Let $f:X \to Y$ be a surjective morphism of smooth projective varieties, $D$ be a simple normal crossings divisor on $X$ and $U_Y \subset Y$ be an open subset over which $(X,D)$ is log smooth (in the sense of the restriction of $f$ to any stratum is smooth over $U$). Then, I think by definition (if I was not mistaken about the definition), denoting $U=f^{-1}U_Y$, the morphism $f|_U:(U,D|_U) \to U_Y$ is toroidal.

Do we have modifications $\pi: X' \to X,\phi: Y' \to Y$ and an induced morphism $f':(U' \subset X') \to (U_{Y'} \subset Y')$

which satisfies:

1 $f':(U' \subset X') \to (U_{Y'} \subset Y')$ is toroidal.

2 $\pi^{-1}D \subset X' \backslash U'$.

3 $\phi: U_{Y'} \to U_Y, \pi : f'^{-1} U_{Y'} \to U$ are isomorphisms.

Roughly speaking, can we toroidalize varieties and morphisms only on ``non-toroidal" places?



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