On Uniformization of Complex Manifolds: The Role of Connections (Mathematical Notes)

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  • Serie: Mathematical Notes.
  • On Uniformization of Complex Manifolds: The Role of Connections (MN-22).
  • On Uniformization of Complex Manifolds: The Role of Connections (MN-22)!

Higher Homotopy Types One active direction of research is how to extend Hodge theory to take into account higher homotopy types. Closely related are topics about cohomology support loci. 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. Lazarsfeld, M. Derivative complex, BGG correspondence, and numerical inequalities for compact Kahler manifolds.

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Arxiv preprint Non Abelian Cohomology. Pure Appl. Algebra , Katzarkov, T. Pantev, B. Schematic homotopy types and non-abelian Hodge theory. Compositio Math.

Algebraic aspects of higher nonabelian Hodge theory. Press Lect. Toen, G. Homotopical algebraic geometry II: Geometric stacks and applications. Eyssidieux, C. Variations of mixed Hodge structure attached to the deformation theory of a complex variation of Hodge structures. Goldman, J. Millson, The deformation theory of representations of fundamental groups of compact Kahler manifolds, Publ.

Millson, The homotopy invariance of the Kuranishi space. Illinois J. Geometry of cohomology support loci for local systems. Algebraic Geom. The Hodge theoretic fundamental group and its cohomology. Pencils of plane curves and characteristic varieties. Preprint math.

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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.

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A major recent direction has been to look at irregular connections. Kobayashi-Hitchin correspondence for tame harmonic bundles and an application. Memoirs of the AMS Kobayashi-Hitchin correspondence for tame harmonic bundles, II. Harmonic bundles on noncompact curves. Jost, K. Chtoucas de Drinfeld et correspondance de Langlands. Inventiones , On a conjecture of Kashiwara. On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero. Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity.

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Inaba, K. Iwasaki, Masa-Hiko Saito. Part II. Moduli spaces and arithmetic geometry. Japan , Rigid local systems. Dettweiler, S.

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Middle convolution of Fuchsian systems and the construction of rigid differential systems. Riemann-Hilbert for tame complex parahoric connections. Periods for irregular singular connections on surfaces. Harmonic metrics and connections with irregular singularities. Fourier 49 , Biquard, P. Wild nonabelian Hodge theory on curves. Jost, Y. Yang, K. The cohomology of a variation of polarized Hodge structures over a quasi-compact Kahler manifold.

Cohomologies of unipotent harmonic bundles over noncompact curves. Reine Angew. Cecotti, C. Topological--anti-topological fusion. Nuclear Phys. B , 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.

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Compactification of the moduli space of local systems. There are a lot of things about rigid local systems, middle convolution etc. A growing subject is the theory of homotopy types of complements of hyperplane arrangements. 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.

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. 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.

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