2019/05/15 by Stephen Pietromonaco, Pietromonaco, Stephen
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1905.07085
openalex publication_date 2019/05/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The banana manifold Xban is a smooth projective Calabi-Yau threefold fibered over ℙ1 by abelian surfaces. Each singular fiber contains a "banana configuration of curves" which generates the rank-three lattice Γ of curve classes supported in the fibers of Xban→ℙ1. The Donaldson-Thomas partition function of Xban in fiber classes was computed by J. Bryan (arXiv:1902.08695) to be the infinite product ZDT(Xban)Γ= ∏_d1,d2,d3≥0∏k∈ℤ(1-Q1^d1Q2^d2Q3^d3tk)^-12c(||\underline\bfd||,k) where ||\underline\bfd||=2d1d2+2d1d3+2d2d3-d12-d22-d32, and c(||\underline\bfd||,k) are coefficients of the equivariant elliptic genus of ℂ2. We observe that under a change of variables, ZDT(Xban)Γ behaves formally like a Borcherds lift of the equivariant elliptic genus. The main result of this thesis is that the associated Gromov-Witten potentials Fg in genus g≥2 are meromorphic genus two Siegel modular forms of weight 2g-2. They arise as Maass lifts of weak Jacobi forms of weight 2g-2 and index 1 arising in an expansion of the elliptic genus in the equivariant parameter. We show the equivariant elliptic genus of ℂ2 encodes the Gopakumar-Vafa invariants of Xban. Therefore, one can regard Xban as an example where the generating functions of Gromov-Witten and Donaldson-Thomas invariants in fiber classes are produced by standard lifts of a modular object encoding the Gopakumar-Vafa invariants. We note that because this is a Masters thesis, the first six chapters offer an extended introduction to the relevant background material, while the original results are presented in the final chapter.