Let me recall some quick definitions. A projective hyperkahler manifold is a simply connected smooth projective variety $M$ such that $H^0(M,\Omega_M^2)=\mathbb C\sigma$, with $\sigma$ an everywhere non-degenerate holomorphic 2-form.

A normal projective variety $M$ is said to have $\textit{symplectic singularities}$ if the smooth part $M_{reg}$ admits a symplectic 2-form $\omega$ such that for any resolution $\pi:\tilde{M}\rightarrow M$, $(\pi|_{\pi^{-1}(M_{reg})})^*\omega$ extends to a holomorphic 2-form on all of $\tilde{M}$.

I've seen it stated that given a hyperkahler manifold $M$ and a dominant rational map $M \dashrightarrow \overline{M}$, with $\overline{M}$ a normal projective variety with symplectic singularities, then $\dim M=\dim \overline{M}$. Moreover, supposedly this follows from the definitions.

I was wondering why this fact is true?