The holomorphic density property of a complex manifold 
means that the Lie algebra generated by its complete
holomorphic vector fields is dense in the Lie
algebra of all holomorphic vector fields, in
the compact-open topology. A holomorphic
vector field is complete here if its flow exists
for every complex time. This is a condition on the abundance of holomorphic transformations,
not on the topological denseness of a subset.
The Tóth-Varolin conjecture asks whether a Stein manifold diffeomorphic
to 
and having the holomorphic density property must be biholomorphic
to 
(Gorai 2026).
