Papers › Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds
Charles Doran, Boris Pioline, Thorsten Schimannek
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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 h_(1,1)(X)=h_(2,1)(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 Mₘ-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 Γ₀(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.
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