I have the following question?

Let $f:X\rightarrow Y$ be surjective projective morphism between smooth projective variety. 
I learned that if $dimY=1$, then $R^if_*\mathcal O_X$ is torsion free $\forall i\geq0$.
I think smoothness of $X,Y$ is important here, and I wonder what about the condition $dim Y=1$?

***Q1***  Is it still true that $R^if_*\mathcal O$ is torsion free $\forall i\geq0$ when $dim Y>1$? If false, what's a natural counter example? (To show how dimension of $Y$ matters).

Let $f:X\rightarrow Y$ be surjective projective morphism between  projective variety ($Y$ can be singular).
 
***Q2*** Is it true that the direct image $f_*\mathcal O_X$ is always torsion free? (I feel it's true because the product of 2 nonzero holomorphic function is nonzero?)