Explicit Ricci-flat metrics on a special Schoen threefold

Abstract

We give an explicit convergent representation of the Ricci-flat Kähler metric in a one-parameter family of classes on a fixed special Schoen threefold. The threefold resolves a shifted square-torus quotient with sixteen disjoint elliptic singular curves. Radial Eguchi--Hanson product metrics and fixed flat collars define a positive background with an exact $a^4$ volume defect. Spherical harmonics, tangential Fourier modes, Frobenius recurrences, radial ordered integrals, and the flat quotient heat kernel specify the inverse of the background scalar linearized operator. A boundary transmission series has contraction factor $1/15$. Independent weighted zeroth-order and six-dimensional Schauder estimates give the inverse bound needed for the quadratic and cubic Monge--Ampère terms. The resulting source-degree recursion converges for $0<a\le 2^{-2260}$, with explicit coefficient, potential-tail, and relative metric-tail bounds. The representation uses specified kernels, prescribed integrals, and convergent series; its parameter interval is a sufficient quantitative bound on this fixed model.

36 pages