Simpson correspondence

In algebraic geometry, the Nonabelian Hodge correspondence or Corlette–Simpson correspondence (named after Kevin Corlette and Carlos Simpson) is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, complex algebraic curve.

Unitary representations

For a compact Riemann surface X of genus at least 2, M. S. Narasimhan and C. S. Seshadri established a bijection between irreducible unitary representations of and the isomorphism classes of stable vector bundles of degree 0.[1] This was then extended to arbitrary complex smooth projective varieties by Karen Uhlenbeck and Shing-Tung Yau, and by Simon Donaldson.[2]

Arbitrary representations

These results were extended to arbitrary rank two complex representations of fundamental groups of curves by Nigel Hitchin[3] and Donaldson[4] and then in arbitrary dimension by Corlette[5] and Simpson,[6][7] who established an equivalence of categories between the finite-dimensional complex representations of the fundamental group and the semi-stable Higgs bundles whose Chern class is zero.

References
  1. Narasimhan, M. S.; Seshadri, C. S. (1965). "Stable and unitary vector bundles on a compact Riemann surface". Annals of Mathematics. 82: 540–567.
  2. Donaldson, Simon (1987). "Infinite determinants, stable bundles and curvature". Duke Mathematical Journal. 54: 231–247.
  3. Hitchin, Nigel J. (1987). "The self-duality equations on a Riemann surface". Proceedings of the London Mathematical Society. 55 (1): 59–126.
  4. Donaldson, Simon K. (1987). "Twisted harmonic maps and the self-duality equations". Proceedings of the London Mathematical Society. 55 (1): 127–131.
  5. Corlette, Kevin (1988). "Flat G-bundles with canonical metrics". Journal of Differential Geometry. 28 (3): 361–382.
  6. Simpson, Carlos T. (1990). "Nonabelian Hodge Theory" (PDF). Kyoto: International Congress of Mathematicians talk.
  7. Simpson, Carlos T. (1992). "Higgs bundles and local systems". Publications Mathématiques de l'IHÉS. 75: 5–95.
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