Suppose $X$ is a normal projective variety over $\mathbb C$. In the case $X$ is smooth according to Hodge theory $h^1(X,O(X))$ is the dimension of the space of holomorphic $1$-forms on $X$ and this number is equal as well to the half of the first Betti number $b_1(X)/2$ .
I would like to know what happen in the case when $X$ is singular and normal.
1) Is there some relation (equality or inequality) between $h^1(X,O(X))$ and $b_1(X)$? For example does $b_1(X)=0$ imply $h^1(X,O(X))=0$?
2) Suppose that $h^1(X,O(X))=n$ is it true that there is a canonical $n$-dimensional space of $1$-forms on $X$, holomorphic outside of its singularities? (if yes, can something be said about their behaviour at singularities?)
Is there some pedagogical reference treating these questions?