Biholomorphic classification of admissible period parameters

Determine whether distinct admissible values of the period parameter c_0 produce biholomorphically inequivalent compact complex threefolds Y_{c_0} with the fixed logarithmic twists and cusp gluing.

Background

The paper constructs an admissible family of compact complex threefolds Y_{c_0}, parametrized by constants c_0 satisfying Im(c_0)<0, and proves that every member of this family is Oka. The construction uses the same monodromy, logarithmic twists, and untranslated cusp gluing for all admissible parameters.

The authors explicitly note that it is not known whether different admissible parameters yield distinct biholomorphism classes. Consequently, the family has not yet been shown to contain more than one biholomorphism class of Oka complex structures on the six-sphere.