Geometry of Albanese image Let $X$ be a compact Kähler manifold and $alb \colon X \to \mathrm{Alb}(X)$ be the Albanese morphism. I am interested in a number of questions about relations between the geometry of $X$ and the geometry of $alb$. There is a number of more or less obious considerations (such as: $alb$ induces isomorphisms on $H^1$ and $Pic^0$; its fibers contain each point together with its rational connected component, etc.) However, I can't find even particular answers to plenty of natural questions. For example here are the simplest: 


*

*When is the Albanese map surjective?

*And when is it injective?

*When is the image smooth? I've heard that it can be singular, though I am not able to construct a counter-example nor to find it in the literature.
I suspect that no simple answers can be giving to these questions, but I'd be glad to hear any necessar and/or sufficient conditions on  $X$ for the questions above. 
As I have mentioned, these are only the simplest questions and I am interested in any non-trivial results on Albanese mappings.
 A: The first natural observation is the following. Set as usual $q(X):=h^{1, \,0}(X)$, so that $q(X)= \dim  \mathrm{Alb}(X)$. Then  the Albanese map of $X$ cannot be surjective if $\dim X < q(X)$, and it cannot be injective if $\dim X > q(X)$. 
Moreover, the Albanese map is never injective if $X$ contains some rational curve, because any map from $\mathbb{P}^1$ to an abelian variety is necessarily constant.
Let me now give an example related to your third question, showing that the answer can be quite subtle in general.
Let $C$ be a smooth curve of genus $3$ and let $X:= \mathrm{Sym}^2(C)$ be its second symmetric product. Then $X$ is a smooth, minimal surface of general type with $p_g=q=3$, $K^2=6$.
Therefore $\mathrm{Alb}(X)$ is an abelian threefold, and we can show that the Albanese map $$a_X \colon X \longrightarrow \mathrm{Alb}(X)$$ is a birational morphism onto its image $\Sigma \subset \mathrm{Alb}(X)$, which is a principal polarization.
Moreover:


*

*if $C$ is non-hyperelliptic then $a_X$ is an immersion and so $\Sigma$ is smooth; 

*if $C$ is hyperelliptic then $a_X$ contracts the unique
$(-2)$-curve in $X$ corresponding to the $g_2^1$ on $C$; in this
case, $\Sigma$ has an ordinary double point as its unique
singularity.

A: Let me give an answer to your first question.
Let $\pi: X \to Y$ be a dominant morphism of smooth projective varieties over $\mathbf C$ with connected fibers; $E$ a prime divisor on $Y$. The multiplicity of $\pi$ along $E$ is defined by
$$
m(E) \overset{\text{def}}= \inf\{m_j\}, \quad \pi^\ast(E) = \sum_j m_j D_j, 
$$
and
$$
\Delta_\pi \overset{\text{def}}= \sum_i \left(1 - \frac1{m(E_i)}\right) E_i
$$
is called the multiplicity divisor associated to $\pi$.
Let $\pi: X \dashrightarrow Y$ be a dominant rational map of smooth projective varieties over $\mathbf C$ with connected fibers. The Kodaira dimension of $\pi$, denoted by $\kappa(\pi)$, is defined to be $\inf\{\kappa(Y^\prime, K_{Y^\prime} + \Delta_{\pi^\prime})\}$, where $\pi^\prime: X^\prime \to Y^\prime$ is taken over all dominant morphisms such that there exist birational maps $u: X \dashrightarrow X^\prime$ and $v:Y \dashrightarrow Y^\prime$ satisfying $\pi^\prime \circ u = v \circ \pi$.
Let $\pi: X \dashrightarrow Y$ be a dominant rational map of smooth projective varieties over $\mathbf C$ with connected fibers. $\pi$ is said to be of general type if $\kappa(\pi) = \dim(Y)$.
Let $X$ be a smooth projective variety over $\mathbf C$. $X$ is said to be special if there is no dominant rational map of general type with connected fibers from $X$ to any smooth projective variety $Y$ with $\dim(Y) > 0$.
Theorem. If $X$ is special, then the Albanese morphism $\alpha: X \to A$ is dominant with connected fibers and $\Delta_\alpha = 0$.
Proof. [CAM] Proposition 5.3.
Theorem. If $X$ is rationally connected, then $X$ is special (but of course the Albanese morphism is trivial in this case).
Theorem. If $\kappa(X) = 0$, then $X$ is special.
Theorem. If $-K_X$ is nef, then $X$ is special.
Theorem. If $X$ is special, then any finite étale covering of $X$ is also special.
Theorem. For any $n > 0$ and $\kappa \in \{-\infty, 0, \dots, n - 1\}$, there exists a special variety with dimension $n$ and Kodaira dimension $\kappa$.
Conjecture. $X$ is special if and only if the Kobayashi pseudo-metric on $X$ is trivial.
Conjecture. If $X$ is defined over a number field $K$, then $X_{\mathbf C}$ is special if and only if $X(L)$ is Zariski dense for some finite extension $L \mid K$.
[CAM] Frédéric Campana. Orbifolds, Special Varieties and Classification Theory.
