Consider an algebraic elliptic surface $f :X \rightarrow C$ with $\chi(X) > 0$
or equivalently the fibration has reduced singular fibers (the field under consideration
is $\mathbb{C}$).
Now Hodge theory gives us 
\begin{equation} 
q(X) = dim_{\mathbb{C}}(H^{0}(X; \Omega^1_X)) = dim_{\mathbb{C}}(H^1(X,\mathcal{O}_X)).
\end{equation}
On the other hand the $dim_{\mathbb{C}}(H^1(X,\mathcal{O}_X))$ can be seen to be
equal to the genus $g$ of $C$. We also have 
$dim_{\mathbb{C}}H^0(C,K_C) = g$ and hence the canonical inclusion map of sheaves
\begin{equation}
0 \rightarrow f^{*}K_c \rightarrow \Omega^1_X \implies H^{0}(C,K_c) = H^{0}(X,\Omega^1_X).
\end{equation}
Now what I seek is to understand this "equality" in a more geometric way. Is there
a better way to see (apart from the above method) to see that all the 1 forms
in fact come from the base curve?.