Speaker: Stefan Kebekus (Freiburg)

Title of the talk: Families of varieties of general type over compact bases

Abstract: Generalizing the well-known Shafarevich hyperbolicity conjecture, it has been conjectured by Viehweg that a quasi-projective manifold that admits a generically finite morphism to the moduli stack of canonically polarized varieties is necessarily of log general type. Given a projective threefold Y that admits a non-constant map to the moduli stack, we establish a strong relationship between the moduli map and the minimal model program of Y: in all relevant cases the minimal model program leads to a fiber space whose fibration factors the moduli map. A much refined affirmative answer to Viehweg's conjecture for families over threefolds follows as a corollary.