Let $M$ be an affine complex manifold, let $A$ be an abelian scheme over $M$. Let $\mathcal{A}$ be the sheaf of local sections of $A$ over $M$. We can equip $M$ with etale topology $M\_{et}$ or complex topology $M\_{an}$. There is a natural comparison map $$\gamma\colon H^1(M\_{et},\mathcal{A})\to H^1(M\_{an},\mathcal{A})$$ between the corresponding sheaf cohomologies. 


(a) Are there explicit examples such that $\mathrm{ker}(\gamma)\neq 0$?


(b) Are there explicit examples such that $\mathrm{coker}(\gamma)\neq 0$?


[In case (a) we need to find an algebraic $A$-torsor $T$, such that $T/M$ admits an analytic section but not an algebraic section. Suppose such $T$ exist, we can consider its relative albanese map $a:T\to \mathrm{Alb}(T)$, this is a family of algebraic maps over $M$, I think there should be plenty of non-algebraic family of algebraic maps, but I am not sure how to make such an example..]


[In case (b), I think we need to find a non-algebraic family of complex torus $T'/M$, whose albanese is algebraic, not sure how to find one...]



Analytic and algebraic torsor of abelian scheme
Here is an example when $\gamma $ is not injective.


In general, if $A=A\_0\times M$ is a constant abelian scheme, choose a presentation for $(A\_0)\_{an}$ as $\mathbb{C}^g/\mathbb{Z}^{2g}$. This induces a short exact sequence of sheaves on $M\_{an}$: $$0\to\underline{\mathbb{Z}}^{2g}\to\mathcal{O}\_{M\_{an}}^{\oplus g}\to\mathcal{A}\to 0$$


It induces a long exact sequence, part of which looks like $H^1(M\_{an},\mathcal{O})^{\oplus g}\to H^1(M\_{an},\mathcal{A})\to H^2(M\_{an},\mathbb{Z})^g$. In particular, if $M$ is affine and $H^2(M\_{an},\mathbb{Z})=0$, then $H^1(M\_{an},\mathcal{A})=0$.


Let now $A\_0=E$ be an elliptic curve and $M=E'\setminus\{p\}$ be the complement to a point in an elliptic curve $E'$ which is not isogenous to $E$. We have a long exact sequence of abelian groups $$Maps(M,E)\xrightarrow{[n]}Maps(M,E)\to H^1(M\_{et},\mathcal{A}[n])\to H^1(M\_{et},\mathcal{A})$$ Here $\mathcal{A}[n]$ is the sheaf of $n$-torsion in our abelian scheme, which is isomorphic to the constant sheaf $\underline{(\mathbb{Z}/n)^2}$. Any morphism $M\to E$ extends to a morphism $E'\to E$ (this fails for morphisms $M\_{an}\to E\_{an}$ of analytic spaces) which has to be constant by the assumption, so $Maps(M,E)=0$. Thus we get an injection $$(\mathbb{Z}/n)^4=H^1(M,(\mathbb{Z}/n)^2)\hookrightarrow H^1(M\_{et},\mathcal{A})$$ which gives a non-zero element in $H^1(M\_{et},\mathcal{A})$ while the source of the map $\gamma$ is zero.


