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Deformations of elliptic Calabi-Yau manifolds. (English) Zbl 1326.14081
Hacon, Christopher D. (ed.) et al., Recent advances in algebraic geometry. A volume in honor of Rob Lazarsfeld’s 60th birthday. Based on the conference, Ann Arbor, MI, USA, May 16–19, 2013. Cambridge: Cambridge University Press (ISBN 978-1-107-64755-8/pbk; 978-1-107-41600-0/ebook). London Mathematical Society Lecture Note Series 417, 254-290 (2014).
Summary: We investigate deformations and characterizations of elliptic Calabi-Yau varieties, building on earlier works of Wilson and Oguiso. We show that if the second cohomology of the structure sheaf vanishes, then every deformation is again elliptic.
More generally, all non-elliptic deformations derive from abelian varieties or $$K3$$ surfaces. We also give a numerical characterization of elliptic Calabi-Yau varieties under some positivity assumptions on the second Todd class. These results lead to a series of conjectures on fibered Calabi-Yau varieties.
For the entire collection see [Zbl 1318.14002].
Reviewer: Reviewer (Berlin)

MSC:
 14J10 Families, moduli, classification: algebraic theory 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14D06 Fibrations, degenerations in algebraic geometry