Let $(M,\omega)$ be a Kahler manifold. How can we define simply the Picard number for the special case where $M$ is also projective? Wikipedia defines it as the rank of the Neron-Severi group.
In particular, when we say that it has Picard number equal to 1, do we just mean that $Pic(M)=\mathbb{Z}$?
I would be appreciate any help/references.
Let $X$ be a compact complex manifold, then the rank of the image $\text{Pic}(X)\to H^2(X,\mathbb R)$ is called the Picard number $\rho(X)$.
Note that if $X$ be a compact Kahler manifold then the Picard number satisfies $$\rho(X)=rk(NS(X))=rk(H^{1,1}(X,\mathbb Z))$$
By another method also we can define the Picard number.
A line bundle $L$ is numerically trivial if $(L.L') = 0$ for all line bundles $L'$.
The subgroup of all numerically trivial line bundles is denoted $Pic(X)^τ ⊂ Pic(X)$, define
$$Num(X) := Pic(X)/Pic^τ(X).$$
we have the following result for the Picard number (see page 10 )
$$ρ(X) = rk NS(X) = rk Num(X).$$
We define the Neron-Severi lattice and the associated Neron-Severi space as follows
$$NS(X) := H^{1,1}_{\mathbb R}(X) ∩(H^2(X, \mathbb Z)/{\text\{torsion\}})$$
$$ NS_{\mathbb R}(X) := NS(X) ⊗_{\mathbb Z} \mathbb R$$
Hence, we have
$$ρ(X) = rank_{\mathbb Z} NS(X) = \dim_{\mathbb R }NS_{\mathbb R}(X)$$
If $f : X −→ Y$ is a proper morphism between normal varieties, then the relative Picard number of $f$ is given by $$ρ(X/Y ) = ρ(X) - ρ(Y )$$.