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

1$\begingroup$ What do you mean by relatively compact? If $X$ is a normal projective variety which is $\mathbb Q$Gorenstein, then for any local trivialization $\sigma$ of $mK_X$ defined on $U\subset X$, the integral $\int_{U_{\rm reg}}(\sigma\wedge \bar \sigma)^{1/m}$ is finite if and only if $U$ has klt singularities. In particular, as quotient singularities are klt, the integral you wrote is finite. But klt singularities are not quotient in general. $\endgroup$ – Henri Jan 12 '17 at 17:48

$\begingroup$ relatively compact means its closure be compact. Can you give a reference? $\endgroup$ – pickasa Jan 12 '17 at 18:05

$\begingroup$ If $X$ is projective, it is compact in the analytic topology. Maybe you meant quasiprojective? anyway it is not really important for the question here. $\endgroup$ – Henri Jan 12 '17 at 19:13
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 nform $Ω$, $∫_XΩ∧\bar Ω<∞$ 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