Geometric invariant theory and nonabelian Hodge correspondence

• Dates:

1. Lecture          Tuesdays      (every week)                                                                           9:15-10:45

2. Lecture          Thursdays   (even week, beginning from 2rd week)                 9:15-10:45

3. Exercise        Thursdays   (odd week, beginning from 3rd week)                   9:15-10:45

      Please register the course via Muesli.

• Place:

INF 205 / SR 4.

• Description of the course

1. Who are suitable for this course?

This course is suitable for senior undergraduate students, master students, and young Ph.D students who have a basic knowledge on complex geometry and algebraic geometry. 

2. What is nonabelian Hodge?

Nonabelian Hodge theory is a theory aimed on studying the interrelationship between Higgs bundles and local systems. More precisely,  over a base compact K\"ahler manifold, in the categorical point of view, we have a correspondence between the (dg-)category of polystable Higgs bundles of fixed rank with vanishing Chern classes and the (dg-)category of semisimple local systems of the same rank; in the moduli point of view, the moduli space of semistable Higgs bundles of fixed rank with vanishing Chern classes and the moduli space of all local systems of the same rank are homeomorphic as topological spaces. But as algebraic varieties, they are very different, the first one is a singular quasi-projective variety, while the latter one is an affine variety, and is complex analytic isomorphic to a quasi-projective variety that parametrizes the equivalence classes of flat bundles of the same rank by Riemann--Hilbert correspondence. These three moduli spaces are called Dolbeault, Betti, and de Rham, respectively.

In 1965, Narasimhan and Seshadri showed a celebrating result: over a compact Riemann surface, every stable vector bundle is exactly comes from an irreducible projectively unitary representation of the fundamental group. And in particular, every stable vector bundle of degree 0 is exactly comes from an irreducible unitary representation of the fundamental group. In 1983, Donaldson found this correspondence can be interpreted as the existence of Hermitian-Einstein metric on a stable vector bundle, this metric gives rise to a projectively flat unitary Chern connection, and in particular, if the vector bundle is of zero degree, then the unitary Chern connection is flat. The generalization to higher dimensional projective varieties (or more general, to compact K\"ahler manifolds) also make sense, this was achieved by Donaldson for projective manifolds, and Uhlenbeck-Yau for general compact K\"ahler manifolds. But we should notice that to obtain flat unitary connections, one should impose the condition of vanishing Chern classes on the vector bundle.

The generalization of the Narasimhan--Seshadri correspondence to the topological side of all representations of the fundamental group is not a trivial thing, and in fact it is very hard to go through. The corresponding holomorphic side of stable vector bundles should involve more data, in Riemann surface case, stable vector bundles are replaced by stable pairs such that each pair consists of a holomorphic vector bundle and a holomorphic section of the endomorphism bundle twisted by the holomorphic cotangent bundle of the Riemann surface. In higher dimensional case, an extra integrable condition should be imposed for these holomorphic sections. Such pairs are called ''Higgs bundles'', and such generalized correspondence is what we describe in the first paragraph, and is usually referred as the ''Corlette--Simpson correspondence''.

The bridge between stable vector bundle and irreducible projectively unitary representations, and the bridge between stable Higgs bundles with vanishing Chern classes and irreducible representations, are built  by the existence of good metrics (Hermitian-Einstein metrics and  pluri-harmonic metrics,  respectively), which are achieved by solving highly fully nonlinear partial differential equations with the method of heat flow, or the method of continuity.

Categorical correspondence only gives us the isomorphism between sets, which to some extend, is not of the most interesting. We expect, first of all, a variety instead of a set that can parametrize these objects such that the closed points are exactly the equivalent classes of these objects, this variety is called ''moduli space''. Then we expect the correspondence gives rise to an ''identification''  of the corresponding moduli spaces.

 The categorical correspondence and the moduli correspondence mentioned above are called the ''nonabelian Hodge correspondence''.

In this course, I hope to describe the theory from these two aspects. More explicitly, I will divide it into the following parts:

1. Overview of the course (historical description); 1 course

1. A crash course to complex/Kaehler geometry; 2 courses, at most 3

2. Introduction to geometric invariant theory; 2 courses, at most 3

3. Representations and its moduli space; 2 course

4. Flat bundles and the Riemann--Hilbert correspondence; 2 courses, at most 3

5. Higgs bundles and (pluri-)harmonic metrics; 3 courses

6. Nonabelian Hodge correspondence and possible generalizations; 2 courses, at most 3

• Lecture notes and exercise sheets

• Notes: write me an email if you need.

• Exercise sheets: 

1 (please send the answer sheet to me before 05/05/2022)

2 (please send the answer sheet to me before 12/05/2022)

3 (please send the answer sheet to me before 24/05/2022)

4 (please send the answer sheet to me before 02/06/2022)

5 (please send the answer sheet to me before 28/06/2022)

6 (please send the answer sheet to me before 28/06/2022)

• Exam

For students who want the final ''Leistungspunkte'': you need to finish at least 50% of the exercise sheets to participate the final oral exam, they will be partially graded. 

I encourage everyone to present solutions to 2 questions on the exercise courses, these 2 problem should not be on the same exercise sheet.  

For detailed description, see the course information of the lecture note 1.

• References

• Preliminaries on complex/Kaehler geometry:

1. D. Huybrechts, Complex geometry : an introduction, Universitext, Springer-Verlag Berlin Heidelberg, 2005.

2. R. O. Wells, Differential analysis on complex manifolds, GTM 65, Springer-Verlag, New York, 2008.

3. C. Voisin, Hodge theory and complex algebraic geometry I, Cambridge studies in advanced mathematics 76, Cambridge University Press, 2002.

• References on geometric invariant theory:

1. D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory, Springer-Verlag Berlin Heidelberg, 1994.

2. A. H.W. Schmitt, Geometric invariant theory and decorated princpal bundles, Zurich Lectures on Advanced Matheamtics, EMS, 2008.

3. P. E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute lectures, 1978.

• References on nonabelian Hodge theory:

  1. S. K. Donaldson, Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55, 127-131 (1987).

  2. K. Corlette, Flat G-bundles with canonical metrics, Jour. Diff. Geom. 28, 361-382 (1988).

  3. N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. 1, 59-126 (1987).

  4. C. T. Simpson, Constructing of variations of Hodge structure using Yang-Mills theory and applications to uniformization, Jour. Amer. Math. Soc. 1, 867-918 (1987).

  5. C. T. Simpson, Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75, 5-95 (1992). 

  6. C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety I, Inst. Hautes Études Sci. Publ. Math. 79, 47-129 (1994). 

  7. C. T. Simpson, Moduli of representations of the fundamental group of a smooth projective variety II, Inst. Hautes Études Sci. Publ. Math. 80, 5-79 (1994).