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



> 
> **Question** Let $k=\mathbb{C}$. 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>.



Maximality of Albanese dimension
I think that the varieties you are interested in are called *of maximal Albanese dimension* or *of Albanese general type*.


As remarked in the comments, their Kodaira dimension must be non-negative; however, since the behavior of the Albanese map is related to the $1$-forms and not to the top forms (canonical divisor), there is no clear relation between maximality of the Albanese dimension and maximality of the Kodaira dimension.


For instance, Abelian varieties are of maximal Albanese dimension but not of maximal Kodaira dimension. On the other hand, a surface of general type with irregularity $0 \leq q\leq 1$ is of maximal Kodaira dimension, but not of maximal Albanese dimension.


