Around Kollár-Pardon Conjecture

The operation of taking the universal cover, despite being fundamental in the context of topology and differential geometry, remains rather mysterious in the context of complex algebraic geometry. One of the main difficulties is that the universal cover of a smooth complex algebraic variety is merely a complex manifold usually lacking any algebraic structure. Moreover, it is usually ‘’wild’’ in a certain sense. This is made precise by the Kollár-Pardon Conjecture, which says that the universal cover of a smooth complex projective variety is almost never a semialgebraic subset of an algebraic variety, except for several classical cases.

I will discuss the context of the Kollár-Pardon Conjecture and the known results in its direction. After that, I will explain how one can see Kollár-Pardon Conjecture as a special case of a general Meta-Conjecture, which can be applied to a huge set of various geometric situations and discuss its version for Riemannian manifolds.