Wojciech J. Gajda

The Shafarevich conjecture for canonically polarized varieties, a special case of the Lang-Vojta conjecture in diophantine geometry, predicts that over any number field there are only finitely many isomorphism classes of smooth projective canonically polarized varieties with given Hilbert polynomial and good reduction outside a given finite set of primes. In the case of curves this was proven by Faltings on his way to the Mordell conjecture. Since then, many further cases have been established, but all of them rely on some classification of the varieties under consideration. In the talk I will discuss recent work with Marco Maculan in which we prove the Shafarevich conjecture for all varieties with globally generated cotangent bundle, subject to mild numerical conditions. We combine the Lawrence-Sawin-Venkatesh method with the big monodromy theorem from our work with Javanpeykar, Lehn and Maculan; the key input is the convolution of perverse sheaves on abelian varieties.