2019/02/22 by Jim Bryan, Bryan, Jim · 1 citation
Mathematics · #11F46 #11F50 #14N35 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.1902.08695
openalex publication_date 2019/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A banana manifold is a compact Calabi-Yau threefold, fibered by Abelian surfaces, whose singular fibers have a singular locus given by a "banana configuration of curves". A basic example is given by Xban, the blowup along the diagonal of the fibered product of a generic rational elliptic surface S→ ℙ1 with itself. In this paper we give a closed formula for the Donaldson-Thomas partition function of the banana manifold Xban restricted to the 3-dimensional lattice Γ of curve classes supported in the fibers of Xban→ ℙ1. It is given by ZΓ(Xban) = ∏_d1,d2,d3≥ 0 ∏k (1-pkQ1^d1Q2^d2Q3^d3)-12c(||d ||,k) where ||d || = 2d1d2+ 2d2d3+ 2d3d1-d12-d22-d32, and the coefficients c(a,k) have a generating function given by an explicit ratio of theta functions. This formula has interesting properties and is closely realated to the equivariant elliptic genera of Hilb (ℂ2). In an appendix with S. Pietromonaco, it is shown that the corresponding genus g Gromov-Witten potential Fg is a genus 2 Siegel modular form of weight 2g-2 for g≥ 2; namely it is the Skoruppa-Maass lift of a multiple of an Eisenstein series: \frac6|B2g|g(2g-2)! E2g(τ).