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?.



sections of the cotangent bundle of elliptic surfaces
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](http://www.math.u-bordeaux1.fr/~qliu/articles/conduct-ineg.ps) (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. 


