The de Jong fundamental group of a non-trivial abelian variety is non-abelian

Key Points

• To demonstrate the non-abelian nature of the de Jong fundamental group for abelian varieties over complete algebraically closed fields.
• Analyzed the structure of the de Jong fundamental group in the context of abelian varieties.
• Generalized existing arguments related to the fundamental group for projective varieties.
• Examined properties of connected rigid analytic varieties with non-constant maps to projective spaces.
• Established that the de Jong fundamental group of any non-trivial abelian variety is non-abelian.
• Showed that the fundamental group depends on the complete algebraically closed extension.
• Provided insights into the structure of fundamental groups for a broader class of rigid analytic varieties.