Abstract
We study threefolds fibred by K3 surfaces admitting a lattice polarization by a certain class of rank 19 lattices. We begin by showing that any family of such K3 surfaces is completely determined by a map from the base of the family to the appropriate K3 moduli space, which we call the generalized functional invariant. Then we show that if the threefold total space is a smooth Calabi–Yau, there are only finitely many possibilities for the polarizing lattice and the form of the generalized functional invariant. Finally, we construct explicit examples of Calabi–Yau threefolds realizing each case and compute their Hodge numbers.
Highlights
The primary aim of this paper is to study threefolds fibred by K3 surfaces polarized by a certain class of rank 19 lattice, with a particular focus on Calabi–Yau threefolds
B Alan Thompson a.m.thompson@lboro.ac.uk Extended author information available on the last page of the article expect mirror symmetry for Calabi–Yau threefolds fibred by Mn-polarized K3 surfaces to be closely linked to the Fano-LG correspondence for smooth Fano threefolds of Picard rank 1
Be noted that several Calabi–Yau threefolds fibred by M1-polarized K3 surfaces are known to exist, but no classification for them is currently known; the authors intend to address this in future work
Summary
The primary aim of this paper is to study threefolds fibred by K3 surfaces polarized by a certain class of rank 19 lattice, with a particular focus on Calabi–Yau threefolds. This means that the analogue of Theorem 2.2 does not hold for M1-polarized families of K3 surfaces (see Remark 2.3); we primarily restrict our attention to the cases where n ≥ 2 It should, be noted that several Calabi–Yau threefolds fibred by M1-polarized K3 surfaces are known to exist (see [8, Theorem 5.20]), but no classification for them is currently known; the authors intend to address this in future work. Following on from this, the main results of the paper (Theorems 2.12, 2.13) show that a Calabi–Yau threefold may only admit a non-isotrivial fibration by Mn-polarized K3 surfaces if n ∈ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 23}. This gives a substantial extension of the results of [9]
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