3 Changed Question 2, improved the exposition and fixed some imprecisions.
By an entire curve we I mean a non constant holomorphic map $f\colon\mathbb C\to X$, and analytically degenerate means that there exists a closed analytic subset $Y\subsetneq X$ such that $f(\mathbb C)\subset Y$.

This conjecture has been proven, thanks to the works of (among other people) Ochiai, by Kawamata and, independently, by Wongin the late 70's.

Let $A$ be an abelian variety and $f\colon\mathbb C\to A$ an entire curve. Then, the Zariski closure $\overline{f(\mathbb C)}$ is a translate of a subtorus.

Here More generally, if a subvariety of an abelian variety is my first question:not a translate of a subtorus, then every entire curve in $X$ is analytically degenerate.

Question 1. Is there any geometric characterization or sufficient condition in order to insure that the Albanese variety of an a projective algebraic variety manifold is simple?

Next

Question 2. Is there any geometric characterization or sufficient condition, I would like other than having big irregularity, in order to exploit insure that the links between Bloch's conjecture and image of a projective algebraic manifold via the Green-Griffiths conjecture. The latter states Albanese map is a proper subvariety?

Notice that every entire curve in by the universal property of the Albanese map, if the image of a projective algebraic manifold of general type via the Albanese map is analytically degeneratea proper subvariety then this image is necessarily not a translate of a subtorus.In particular,

N.B. I would like to understand to what extent changed the Bloch conjecture can be thought last part of my post. Now, Question 2 is no more as an evidence toward in the Green-Griffiths conjecture.

Of coursepreviously, having big irregularity does not imply being of general type: take reedited post. In particular the product comment of $\mathbb P^1$ and a genus $g$ curve $C$. Then, $q(\mathbb P^1\times C)=g$ but $\kappa(\mathbb P^1\times C)=-\infty$. So,ulrich refers to my previous Question 2. Let $X$ be a projective manifold. Is there any sufficient condition involving (among possibly other things) its irregularity which insures that $X$ is of general type? What if $X$ is a subvariety of an abelian variety?

2 deleted 5 characters in body

The Bloch conjecture states the following:

Bloch's conjecture. Let $X$ be a compact complex Kähler variety such that the irregularity $q = h^0(X,\Omega^1_X)$ is larger than the dimension $n = \dim X$. Then, every entire curve drawn in $X$ is analytically degenerate.

Here $X$ may be singular and $\Omega^1_X$ can be defined in any reasonable way (direct image of the $\Omega^1_{\widetilde X}$ of a desingularization $\widetilde X$ or direct image of $\Omega^1_U$ where $U$ is the set of regular points in the normalization of $X$). By an entire curve we mean a non constant holomorphic map $f\colon\mathbb C\to X$, and analytically degenerate means that there exists a closed analytic subset $Y\subsetneq X$ such that $f(\mathbb C)\subset Y$.

This conjecture has been proven thanks to the works of (among other people) Ochiai, Kawamata, Wong in the late 70's.

A standard Albanese map argument permits to reduce the conjecture to the following statement:

Let $A$ be an abelian variety and $f\colon\mathbb C\to A$ an entire curve. Then the Zariski closure $\overline{f(\mathbb C)}$ is a translate of a subtorus.

In particular a subvariety of an abelian variety does not have any entire curve (Brody hyperbolicity) if and only if it does not contain any translate of a subtorus. Thus, in a simple abelian variety every subvariety is hyperbolic.

Here is my first question:

Question 1. Is there any geometric characterization or sufficient condition in order to insure that the Albanese variety of an algebraic variety is simple?

Next, I would like to exploit the links between Bloch's conjecture and the Green-Griffiths conjecture. The latter states that every entire curve in a projective manifold of general type is analytically degenerate. In particular, I would like to understand to what extent the Bloch conjecture can be thought as an evidence toward the Green-Griffiths conjecture.