First cohomological support locus of a fibration 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 a subgroup of $\mathrm{Pic}^0(C)$?

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).  
 A: Francesco,
 I don't mean to be picky but I suspect a typo in the way you defined $V^1(\omega_S)$. Do you really only consider $\eta\in Pic^0( C)$? But let me stick to that for now. The post edit version of your question is whether $V^1(\omega_S)=Pic^0( C)$, when $g( C) =1$ and $S$ has general type? 
Following abx, dualizing and using Leray gives 
$$V^1(\omega_S) = \{\eta\in Pic^0(C)\mid H^1(f_*\omega_S\otimes \eta)=H^0(R^1f_* O_S\otimes \eta^{-1})\not=0\}\cup \{O_C\}$$

You just need $\deg f_*\omega_S > 0$ by Riemann-Roch (remember $g=1$), but I'm pretty sure it's true in your case where $S$ has general type -- this probably goes back to Viehweg. I'm away from my books, otherwise I would check. 
A: It is a general fact that  $f_*\omega _S\cong A\oplus F$, where $A$ is an ample vector bundle and $F$ a flat unitary vector bundle (see T. Fujita, The sheaf of relative canonical forms of a Kähler fiber space over a curve. Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 7, 183–184). Since $\pi _1(E)$ is abelian $F$ is a direct sum of flat line bundles $L_i$, so $V^1(\omega _S)=\{\mathscr{O}_E,L_i\} $. But I don't see why this should be a subgroup of $\mathrm{Pic^o}(E)$ in general.
