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Work over the complex numbers. Let $(B, \Delta)$ be a normal irreducible variety of log general type, i.e., $K_B + \Delta$ is ample. Let $f : (\widetilde{B}, \widetilde{\Delta}) \to (B, \Delta)$ be a log resolution of $(B, \Delta)$, i.e., $\widetilde{B}$ is smooth and $\widetilde{\Delta}$ has simple normal crossing support. I am new to birational geometry, and would like to ask:

Question: What can we say about the pair $(\widetilde{B}, \widetilde{\Delta})$? Is it still of log general type?

Please let me know if the question needs further clarification or is not well formulated.

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    $\begingroup$ If $K_{\tilde B}+\tilde \Delta =f^*(K_B+\Delta )+E$ where $E$ is effective and exceptional, then $h^0(m(K_{\tilde B}+\tilde \Delta)) =h^0(m(K_B+\Delta ))$ for all $m\geq 0$ hence also the Kodaira dimensions agree. In particular this works if $(B,\Delta )$ is log canonical and $\tilde \Delta$ is the strict transform of $\Delta$ plus the reduced exceptional divisor. $\endgroup$ Commented Sep 21, 2020 at 3:51
  • $\begingroup$ @Hacon Thank you, if you would like to post this as an answer, I will be happy to accept it. $\endgroup$ Commented Sep 21, 2020 at 5:39

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If $K_{\tilde B}+\tilde \Delta=f^*(K_B+\Delta)+E$ where $E$ is effective and exceptional, then $h^0(m(K_{\tilde B}+\tilde \Delta))=h^0(m(K_B+\Delta))$ for any $m\geq 0$ and hence also the Kodaira dimensions agree. In particular this works if $(B,\Delta )$ is log canonical and $\tilde \Delta$ is the strict transform of $\Delta$ plus the reduced exceptional divisor.

Note that $K_B+\Delta$ being ample is not important here. However it is true that if $(B,\Delta )$ is klt and of log general type, then it has a canonical model where $K_B+\Delta$ becomes ample.

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