20 Open questions about deformations of compactifiable manifolds

Abstract

Deformation theory of complex manifolds is a classical subject with recent new advances in the noncompact case using both algebraic and analytic methods. In this note we recall some concepts of the existing theory and introduce new notions of deformations for manifolds with boundary, for compactifiable manifolds, and for q-concave spaces. We highlight some of the possible applications and give a list of open questions which we intend as a guide for further research in this rich and beautiful subject.

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Acknowledgements

This worked started during a meeting of Research in Pairs supported by the Centro Internazionale per la Ricerca Matematica, of Fondazione Bruno Kessler, Trento (Italy). F. R. was supported by Beca Doctorado Nacional Conicyt Folio 21170589. E. G. and F. R. thank the Office of External Activities of ICTP for the support under Network grant NT8. E.B. was partially supported by MIUR and GNSAGA of INdAM (Italy). The authors thank Severin Barmeier for suggesting several improvements to the original text. Finally, we gladly acknowledge the help of the referee, who made us not only improve the paper, but also learn more about the subject.

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Correspondence to Francisco Rubilar.

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Ballico, E., Gasparim, E. & Rubilar, F. 20 Open questions about deformations of compactifiable manifolds. São Paulo J. Math. Sci. (2021). https://doi.org/10.1007/s40863-021-00213-8

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