Two decades ago, Katzarkov et al. conjectured that a small deformation of a projective variety with big fundamental group still has big . This conjecture was previously known only for surfaces and in some partial cases for threefolds due to Claudon. Recently, in joint work with Mese and Wang, we proved this conjecture when is linear. In this talk, I will outline the main ideas of the proof and discuss related results on Campana’s broader conjecture concerning the deformation invariance of -dimension.
