Motives under de-singularization

MathOverflow Asked by user161697 on January 1, 2022

Let $X$ be a singular variety over a field $k$ of characteristic 0.

Suppose a minimal resolution of singularities of $X$, $f:tilde{X} rightarrow X$, exists, i.e., $tilde{X}$ is a smooth projective variety, $f$ is a proper birational morphism and every resolution of singularities of $X$ factors through $f$.

What is the most one can say about the relationship between the motives and motivic cohomology of $X$ and $tilde{X}$?

A simple illustrative example would be very helpful as would any references.

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