  We prove the Tate conjecture for divisor classes and the Mumford-Tate
conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a
finitely generated field of characteristic 0, under a mild assumption on their
moduli. As an application of this general result, we prove the Tate and
Mumford-Tate conjectures for some classes of algebraic surfaces with $p_g=1$.
