# Question about definition of non-compact Calabi-Yau manifolds

Using the following definition:

Definition $(X, J, \omega, \Omega)$ is a Calabi-Yau manifold if $g(\cdot, \cdot)= \omega(\cdot, J \cdot)$ and $\Omega$ is a nowhere vanishing holomorphic $(n,0)$-form satisfying $\Omega \wedge \bar{\Omega}= \omega^n/n!$.

My questions are:

1) Does this mean the symplectic form completely determines $\Omega$? Sorry if this is obvious

3) I've also seen a definition where $\Omega \wedge \bar{\Omega}= \omega^n/n!$ (or something very similar) only needs to be satisfied asymptotically. Is this is an important distinction/ would that mean Ricci curvature in no longer vanishing?