I am new to the Albanese map so am not sure about its properties.

QuestionLet $k=\mathbb{C}$. Suppose $X$ smooth projective variety, let $\alpha$ be the Albanese map of $X$. Is there a descripition of $X$ such that $\dim(X)=\dim(α(X))$?

Any remarks or references would be appreciated.

Cross-Posted on Stack-Exchange https://math.stackexchange.com/questions/4400133/equidimensional-albenese-map.

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