A complex manifold is a manifold 
whose coordinate charts
are open subsets of 
and the transition functions
between coordinate 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 a holomorphic map
if its local expressions 
in source and target coordinate
charts 
and 
are holomorphic on their domains.
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 hyperspheres 
. 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. 
." Sep. 7, 2026.