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.
To Robert Lazarsfeld on the occasion of his sixtieth birthday
The aim of this paper is to answer some questions about Calabi-Yau manifolds that were raised during the workshop String Theory for Mathematicians, which was held at the Simons Center for Geometry and Physics.
F-theory posits that the “hidden dimensions” constitute a Calabi-Yau 4-fold X that has an elliptic structure with a section. That is, there are morphisms g: X → B whose general fibers are elliptic curves and σ: B → X such that g ∘ σ = 1B (see [Vaf96, Don98]). In his lecture, Donagi asked the following:
Question 1 Is every small deformation of an elliptic Calabi-Yau manifold also an elliptic Calabi-Yau manifold?
Question 2 Is there a good numerical characterization of elliptic Calabi-Yau manifolds?
Clearly, an answer to Question 2 should give a solution of Question 1. The answers to these problems are quite sensitive to which variant of the definition of Calabi-Yau manifolds one uses. For instance, a general deformation of the product of an Abelian variety and of an elliptic curve has no elliptic fiber space structure and every elliptic K3 surface has non-elliptic deformations. We prove in Section 5 that these are essentially the only such examples, even for singular Calabi-Yau varieties (Theorem 31). In the smooth case, the answer is especially simple.
Theorem 3Let X be an elliptic Calabi-Yau manifold such that H2 (X, OX) = 0. Then every small deformation of X is also an elliptic Calabi-Yau manifold.