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Let $X$ be a normal projective variety and $D$ a prime divisor such that $mD$ is Cartier for some integer $m>0$. Suppose $H^1(X,\mathcal{O}_X)=0$ and $mD|_D\sim 0$.

My questions are the following:

Main Question 1. Is the Iitaka dimension $\kappa(X, D)=\kappa(X, mD)>0$?

To solve above, one may come up with the following question.

Question 2. Is $\mathcal{O}_{mD}(mD)\cong \mathcal{O}_{mD}$?

Question 3. Can we remove the assumption $H^1(X,\mathcal{O}_X)=0$ for the above questions?

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  • $\begingroup$ If the question is too general, one may focus on the case when $X$ is a normal projective surface. $\endgroup$
    – Sheng Meng
    Commented Jun 3, 2019 at 12:03
  • $\begingroup$ What about the case of $P^1 \times P^1$ and D is a fiber of either projection. $\endgroup$
    – meh
    Commented Jun 3, 2019 at 15:34
  • $\begingroup$ @aginensky In your example one has $\kappa(D)=1$ unless I'm mistaken $\endgroup$
    – Henri
    Commented Jun 3, 2019 at 17:21
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    $\begingroup$ If you remove the assumption $H^1(X,\mathcal O_X)=0$, then there is a counter-example due to Demailly-Peternell-Schneider (Ex 1.7 in "Compact complex manifolds with nef tangent bundles"). There, $X$ is a projectivization of a rank 2 vector bundle $E$ over an elliptic curve (a non-trivial extension of two trivial line bundles) and $D$ will be the zero locus of the only section of $\mathcal O_{\mathbb P(E)}(1)$. $\endgroup$
    – Henri
    Commented Jun 3, 2019 at 18:02
  • $\begingroup$ @ Henri, my bad, I misread what is asked. $\endgroup$
    – meh
    Commented Jun 3, 2019 at 18:08

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