# What is the Albanese map good for?

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

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

A crystal lattice is an infinite graph with a diagram in d-space which is periodic with respect to a lattice group action by translations. Kotani and Sunada studied asymptotics of simple random walks on crystal lattices and obtained formulae in terms of (a generalization of) discrete Albanese maps. They suggest the following heuristic statement to summarize their result: "A random walker detects the most natural way for his crystal lattice to sit in space".

The Albanese maps arise here as maps from graphs to graph diagrams in the torus which minimize certain energy functionals. Heuristically, the Albanese maps are thus "ways to take crystal lattices induced by the graph and to nicely seat them in d-space".

The original papers:

1. M. Kotani and T. Sunada, Albanese maps and an off diagonal long time asymptotic for the heat kernel, Comm. Math. Phys. 209 (2000), 633-670.
2. M. Kotani and T. Sunada, Standard realizations of crystal lattices via harmonic maps. Trans. AMS, 353(1) (2001), 1-20.

A survey paper:

• T. Sunada, Discrete geometric analysis. In Geometry on Graphs and Its Applications, Proceedings of symposia in pure mathematics (Vol. 77, pp. 51-86) 2008.
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