Moduli spaces of local systems and higher Teichmuller theory. Publ. Math. I.H.E.S. 103 (2006), 1-211. math/0311149.
- [Ma-Xi-02]{Markman-Xia-02} E. Markman, E. Xia. The moduli of flat ${\rm PU}(p,p)$-structures with large Toledo invariants. Math. Z. 240 (2002), 95-109.
- [GP-Go-Mu-08]{Garcia-Prada-Gothen-Munoz-08} O. Garcia-Prada, P. Gothen, V. Mu\~noz. Betti numbers of the moduli space of rank $3$ parabolic Higgs bundles. Memoirs A.M.S. 879 (2007).
- [Si-09]{Sikora-09} A. Sikora. Character varieties. Preprint \verb+arXiv:0902.2589+ (2009).
- [Br-GP-Go-03]{Bradlow-GarciaPrada-Gothen-03} S. Bradlow, O. Garc\'ia-Prada, P. Gothen. Surface group representations and $U(p,q)$-Higgs bundles. J. Diff. Geom. 64 (2003), 111-170.

### Higher Homotopy Types

One active direction of research is how to extend Hodge theory to take into account higher homotopy types.There is the schematic homotopy type approach \cite{Katzarkov-Pantev-Toen-08}, and the nonabelian cohomology approach \cite{Simpson-02}.

Derived stacks should play an important role \cite{Kapranov-01}, \cite{Toen-Vezzosi-08}.

Closely related are topics about cohomology support loci. In this direction it would be good to look at the recent paper \cite{Lazarsfeld-Popa-09}.

For fundamental groups, there is mixed Hodge theory for the relative Malcev completion \cite{Hain-98} as well as for the representation spaces \cite{Eyssidieux-Simpson-09}.

Pridham has several original points of view.

In all of the above, one interesting object to study is the Gauss-Manin connection, which satisfies Griffiths transversality with respect to the Hodge filtration.

- [La-Po-09]{Lazarsfeld-Popa-09} R. Lazarsfeld, M. Popa. Derivative complex, BGG correspondence, and numerical inequalities for compact Kahler manifolds. Arxiv preprint 0907.0651 (2009).
- [Gi-64]{Giraud-64} J. Giraud. Non Abelian Cohomology. (1964).
- [Ka-01]{Kapranov-01} M. Kapranov. Injective resolutions of $BG$ and derived moduli spaces of local systems. J. Pure Appl. Algebra 155 (2001), 167-179.
- [Ka-Pa-To-08]{Katzarkov-Pantev-Toen-08} L. Katzarkov, T. Pantev, B. Toen. Schematic homotopy types and non-abelian Hodge theory. Compositio Math. 144 (2008), 582-632.
- [Si-02]{Simpson-02} C. Simpson. Algebraic aspects of higher nonabelian Hodge theory. In Motives, polylogarithms and Hodge theory, Part II (Irvine, CA, 1998), Int. Press Lect. Ser., 3, II, Int. Press, Somerville (2002), 417�604.
- [To-Ve-08]{Toen-Vezzosi-08} B. Toen, G. Vezzosi. Homotopical algebraic geometry II: Geometric stacks and applications. Mem. Amer. Math. Soc. 193 (2008).
- [Ey-Si-09]{Eyssidieux-Simpson-09} P. Eyssidieux, C. Simpson. Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures. Preprint \verb+arXiv:0902.2626+ (2009).
- [Go-Mi-88]{GM} W. Goldman, J. Millson, The deformation theory of representations of fundamental groups of compact Kahler manifolds, Publ. Math. IHES. 67 (1988), no. 1, 43--96.
- [Go-Mi-90]{GM2} W. Goldman, J. Millson, The homotopy invariance of the Kuranishi space. Illinois J. Math. 34 (1990), 337--367.
- [Ha-87]{Hain-87} R. Hain, The De Rham Homotopy Theory of Complex Algebraic Varieties, I, K-theory 1 (1987), 271--324.
- [Ha-98]{Hain-98} R. Hain, The Hodge De Rham theory of relative Mal\v{c}ev completion,Ann. Sci. Ec. Norm. Sup. 31 (1998), 47--92.
- [Pr-04]{Pri1} J. Pridham The deformation theory of representations of the fundamental group of a smooth variety, arxiv:math/0401344 (2004).
- [Pr-06]{Pri3} J. Pridham Non-abelian real Hodge theory for proper varieties, arxiv/math:0611683 (2006).
- [Pr-09]{Pri5} J. Pridham Formality and splitting of real non-abelian mixed Hodge structures, arxiv/math: 0902.0770 (2009).
- [Ar-97]{Arapura-97} D. Arapura. Geometry of cohomology support loci for local systems. I. J. Algebraic Geom. 6 (1997), 563-597.
- [Ar-10]{Arapura-10} D. Arapura. The Hodge theoretic fundamental group and its cohomology. In The Geometry of Algebraic Cycles: Proceedings of the Conference on Algebraic Cycles, Columbus, Ohio, 2008, Clay Mathematics Proceedings (2010).
- [Di-06]{Dimca-06} A. Dimca. Pencils of plane curves and characteristic varieties. Preprint math.AG/0606442.

### The noncompact case, Kashiwara conjecture

With work of T. Mochizuki and C. Sabbah, we now have a much better understanding of how things work over a smooth quasiprojective base variety.It will be interesting to compare with the arithmetic proof of the Kashiwara conjecture \cite{Drinfeld-01} \cite{Lafforgue-01}

The basic example of local systems on complements of a finite set of points in $\pp ^1$ has turned out to be very rich.

A major recent direction has been to look at irregular connections.

This is related to $tt^{\ast}$-geometry.

- [Mo-06]{Mochizuki-06} T. Mochizuki. Kobayashi-Hitchin correspondence for tame harmonic bundles and an application. Ast\'erisque 309 (2006).
- [Mo-07]{Mochizuki-07} T. Mochizuki. Asymptotic behaviour of tame harmonic bundles and an application to pure twistor $D$-modules, parts I and II. Memoirs of the AMS 869-870 (2007).
- [Mo-09]{Mochizuki-09} T. Mochizuki. Kobayashi-Hitchin correspondence for tame harmonic bundles, II. Geom. Topol. 13 (2009), 359-455.
- [Sa-05]{Sabbah-05} C. Sabbah. Polarizable twistor ${\mathcal D}$-modules. Ast\'erisque 300, (2005).
- [Si-90]{Simpson-90} C. Simpson. Harmonic bundles on noncompact curves. J. Amer. Math. Soc. 3 (1990), 713-770.
- [Bi-96]{Biquard-96} O. Biquard. Sur les fibr\'es paraboliques sur une surface complexe. J. London Math. Soc. 53 (1996), 302-316.
- [Jo-Zu-96]{Jost-Zuo-96} J. Jost, K. Zuo. Harmonic maps and $SL(r,\cc )$ representations of fundamental groups of quasi-projective manifolds. J. Algebraic Geom. 5 (1996), 77-106.
- [La-01]{Lafforgue-01} L. Lafforgue.Chtoucas de Drinfeld et correspondance de Langlands. Inventiones 147 (2001), 1-241.
- [Dr-01]{Drinfeld-01} V. Drinfeld. On a conjecture of Kashiwara. Math. Res. Lett. 8 (2001), 713-728.
- [CB-03]{CrawleyBoevey-03} W. Crawley-Boevey. On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero. Duke Math. J. 118 (2003), 339-352.
- [CB-04]{CrawleyBoevey-04} W. Crawley-Boevey. Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity. Publ. Math. I.H.E.S. 100 (2004), 171-207.
- [Bo-05]{Boalch-05} P. Boalch. From Klein to Painlev\'e via Fourier, Laplace and Jimbo. Proc. London Math. Soc. 90 (2005), 167-208.
- [In-Iw-Sa-06]{Inaba-Iwasaki-Saito-06} M. Inaba, K. Iwasaki, Masa-Hiko Saito. Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlev\'e equation of type VI: \newline Part I.} Publ. Res. Inst. Math. Sci. 42 (2006), 987-1089. Part II. Moduli spaces and arithmetic geometry. Adv. Stud. Pure Math. 45, Math. Soc. Japan (2006), 387--432.
- [Ka-96]{Katz-96} N. Katz. Rigid local systems. Annals of Mathematics Studies 139, Princeton University Press (1996).
- [De-Re-07]{Dettweiler-Reiter-07} M. Dettweiler, S. Reiter. Middle convolution of Fuchsian systems and the construction of rigid differential systems. J. Algebra 318 (2007), 1-24.
- [Bo-10]{Boalch-10} P. Boalch. Riemann-Hilbert for tame complex parahoric connections. Preprint \verb+arXiv:1003.3177+ (2010).
- [Hi-07]{Hien-07} M. Hien. Periods for irregular singular connections on surfaces. Periods for irregular singular connections on surfaces. Math. Ann. 337 (2007), 631-669.
- [Sa-99]{Sabbah-99} C. Sabbah. Harmonic metrics and connections with irregular singularities. Ann. Inst. Fourier 49 (1999), 1265-1291.
- [Bi-Bo]{Biquard-Boalch-} O. Biquard, P. Boalch. Wild nonabelian Hodge theory on curves. ??? math/0111098.
- [Jo-Ya-Zu-07a]{Jost-Yang-Zuo-07a} J. Jost, Y. Yang, K. Zuo. The cohomology of a variation of polarized Hodge structures over a quasi-compact Kahler manifold. J. Algebraic Geom. 16 (2007), 401-434.
- [Jo-Ya-Zu-07b]{Jost-Yang-Zuo-07b} J. Jost, Y. Yang, K. Zuo. Cohomologies of unipotent harmonic bundles over noncompact curves. J. Reine Angew. Math. 609 (2007), 137-159.
- [Ce-Va-91]{Cecotti-Vafa-91} S. Cecotti, C. Vafa. Topological--anti-topological fusion. Nuclear Phys. B 367 (1991), 359-461.
- [He-03]{Hertling-03} C. Hertling. $tt^{\ast}$ geometry, Frobenius manifolds, their connections, and the construction for singularities. J. Reine Angew. Math. 555 (2003), 77-161.

### Other Topics

Some choices had to be made. Here is a brief description of some of the topics which are left out of the above.- Factorization results and the Shafarevich conjecture
- Constructions of moduli of logarithmic connections, parabolic bundles, parabolic Higgs bundles and the like.
- The relationship between local systems with finite order monodromy, and local systems on DM-stacks.
- Compactification of the moduli space of local systems.
- There are a lot of things about rigid local systems, middle convolution etc. which are only briefly mentioned above.
- A growing subject is the theory of homotopy types of complements of hyperplane arrangements. Things like the cohomology support loci show up here (cf \cite{Dimca-06} for example).
- Reznikov's theory of Chern-Simons regulators can be extended to the quasiprojective case, this is a topic of current work of myself and Iyer.
- Donaldson's theory of $K$-stability has led to techniques for proving theorems about GIT approximation of Yang-Mills solutions, see Keller's thesis.
- There is the whole theory of nc-Hodge structures, this is related to $tt^{\ast}$-geometry.
- The work of Zuo and Viehweg applies Higgs bundles to classification of special subvarieties of Shimura varieties.
- One can consider the asymptotic behavior of the Riemann-Hilbert correspondence at infinity.
- There is a currently developing theory of ``BPS states'' and wallcrossing; possibly related to the previous item. This looks extremely interesting but I don't know of very many good references available.

### Geometric Langlands theory

While this is not a topic being covered at this Talbot, we include references here.Many workers have proposed approaches to the geometric Langlands correspondence related to nonabelian Hodge theory.

I would like to concentrate on trying to understand how the correspondence itself actually works. That might well take us into questions about irregular connections.

The main subject of many of the works is, on the other hand, about how to establish a general framework relating geometric Langlands with other subjects such as mirror symmetry.

- [Be-Dr]{Beilinson-Drinfeld} A. Beilinson, V. Drinfeld. Quantization of Hitchin's integrable system and Hecke eigensheaves.
- [Ar-01]{Arinkin-01} D. Arinkin. Orthogonality of natural sheaves on moduli stacks of $SL(2)$-bundles with connections on $\pp ^1$ minus $4$ points. Selecta Mathematica 7 (2001), 213-239.
- [Fr-Ga-Vi-01]{Frenkel-Gaitsgory-Vilonen-01} E. Frenkel, D. Gaitsgory, K. Vilonen. On the geometric Langlands conjecture. J. A.M.S. 15 (2001), 367-417.
- [Ng-10]{Ngo-10} B.-C. Ngo. Le lemme fondamental pour les alg\`ebres de Lie. Publ. Math. I.H.E.S. 111 (2010), 1-169.
- [Ka-Wi-06]{Kapustin-Witten-06} A. Kapustin, E. Witten. Electric-magnetic duality and the geometric Langlands program. Arxiv preprint hep-th/0604151 (2006).
- [Gu-Wi-06]{Gukov-Witten-06} S. Gukov and E. Witten. Gauge theory, ramification, and the geometric Langlands program. hep-th/0612073.
- [Do-Pa-06]{Donagi-Pantev-06} R. Donagi, T. Pantev. Langlands duality for Hitchin systems. Arxiv preprint math/0604617 (2006).
- [Ha-Th-03]{Hausel-Thaddeus-03} T. Hausel, M. Thaddeus. Mirror symmetry, Langlands duality, and the Hitchin system. Inventiones 153 (2003), 197-229.
- [Na-10]{Nadler-10} D. Nadler. The Geometric Nature of the Fundamental Lemma. Arxiv preprint arXiv:1009.1862 (2010).

### Crystalline and other $p$-adic versions

A number of $p$-adic nonabelian Hodge theory approaches have been proposed.- [Og-Vo]{Ogus-Vologodsky} A. Ogus, V. Vologodsky.
- [Ol]{Olsson} M. Olsson.
- [Fa]{Faltings} G. Faltings.
- [Sc]{Schepler} D. Schepler.