By a K3 fibered Calabi-Yau threefold, I mean a smooth projective threefold $X$ with trivial canonical class and $h^{1,0}(X) =h^{2,0}(X) = 0$ that has a fibration $X \rightarrow \mathbb P^1$ whose generic fibers are smooth K3 surfaces (to be denoted by $S_X$)

I gathered some examples of mirror pairs of those K3-fibered CY threefolds.

Borcea--Voisin CY's,

CY double coverings over some quasi-Fano threefolds in this paper.

From these examples, I observed:

In these examples of mirror pairs of K3-fibered CY threefolds X, Y. Their generic K3 fibers are K3 mirrors (in the definition of this paper). In concrete words, there are embeddings lattices $\phi_X: Pic(S_X) \rightarrow L$, $\phi_Y: Pic(S_Y) \rightarrow L$ to K3 lattice $L$ of rank 22 and signature $(3, 19)$ such that $L = im(\phi_X) \oplus im(\phi_Y) \oplus U$ ,where $U$ is a unimodular hyperbolic plane in $L$.

My question is:

  1. Are there other examples of mirror pairs of K3-fibered CY threefolds that support or fail this observation.
  1. If other known examples also support this observation, Is this observation expected from any kind of theories of mirror symmetry of Calabi-Yau threefolds with K3 fibration?

1 Answer 1


Morally speaking, a K3 fibration on one side of the mirror correspondence should correspond to a K3 degeneration on the other side. In the physics literature, I think this observation goes back to studies of the F-theory/heterotic duality in the '90s; some coauthors and I wrote a paper with a more combinatorial take that describes how to understand and generate toric examples (though in the toric case, identifying the appropriate polarizing lattice can be subtle). Studies of Tyurin degeneration, including the Nam-Hoon Lee paper you linked and the work of Doran-Harder-Thompson, should give you another perspective on the degeneration/fibration duality.


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