sections of the cotangent bundle of elliptic surfaces Consider an  elliptic surface $f :X \rightarrow C$ with $\chi(X) > 0$
or equivalently the fibration has a reduced singular fiber apart from possibly multiple fibers
 (the field under consideration
is $\mathbb{C}$,$X$,$C$ are smooth and projective ).
Denote by $\Omega$ the cotangent bundle of $X$ and
$K$ the canonical bundle of $C$. We have an inclusion 
\begin{equation}
0 \rightarrow f^*K \rightarrow \Omega
\end{equation}
Let $q(X)$ denote the dimension of $H^0(X,\Omega)$ over $\mathbb{C}$.
We have $q(X) = g$ where $g$ is the genus of the curve $C$.On the other hand
dimension of $H^{0}(C,K)$ is $g$ as well and hence we get (using the above inclusion of
sheaves)
\begin{equation}
H^{0}(C,K) = H^{0}(X,\Omega).
\end{equation}
Now what I am trying to understand is the above equality in a better way (rather
than just by using the dimension argument as above). Is there a way to see why
all the 1-forms are pull backs of those from the base curve?.
 A: This is a comment starting from a slightly more general context. Most of the following material can be found in  a paper of T. Saito and me (but most dealing with the positive characterisitc case).
Let $f : X\to C$ be a flat morphism of smooth (geometrically connected) projective varieties over a field $k$ of characteristic $0$. Consider the canonical exact sequence 
$$ 0 \to f^*\Omega_{C/k}\to \Omega_{X/k}\to \Omega_{X/C} \to 0. $$ 
We have $f_{*}\mathcal O_X=\mathcal O_C$. Taking $f_*$ we get an exact sequence of sheaves on $C$: 
$$ 0 \to \Omega_{C/k}\to f_{*}\Omega_{X/k} \to f_{*}\Omega_{X/C}\stackrel{\theta}{\to} R^1f_{*}(\mathcal O_X)\otimes\Omega_{C/k}. $$ 
We have $R^1f_{*}(\mathcal O_X)\simeq \omega_{X/C}^{\vee}$ (in characteristic $0$).
Let $T=\Omega_{X/C, \rm{tors}}$ (torsion as $\mathcal O_X$-module). 
A local analysis shows easily that $T$ is an invertible sheaf over the verticla divisor $D:=\sum_{s\in C} D_s$, where $D_s=X_s-(X_s)_{\mathrm{red}}$ (here again we need $k$ of characteristic $0$) and we have an exact sequence
$$ 0 \to T\to \Omega_{X/C}\to \omega_{X/C}(-D) \to S \to 0$$
with $S$ of finite length. Thus the $\mathcal O_C$-torsion part of $f_*\Omega_{X/C}$ is exactly $f_*T$. 
At the generic fiber $\theta$ is the Kodaira-Spencer map. It is non-trivial when $f$ is non-isotrivial, and it is injective if moreover the generic fiber has genus $1$. So under these hypothesis, we have 
$$ 0 \to \Omega_{C/k}\to f_{*}\Omega_{X/k} \to f_{*}T\to 0. $$ 
Therefore the canonical map $H^0(C, \Omega_{C/k})\to H^0(X, \Omega_{X/k})$ is an isomorphism if
$$H^0(X, \Omega_{X/C, \mathrm{tors}})=H^0(C, f_*T)=0.$$ 
In a small neighborhood of a non-multiple fiber $X_s$, one can show that $H^0(f_*T)=0$. Otherwise (especially when $X_s$ is irreducible but not reduced) I don't know. 
A: Consider the short exact sequence
$$
0\to f^*\Omega_C\to \Omega_X\to \Omega_{X/C}\to 0,
$$
and the long exact sequence induced by $f_*$:
$$
0\to \Omega_C\to f_*\Omega_X\to f_*\Omega_{X/C}\to \Omega_C\otimes R^1 f_* \mathscr O_X\to \ldots
$$
If $f$ is not isotrivial, then $\beta: f_*\Omega_{X/C}\to \Omega_C\otimes R^1 f_* \mathscr O_X$ is generically injective (at least where $f$ is smooth $f_*\Omega_{X/C}$ is a line bundle and $\beta$ is non-zero).
If $\Omega_{X/C}$ is torsion-free, then so is $f_*\Omega_{X/C}$ and hence $\beta$ is injective everywhere. That implies, that then $\alpha:\Omega_C\to f_*\Omega_{X}$ is an isomorphism. 
Therefore, $H^0(U,\omega_C)=H^0(f^{-1}U,\Omega_X)$ for every $U\subseteq C$ open.
