$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$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?



Examples of complex manifolds with trivial Néron–Severi group?
There is a Kähler example constructed in [1, Section 1].


Let $\Gamma = \mathbb{Z}^{2n}$ be a lattice, $\phi: \Gamma \to \Gamma$ a $\mathbb{Z} $-linear map with characteristic polynomial $f(\lambda)=\prod\_{i=1}^n(\lambda - \lambda\_i)(\lambda - \overline{\lambda\_i}) $ where $\lambda\_1,\cdots,\lambda\_n,\overline{\lambda\_1},\cdots,\overline{\lambda\_n}$ are distinct non-real complex numbers. Then $\Gamma\_\mathbb{C}=\Gamma \otimes\_\mathbb{Z} \mathbb{C}$ has decomposition $\Gamma\_\mathbb{C} = \Gamma' \oplus \overline{\Gamma'}$ where $\Gamma'$ is the eigenspace of eigenvalues $\{\lambda\_1,\cdots,\lambda\_n\}$. Then we have a complex torus $T=\Gamma\_\mathbb{C}/\Gamma'\oplus \Gamma$ of complex dimension $n$. And [1] proved that,for $n\ge2$, if the Galois group of $f$ is the $2n$-th symmetric group, then ${\rm NS}(T)=0$.


[1] C. Voisin. On the homotopy types of compact Kähler and complex projective manifolds, Inventiones Math. Volume 157, Number 2, 329 - 343.


