I am reading a textbook on complex manifolds and come across the Albanese map. For a compact Kähler manifold $X$,
$$
T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z})
$$
is a complex torus, called the Albanese torus of $X$. Fix a point $p\in X$, one obtains so called the Albanese map $\phi:X\rightarrow T$ via
$$
q\mapsto [\alpha \mapsto \int_{p}^{q}\alpha],
$$
where $\alpha$ is an element of $H^0(X,\Omega_{X}^1)$ and the value $\int_{p}^{q}\alpha$ is defined up to "cycles" $H_1(M,\mathbb{Z})$. As usual, this map satisfies certain universal properties.
The construction is easy but abstract. I now would like to know how the Albanese map is used. Are there any good applications of the Albanese map?
Best Answer
The main virtue of the Albanese variety is its universal property: given any compact torus $A$, any morphism $X\to A$ factors uniquely through $T=Alb(X)$.
The easiest application of Albanese varieties I can think of is that if $H^0(X,\Omega^1_X)=0$, then every morphism $X\to A$ from $X$ into a compact torus $A$ is constant: indeed it must factor through $T=H^0(X,\Omega_{X}^1)^*/H_1(M,\mathbb{Z})$, which is just a point if $H^0(X,\Omega^1_X)=0$.
This applies in particular to $\mathbb P^n_\mathbb C$, whose holomorphic maps into compact tori are thus all constant.
A more sophisticated use of Albanese varieties is in the proof that any non ruled projective surface has a unique minimal model: see Beauville's book, theorem V.19