I believe the following is true and well known.


**Theorem (?).** Let $X$ and $Y$ be smooth, irreducible, projective varieties over $\mathbb{C}$. Let
$$
f\colon X\rightarrow Y
$$
be a surjective map with generic fibers being irreducible and rationally connected. Then $H^0(X,\wedge^k\Omega\_X)\cong H^0(Y,\wedge^k\Omega\_Y)$ for all $k\ge 0$.


I would appreciate a reference for this result or a simple proof (or, obviously, if this is wrong I would like to know that too).



Outer Hodge groups of rationally connected fibrations
By spectral sequence, we just need to prove that $R^if\_\*\mathcal{O}\_X=0$ for all $i>0$. This is a special case of Theorem 7.1 in [Kollár, Higher direct images of dualizing sheaves I]. 


Actually the answer to you question is just Corollary 7.2 in [Kollár, Higher direct images of dualizing sheaves I]. 


