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Motivated in part by the modular properties of enumerative invariants of K3-fibered Calabi-Yau threefolds, we introduce a family of 39 Calabi-Yau mirror pairs (X, Y) with h₁, ₁ (X) =h₂, ₁ (Y) =2, labelled by certain integer quadruples (m, i, j, s) with m 11. On the A-model side, X arises as a complete intersection in a projective bundle over a Fano fourfold Vₘ^i, j, and admits a Tyurin degeneration into a pair of degree m Fano threefolds Fₘ^i Fₘ^j intersecting on an anticanonical K3 divisor of degree 2m. On the B-model side, Y is fibered by M₌-polarized K3-surfaces of Picard rank 19, and determined by a branched covering of P¹, consistent with the Doran-Harder-Thompson mirror conjecture. When s=0, Y itself acquires a Tyurin degeneration, and correspondingly X acquires a fibration by degree 2m K3 surfaces, such that the two K\"ahler moduli control the size of the K3-fiber and base P¹. While the mirror pairs with m 4 can be realized as complete intersections in products of projective spaces or as hypersurfaces in toric varieties, the examples with m 5 are intrinsically non-toric. We obtain uniform formulae for the genus 0 and 1 topological free energies near the Tyurin degeneration (mirror to the large base limit), exhibiting modular properties under the Fricke-extended congruence group ₀ (m) ^+. We use these results to compute the vertical Gopakumar-Vafa and Noether-Lefschetz invariants and check that their generating functions satisfy the expected modular properties. We also compute generating series of Gopakumar-Vafa invariants with fixed non-zero base degree and exhibit their modular properties.
Doran et al. (Tue,) studied this question.