| Résume | Abstract: The classical Shafarevich conjecture, proved by Faltings,
states that there are only finitely many smooth projective curves of a
given genus over a number field with good reduction outside a finite set
of places. Its geometric analogue, known as the hyperbolicity
conjecture, predicts similar finiteness for projective families over a
pointed curve. In this talk, we study this geometric analogue in the
context of primitive symplectic varieties—singular generalizations of
projective hyper-Kähler manifolds. We prove the finiteness of their
locally trivial families (without fixing a polarization type) over a
pointed curve, and deduce unconditional finiteness for all known
deformation types of smooth hyperkähler varieties. I will also explain
that our results are optimal, in the sense that these conditions cannot
be removed. This is a joint work with L. Fu-T. Takamatsu- H. Zou and H.
Zou-C. Jiang.
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