For a smooth variety $X$ when $Pic^0(X)$ is trivial, we get an isomorphism between $N^1(X)$ and Picard group and life become easier.


My question is whether in general there are theorems, criteria ... which results in triviality of $Pic^0$ in some useful examples ?? Like for hyeprsurfaces in projective space, or in toric varieties, or for complete intersections, or in Fano varieties,...


Please tell, if you know any theorem or criteria like this, or if you know any method which helps one to answer this question in a specific example.


Don't get mad, I know it is vague question.



Vague question on $Pic^0$
Working over $\mathbb{C}$:


We have the short exact sequence of sheaves $0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^\* \to 0$ and we thus have
$$H^1(X, \mathbb{Z}) \to H^1(X, \mathcal{O}) \to \mathrm{Pic}(X) \to H^2(X, \mathbb{Z}) \to H^2(X, \mathcal{O}).$$
Note that the kernel of the last map is discrete, while the cokernel of the first map is connected. So $\mathrm{Pic}^0(X) = \mathrm{CoKer}(H^1(X, \mathbb{Z}) \to H^1(X, \mathcal{O}))$ and $NS(X) = \mathrm{Ker}(H^2(X, \mathbb{Z}) \to H^2(X, \mathcal{O}))$.


Moreover, $H^1(X, \mathcal{O})$ is a $\mathbb{C}$-vector space, and $H^1(X, \mathbb{Z})$ is a finitely generated abelian group. So $\mathrm{Pic}^0(X)$ is trivial if and only if $H^1(X, \mathcal{O})=0$.


I know two main situations where you can conclude that $H^1(X, \mathcal{O})$ is $0$. If $X$ is Stein (for example, affine), then $H^1(X, \mathcal{O})$ vanishes because cohomology of any coherent sheaf on $X$ vanishes. Also, if $X$ is compact Kahler (for example, projective) then $H^1(X, \mathcal{O}) \cong H^{1,0}(X)$ and $\dim H^{1,0}(X) = (1/2) \dim H^1(X, \mathbb{C})$. So $H^1(X, \mathcal{O})$ vanishes if $X$ has no first cohomology. Most (I think all) of the examples you mention are of the second type.


