Automorphism groups of compact complex supermanifolds

Article

Abstract

Let \({\mathcal {M}}\) be a compact complex supermanifold. We prove that the set \({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\) of automorphisms of \({\mathcal {M}}\) can be endowed with the structure of a complex Lie group acting holomorphically on \({\mathcal {M}}\), so that its Lie algebra is isomorphic to the Lie algebra of even holomorphic super vector fields on \({\mathcal {M}}\). Moreover, we prove the existence of a complex Lie supergroup \({{\mathrm{Aut}}}({\mathcal {M}})\) acting holomorphically on \({\mathcal {M}}\) and satisfying a universal property. Its underlying Lie group is \({{\mathrm{Aut}}}_{{{\bar{0}}}}({\mathcal {M}})\) and its Lie superalgebra is the Lie superalgebra of holomorphic super vector fields on \({\mathcal {M}}\). This generalizes the classical theorem by Bochner and Montgomery that the automorphism group of a compact complex manifold is a complex Lie group. Some examples of automorphism groups of complex supermanifolds over \({\mathbb {P}}_1({\mathbb {C}})\) are provided.

Keywords

Compact complex supermanifold Automorphism group 

Mathematics Subject Classification

32M05 32C11 54H15 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutAlbert-Ludwigs-Universität FreiburgFreiburgGermany
  2. 2.Fakultät für MathematikRuhr-Universität BochumBochumGermany

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