Short Tops and Semistable Degenerations

Abstract

One may construct a large class of Calabi-Yau varieties by taking anticanonical hypersurfaces in toric varieties obtained from reflexive polytopes. If the intersection of a reflexive polytope with a hyperplane through the origin yields a lower-dimensional reflexive polytope, then the corresponding Calabi-Yau varieties are fibered by lower-dimensional Calabi-Yau varieties. A top generalizes the idea of splitting a reflexive polytope into two pieces. In contrast to the classification of reflexive polytopes, there are infinite families of equivalence classes of tops. Tops may be used to describe either fibrations or degenerations of Calabi-Yau varieties. We give a simple combinatorial condition on tops which produces semistable degenerations of K3 surfaces, and, when appropriate smoothness conditions are met, semistable degenerations of Calabi-Yau threefolds. Our method is constructive: given a fixed reflexive polytope which will lie on the boundary of the top, we describe an algorithm for constructing tops which yields semistable degenerations of the corresponding hypersurfaces. The properties of each degeneration may be computed directly from the combinatorial structure of the top.

Towards generalized mirror symmetry for twisted connected sum G2 manifolds

“…degenerations of K3 fibres [72,73] together with a generalization to fibre components of multiplicity > 1, see also [74,75]. For any pair of the examples of building blocks tabulated above, we may find hyper Kähler rotations for the elliptic K3 fibres such that…”

Mirror symmetry for G 2-manifolds: twisted connected sums and dual tops

“…These two components are nothing but the two building blocks, which can hence be found by setting ζ a = 0 and ζ b = 0 in the family X ζ . Taking inspiration from [65,66], we will describe this whole set-up by introducing a toric ambient space and defining equation for the whole family after the blow-up. Let us first describe the set-up in detail.…”

Heterotic-type IIA duality and degenerations of K3 surfaces

“…Examples in section 4.1 are such that only a single extra vertex ν 6 is introduced besides ν 1,2,3,4,5 in the toric polytope. In sections 4.2 and 4.3 we also relax this condition, and the construction in [40] (and its obvious generalization) is exploited. We will see a rich variety of branches of the type IIA compactification moduli space, even for a single choice of the lattice Λ S = NS K3 .…”

“…They are obtained as toric hypersurfaces for which the polytope ∆ contains all of the ν 6 's with n ≥ k 4 ≥ m (and k 2 = k 3 = 0). An example of a "short top" [40] is found as a part of this polytope ∆. It turns out that Hodge numbers of those threefolds are as follows:…”

“…This graph comes directly from an edge (at the height= +1) of the polytope ∆. This is an example of a theorem in [40].…”