A complex manifold is a manifold 
whose coordinate charts
are open subsets of 
and the transition functions
between charts are holomorphic
functions. Naturally, a complex manifold of complex
dimension 
also has the structure of a real smooth manifold
of dimension 
.
A function 
is holomorphic
if it is holomorphic in every coordinate
chart. Similarly, a map 
is holomorphic
if its restrictions to coordinate charts on 
are holomorphic.
Two complex manifolds 
and 
are considered equivalent if there is a map 
which is a diffeomorphism
and whose inverse is holomorphic.
Assuming the proposed complex structure on 
described by Alpöge (2026), Liang
et al. (2026) constructed a complex manifold diffeomorphic
to the product of spheres 
. The authors state that GPT-5.6 Sol suggested the
construction route and the classification theorem used to identify the underlying
smooth manifold. It also first generated proofs
of three intermediate results that the authors checked and rewrote. Independent review
of either construction had not been reported as of Sep. 7, 2026 (VibeMathed
2026).
and 
." 30 Aug 2026. 
." 2026.