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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?

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I'd also like to know the answer to this. – Michael Albanese Oct 1 '12 at 7:55
up vote 12 down vote accepted

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

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I see. The vanishing of $H^0(X,\Omega_{X}^1)$ a good criterion not to have any non-trivial map to complex torus. I will check the Beauvill's book, too. Thanks a lot! – M. K. Oct 1 '12 at 18:00

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