Compactification thresholds for polynomial asymptotically Calabi Calabi–Yau manifolds

Establish optimal constants λ and μ such that for any decay rates κ1>λ and κ2>μ in the definition of a polynomial asymptotically Calabi Calabi–Yau manifold (X, I, ω, Ω) with respect to a Calabi model space (C, I_C, ω_C, Ω_C) and a diffeomorphism Φ: C \ K → X \ K, the manifold X admits a complex-analytic compactification to a weak Fano manifold and its Calabi–Yau metric arises from the generalized Tian–Yau construction presented in Theorem 1.

Background

The paper constructs complete Calabi–Yau metrics on X = M \ D, where D is a smooth anticanonical divisor with ample normal bundle, that are weak asymptotically Calabi with polynomial rate and proves uniqueness under polynomial closeness. Hein–Sun–Viaclovsky–Zhang previously showed that asymptotically Calabi (exponential rate) Calabi–Yau manifolds compactify to weak Fano manifolds.

The authors propose extending this compactification/classification framework to a broader class with only polynomial decay. They introduce a precise definition of polynomial asymptotically Calabi metrics with separate decay rates κ1 (for complex structure and holomorphic volume form) and κ2 (for the Kähler form) and conjecture that above certain thresholds the same compactification and origin from their generalized Tian–Yau construction should hold.