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On Rigid Manifolds of Kodaira Dimension 1

  • 2025
  • OriginalPaper
  • Chapter
Published in:
Authors
Ingrid Bauer
Christian Gleissner
Julia Kotonski
Published in
Perspectives on Four Decades of Algebraic Geometry, Volume 1
Publisher
Springer Nature Switzerland

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Abstract

The chapter focuses on rigid manifolds of Kodaira dimension 1, addressing the open question of their existence and classification. It begins by discussing the seminal work of Bauer and Catanese on rigid manifolds and their relation to Kodaira dimension. The authors present a positive answer to the open problem using a specific construction involving the Fermat cubic curve and the Klein quartic. They then analyze the geometry of these rigid quotients, showing that they have isolated canonical singularities. The chapter also provides a detailed classification of minimal examples and discusses the resolution of singularities, preserving the rigidity of the manifolds. The authors use methods from toric geometry to construct these resolutions, generalizing previous work. The chapter concludes with an open problem regarding the biholomorphic equivalence of certain complex conjugate varieties.
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This summary of the content was generated with the help of AI.

The second author “Christian Gleissner” wants to thank Stephen Coughlan and Andreas Demleitner for interesting and useful conversations about rigid manifolds.
Dedicated to the memory of Alberto Collino.

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previous chapter Singularities of Bridgeland Moduli Spaces for K3 Categories: An Update
next chapter On the Components of the Main Stream of the Moduli Space of Surfaces of General Type with
Title
On Rigid Manifolds of Kodaira Dimension 1
Authors
Ingrid Bauer
Christian Gleissner
Julia Kotonski
Copyright Year
2025
Book
Perspectives on Four Decades of Algebraic Geometry, Volume 1

Print ISBN: 978-3-031-66229-4

Electronic ISBN: 978-3-031-66230-0

Copyright Year: 2025

DOI
https://doi.org/10.1007/978-3-031-66230-0_2
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