Ricci-flat manifolds of generalized ALG asymptotics

TL;DR

This paper constructs and classifies complete non-compact Ricci-flat Kähler manifolds with ALG asymptotics in complex dimensions ≥ 3, revealing new geometric structures and explicit examples with specific Betti numbers.

Contribution

It provides existence results for generalized ALG Ricci-flat Kähler manifolds with Schwartz decay and constructs explicit examples with prescribed asymptotic angles and Betti numbers.

Findings

Existence of ALG Ricci-flat Kähler manifolds with Schwartz decay.

Explicit examples with 64 Betti number triples.

Classification of asymptotic angles related to automorphisms of K3 surfaces.

Abstract

In complex dimensions $\geq 3$, we provide a geometric existence for generalized ALG complete non-compact Ricci flat K\"ahler manifolds with Schwartz decay i.e. metric decay in any polynomial rate to an ALG model $\mathbb{C}\times Y$ modulo finite cyclic group action, where $Y$ is Calabi-Yau. Consequently, for any $K3$ surface with a purely non-symplectic automorphism $\sigma$ of finite order, a K\"ahler crepant resolution of the orbifold $\frac{\mathbb{C} \times K3}{\langle \sigma \rangle}$ admits ALG Ricci-flat K\"ahler metrics with Schwartz decay. It is known that K\"ahler crepant resolution exists in our case. Hence there are $39$ integers, such that $2\pi$ divided by each of them is the asymptotic angle of an ALG Ricci-flat K\"ahler $3-$fold with Schwartz decay. We also exhibit a 1638 parameters family of ALG Ricci-flat K\"ahler $3-$folds with asymptotic angle $\pi$ that realize $64$ distinct triples of Betti numbers. They are iso-trivially fibred by $K3$ surface with a non-symplectic Nikulin involution. A simple version of local Kunneth formula for $H^{1,1}$/local $i\partial\overline{\partial}-$lemma plays a role in both the Schwartz decay, and the construction of ansatz that equals a Ricci flat ALG model outside a compact set (isotrivial ansatz). The proof of Schwartz decay relies on a non-concentration of the Newtonian potential, and can not be immediately generalized to fibration with higher dimensional base, due to existence of concentrating sequence of $L^{2}$ normalized eigen-functions on unit round spheres of (real) dimension $\geq 2$.