Let $X$ be a relaively compact projective variety and has only quotient singularities then for any n-form $\Omega$ , $$\int\_{X\_{reg}}\Omega\wedge \bar \Omega$$ is bounded? what about the converse 



quotient singularities
Let $X$ has log terminal singularities when $K\_X$ is $\mathbb Q$-Cartier, see Proposition 1.17, of


Log–canonical forms and log canonical singularities, Hubert Flenner, and Mikhail Zaidenberg, Math. Nachr. 254–255, 107 – 125 2003


The interesting part of your question is Kawamata's holomorphic extension theorem, say that if for a n-form $Ω$, $∫\_XΩ∧\bar Ω<∞$ and $Ω|\_{X∖D}$ be holomorphic where $D$ is the reduced normal crossing divisor on $X$, then $Ω\_X$ is holomorphic also


See lemma 0.5.2. such forms gives a nice sheaf, 


<http://archive.numdam.org/ARCHIVE/CM/CM_1987__64_3/CM_1987__64_3_311_0/CM_1987__64_3_311_0.pdf>


