Complete existence theorem for unfoldings of foliated complex analytic structures

Prove the complete form of the theorem of existence for unfoldings of compact foliated complex manifolds defined by locally free subsheaves of cotangent sheaves, extending the established existence result for one-parameter families.

Background

The paper studies deformations and unfoldings of foliated complex analytic structures described by locally free subsheaves of tangent or cotangent sheaves. In Part III, the author establishes an existence theorem for unfoldings defined by locally free cotangent subsheaves only in the case of one-parameter families.

The unresolved issue is to obtain the corresponding complete existence theorem for arbitrary parameter spaces or multiparameter families of compact foliated complex manifolds. The desired result would complement the complete theorem of completeness for global unfoldings of singular holomorphic foliations that the paper claims to prove.

References

In Part III we study unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves. We prove theorem of existence for unfoldings of foliated complex analytic structures defined by locally free subsheaves of cotangent sheaves for one-parameter families. The author could not succeed in proving the complete form of theorem of existence for unfoldings and showed only for one-parameter families. We conjecture the complete form of theorem of existence for unfoldings of compact foliated complex manifolds defined by locally free subsheavse of cotangent sheaves.

— On deformations of foliated complex analytic structures I: Theorems of existence and completeness  (2609.34830 - Kim, 28 Sep 2026) in Section 1, Introduction, discussion of Part III