  We investigate boundedness results for families of holomorphic symplectic
varieties up to birational equivalence. We prove the analogue of Zarhin's trick
by for $K3$ surfaces by constructing big line bundles of low degree on certain
moduli spaces of stable sheaves, and proving birational versions of Matsusaka's
big theorem for holomorphic symplectic varieties.
  As a consequence of these results, we give a new geometric proof of the Tate
conjecture for $K3$ surfaces over finite fields of characteristic at least $5$,
and a simple proof of the Tate conjecture for $K3$ surfaces with Picard number
at least $2$ over arbitrary finite fields -- including characteristic $2$.
