I have a fibration $f \colon S \longrightarrow E$, where $S$ is a compact, complex surface of general type belonging to a special class I'm studying and $E$ is an elliptic curve.

I computed the first cohomological support locus of the fibration, namely $$V^1(\omega_S):=\{\eta \in \mathrm{Pic}^0(E) \; | \; H^1(S, \, \omega_S \otimes f^*\eta) \neq 0 \}.$$

By some general results of Simpson and other authors, I know that $V^1(\omega_S)$ is a finite union of torsion points in $\textrm{Pic}^0 E$.

However, in my case I find something stronger: $V^1(\omega_S)$ is a *subgroup* of $\textrm{Pic}^0(E)$. So I wonder whether this is a consequence of some more general statement.

Perhaps I do not know the relevant literature enough and my question might be considered trivial by the experts, anyway let me state it explicitly:

Question.Consider a fibration $f \colon S \longrightarrow C$, where $S$ is a surface of general type and $g(C) \geq 1$. Under which conditions the first cohomological support locus $$V^1(\omega_S):=\{\eta \in \mathrm{Pic}^0(C) \; | \; H^1(S, \, \omega_S \otimes f^*\eta) \neq 0 \}$$ is asubgroupof $\mathrm{Pic}^0(C)$?

Notice that, by the results quoted above, I know that in any cases $V^1(\omega_S)$ is a finite union of translates of subtori of $\mathrm{Pic}^0(C)$ by torsion points.

**EDIT.** As suggested by abx in the comment below, an easy application of Serre duality and Leray spectral sequence shows that $V^1(\omega_S)=\textrm{Pic}^0(C)$ as soon as $g(C) \geq 2$. So the question makes sense only when $g(C)=1$ (that is precisely my case).