# quotient singularities

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

• 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. – Henri Jan 12 '17 at 17:48
• relatively compact means its closure be compact. Can you give a reference? – pickasa Jan 12 '17 at 18:05
• 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. – 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
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