Introduction to Nonabelian Hodge Theory Flat connections, Higgs bundles and complex variations of Hodge structure Alberto Garc´ıa-Raboso and Steven Rayan The correspondence between Higgs bundles and local systems can be viewed as a Hodge theorem for non-abelian cohomology. | Carlos T. Simpson [67] 1 Introduction . 1 2 Zoology of connections . 5 2.1 Differentials . 5 2.2 Connections . 7 3 The correspondence between Higgs bundles and local systems . 13 4 A sketch of proof . 14 4.1 Hermitian metrics . 15 4.2 Harmonic bundles . 17 4.3 From flat bundles to harmonic bundles . 20 4.4 From Higgs bundles to harmonic bundles . 21 4.5 Extensions . 22 5 The hyperk¨ahlermanifold . 23 6 Properties of the moduli space of Higgs bundles on a curve . 24 6.1 The first example: the cotangent bundle of a Jacobian . 25 6.2 The Hitchin map, spectral curves, and other features of X (r; d) . 26 M 7 The circle action and complex variations of Hodge structure . 30 8 Example calculations . 34 8.1 y(1; d) .................................................... 34 P 8.2 y(2; 1) .................................................... 34 P 8.3 Degree independence of Betti numbers . 37 References . 37 1 Introduction arXiv:1406.1693v4 [math.CV] 19 Dec 2014 Hodge theory bridges the topological, smooth and holomorphic worlds. In the abelian case, these are embodied by the Betti, de Rham and Dolbeault Alberto Garc´ıa-Raboso, e-mail:
[email protected] Steven Rayan, e-mail:
[email protected] 1 2 Alberto Garc´ıa-Raboso and Steven Rayan cohomology groups, respectively, of a smooth compact K¨ahlermanifold, X.