On the Tate conjecture for divisors on varieties with $h^{2,0}=1$ in positive characteristics

We prove that the Tate conjecture for divisors is ‘‘generically true’’ for mod p reductions of complex projective varieties with $h^{2,0}=1$, under a mild assumption on moduli. By refining this general result, we establish a new case of the BSD conjecture over global function fields, and the Tate conjecture for a class of general type surfaces of geometric genus 1.