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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.

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    $\begingroup$ What do you mean by "description"? This includes all subvarieties of abelian varieties, there is of course no hope to classify them in any sense. $\endgroup$
    – abx
    Commented Mar 10, 2022 at 19:12
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    $\begingroup$ There are restrictions on the numerical invariants. The Kodaira dimension is nonnegative. Moreove, the Hodge number $h^{p,0}(X)$ is at least positive as $\binom{n}{p}$ for all $p$ equal to $0,\dots, n = \text{dim}(X)$. $\endgroup$ Commented Mar 10, 2022 at 21:15
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    $\begingroup$ Related discussion: math.stackexchange.com/q/2576878 $\endgroup$
    – AG learner
    Commented Mar 11, 2022 at 3:31
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    $\begingroup$ If $X\subset A$ is a closed subvariety of an abelian variety $A$, then Ueno proved several structure theorems which you can view as part of a "classification". For example, the stabilizer of $X$ is an algebraic subgroup. Modding out its connected component gives the Ueno fibration $X\to Y$ inside $A\to A/Stab(X)$. The variety $Y$ is of general type (possibly equal to $X$ itself) and $Y$ is again a closed subvariety of an abelian variety (namely $A/Stab(X)$). Similar results hold for varieties with maximal Albanese dimension by work of Kawamata and Yamanoi. $\endgroup$ Commented Mar 11, 2022 at 8:16
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    $\begingroup$ I took the liberty of changing the title, since "equidimensional map" often means "with fibres of the same dimension". $\endgroup$ Commented Mar 11, 2022 at 8:57

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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.

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