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ALG gravitational instantons are complete hyper-K”ahler surfaces asymptotic to a twisted product of the complex plane and an elliptic curve. Following the classical work of Tian-Yau and Hein, etc., on Monge-Ampere methods for Ricci flat K”ahler metrics on quasi-projective varieties, we provide a geometric existence for generalized ALG Ricci flat K ̈ahler 3-folds on crepant resolutions, where the K3 fiber admits a purely non-symplectic automorphism of finite order. These metrics decay to the ALG model at any polynomial rate, and the dimension of the tangent cone at infinity is still 2.

A local K ̈unneth formula plays a role in the Schwartz decay and an ansatz that equals a Ricci flat ALG model outside a compact set. The Schwartz’s decay is due to a non-concentration of the Newtonian potential.