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

  • 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 quasi-projective? 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 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,



Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.