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.



Log resolution of a variety of log general type
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.


