One has the following Zariski connectness theorem
If a proper morphism $\pi: X \rightarrow Y$ of locally Noetherian schemes which has $\mathcal{O}_Y \cong \pi_* \mathcal{O}_X$, then $\pi^{-1}(q)$ is connected for every $q \in \mathrm{Y}$.
In Popa's notes (see here Remark 1.2.9) there is a remark which states that
REMARK 1.2.9. (1) Note that for an arbitrary projective morphism $f: X \rightarrow Y$, the assumption $f_* \mathscr{O}_X \simeq \mathscr{O}_Y$ implies that $f$ is surjective, with connected fibers...
(2) Moreover, by a variant Zariski's Main Theorem, under the normality assumption the connectedness of all the fibers is in fact equivalent to $f_* \mathscr{O}_X \simeq \mathscr{O}_Y$ ...
I can't find the source of the second statement, I can only find the version which says that proper birational morphism into normal variety has $f_*\mathscr{O}_X\cong \mathscr{O}_Y$. It seems not to be the one that Popa state in the notes?