\(\partial \overline{\partial }\)-Complex symplectic and Calabi–Yau manifolds: Albanese map, deformations and period maps

  • Ben Anthes
  • Andrea Cattaneo
  • Sönke Rollenske
  • Adriano Tomassini


Let X be a compact complex manifold with trivial canonical bundle and satisfying the \(\partial \overline{\partial }\)-Lemma. We show that the Kuranishi space of X is a smooth universal deformation and that small deformations enjoy the same properties as X. If, in addition, X admits a complex symplectic form, then the local Torelli theorem holds and we obtain some information about the period map. We clarify the structure of such manifolds a little by showing that the Albanese map is a surjective submersion.


\(\partial \overline{\partial }\)-Lemma Complex symplectic manifold Albanese map 



Ben Anthes and Sönke Rollenske gratefully acknowledge support from the DFG via the Emmy Noether program. We enjoyed discussions on parts of this project with Andreas Krug and we thank Nicolas Perrin for some discussions about Graßmannians. Adriano Tomassini is granted with a research fellowship by Istituto Nazionale di Alta Matematica INdAM, and is supported by the Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, by the Project FIRB “Geometria Differenziale e Teoria Geometrica delle Funzioni”, and by GNSAGA of INdAM. Andrea Cattaneo was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX- 0007) operated by the French National Research Agency (ANR), and is member of GNSAGA of INdAM. Andrea Cattaneo and Adriano Tomassini are grateful to Ben Anthes and Sönke Rollenske for their kind hospitality at Marburg University during the preparation of the paper. The authors want to thank the referee for his/her suggestions on the first version of the paper.


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Authors and Affiliations

  1. 1.FB 12/Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany
  2. 2.Institut Camille JordanUniversité Claude Bernard Lyon 1VilleurbanneFrance
  3. 3.Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e InformaticaUniversità di ParmaParmaItaly

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