The Kollár-Pardon conjecture asserts that a connected normal projective complex variety has a semialgebraic presentation of its universal cover precisely when the cover has a product decomposition
 |
where 
is an integer, 
is a bounded symmetric
domain, and 
is a simply connected
normal projective variety. A point is allowed
for either 
or 
.
The presentation means a biholomorphism with a semialgebraic open subset of a projective variety, with openness in the ordinary complex topology. In affine algebraic charts, semialgebraic means a finite Boolean combination of real polynomial equalities and inequalities in real and imaginary coordinates. Neither the biholomorphism nor the deck transformations is required to be algebraic or to preserve the chosen semialgebraic structure. The projective factor may be singular. For curves the three possibilities are the projective line, the complex line, and the unit disk.
