$DeclareMathOperatorNS{NS}DeclareMathOperatorPic{Pic}$Let $X$ be a compact complex manifold, assume projective if you’d like. Define the Néron–Severi group to be the quotient $$NS(X) = Pic(X) / Pic^0(X).$$ Suppose that $Pic(X) = Pic^0(X) neq 0$. So all divisors are algebraically equivalent, and (by definition) the Picard number is zero.

Can we infer any geometric information from this constraint (does this constrain other invariants such as Kodaira dimension, curvature, etc.)? Are there examples of such $X$? Are there plenty of such examples?