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
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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. $\endgroup$– HenriCommented Jan 12, 2017 at 17:48
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$\begingroup$ relatively compact means its closure be compact. Can you give a reference? $\endgroup$– pickasaCommented Jan 12, 2017 at 18:05
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$\begingroup$ If $X$ is projective, it is compact in the analytic topology. Maybe you meant quasi-projective? anyway it is not really important for the question here. $\endgroup$– HenriCommented Jan 12, 2017 at 19:13
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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