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Nonabelian Hodge correspondences for klt varieties and quasi-etale uniformisation

By Stefan Kebekus

Appears in collections : French-German meeting on complex algebraic geometry / Rencontre franco-allemande en géométrie algébrique complexe, Exposés de recherche

Simpson’s classic nonabelian Hodge correspondence establishes an equivalence of categories between local systems on a projective manifold, and certain Higgs sheaves on that manifold. This talk surveys recent generalisations of Simpson’s correspondence to the context of projective varieties with klt singularities. Perhaps somewhat surprisingly, these spaces exhibit two correspondences: one pertaining to local systems on the whole space, and one to local systems on its smooth locus. As one application, we resolve the quasi-étale uniformisation problem for minimal varieties of general type, and to obtain a complete numerical characterisation of singular quotients of the unit ball by discrete, co-compact groups of automorphisms that act freely in codimension one.

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Citation data

  • DOI 10.24350/CIRM.V.19388203
  • Cite this video Kebekus, Stefan (11/04/2018). Nonabelian Hodge correspondences for klt varieties and quasi-etale uniformisation. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19388203
  • URL https://dx.doi.org/10.24350/CIRM.V.19388203

Bibliography

  • Greb, D., Kebekus, S., Peternell, T., & Taji, B. (2018). Harmonic metrics on Higgs sheaves and uniformization of varieties of general type. <arXiv:1804.01266> - https://arxiv.org/abs/1804.01266
  • Greb, D., Kebekus, S., Peternell, T., & Taji, B. (2017). Nonabelian Hodge Theory for klt spaces and descent theorems for vector bundles. <arXiv:1711.08159> - https://arxiv.org/abs/1711.08159
  • Greb, D., Kebekus, S., Peternell, T., & Taji, B. (2016). The Miyaoka-Yau inequality and uniformisation of canonical models. <arXiv:1511.08822> - https://arxiv.org/abs/1511.08822
  • Greb, D., Kebekus, S., & Taji, B. (2016). Uniformisation of higher-dimensional minimal varieties. <arXiv:1608.06644> - https://arxiv.org/abs/1608.06644

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