A special variety, in the smooth compact Kähler setting, is a manifold with no Bogomolov sheaf. Thus no holomorphic line
bundle 
maps nontrivially to 
with 
for 
.
A point is special. A compact smooth curve is special
precisely when its genus is at most 1.
For a smooth complex quasi-projective variety, specialness uses orbifold bases of algebraic fibrations after proper birational modifications.
Every dominant fibration with connected general fiber onto a positive-dimensional
normal base must have orbifold-base Kodaira dimension
smaller than the base dimension. This is computed on
smooth compactifications with infimum fiber multiplicities, and a removed divisor
has multiplicity infinity. For 
with reduced boundary 
, the curve criterion is
 |
Hence an elliptic curve is special, while a projective line with three punctures is not.
The abelianity conjecture concerns ordinary fundamental groups of compact special manifolds.
