What is the Albanese map good for?
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
The following structural definition(sometimes called Hard 's theorem) which is due to Serre and Iitaka, gives us the main motivations of the study of Albanese map and quasi-Albanese map.
Definition: Let $X$ be a smooth projective variety. Then,
1) we denote by $P_m(X) = h^0(X, mK_X)$ the $m$-th plurigenera of $X$, $q(X) =h^1(X, \mathcal O_X)$ the irregularity of $X$, and $$\kappa(X) = \kappa(X, K_X)=\limsup_{m\to \infty}\frac{\log h^0(X,K_X^{m})}{\log m}$$ the Kodaira dimension of $X$.
2) there exists a morphism $a_X : X \to Alb(X)$ from $X$ to an abelian variety called the Albanese morphism of $X$ satisfying the following universal property: for any morphism $f : X \to A$ from $X$ to an abelian variety $A$, there exists a morphism $g : Alb(X) \to A$ such that the following diagram commutes:
$$ \matrix{ X & \xrightarrow{a_X} & Alb(X)\\ & f \searrow & \downarrow{ g} \\ & & A } $$
3) we call a morphism $f : X \to Y$ from $X$ to a smooth projective variety $Y$ an algebraic fiber space, if $f$ is surjective and $f_*\mathcal O_X = \mathcal O_Y$ .
We say that $X$ is irregular if it admits a non-constant morphism to an abelian variety, or equivalently if $q(X)> 0$.
We denote by $$Alb(X) := H^0(X;\Omega^1_X)^\vee/H_1(X;Z)$$ the Albanese variety of $X$, and by $alb_X : X \to Alb(X)$ the Albanese map of $X$ defined via integration of holomorphic one-forms. We note that $Alb(X)$ is an abelian variety of dimension $q(X)$ and that $alb_X $ induces an isomorphism $H^0(X;\Omega_X^1) \simeq H^0(Alb(X);\Omega^1_{Alb(X)})$. Furthermore, we denote by $\dim alb_X(X)$ the Albanese dimension of $X$ which is the dimension of the image of the Albanese map. Finally, we say that $X$ is of maximal Albanese dimension if $\dim alb_X(X) = \dim X,$ or equivalently if $alb_X$ is generically finite onto its image.
Fix a point $p$ in $X$. Let $c_x$ be a path joining $p$ to a point $x \in X$, and let $a(c_x)$ be the linear form $\omega\to\int_{c_x}\omega $. If we replace $c_x$ by another path $c_x'$ joining $p$ to $x$, we change $a(c_x)$ by an element of $H_1(X,\mathbb Z)$. Thus the class of $a(c_x)$ depends only on $x$. We define this class to be $alb(x)$.
It was a longstanding question that "the Albanese map is indeed a fibration," i.e. surjective and with connected fibres. But this has been recently settled. The second fundamental problem is the existence of Albanese map.
Let me start with the following conjecture which some years ago was solved:
Conjecture: Let $X$ be a compact Kahler manifold with nef Ricci class. Then the Albanese map of $X$ is surjective. In 2012 Paun proved this conjecture
Cao in 2013 showed that: Let $X$ be a compact Kahler manifold with nef anticanonical bundle. Then the Albanese map $\alpha_X: X \to Alb(X)$ is a submersion on the complement of the Harder-Narasimhan filtration singular locus in $X$, hence $\alpha_X$ surjects onto $Alb(X)$
We have the following Isotriviality of the Albanese map recently solved by J.Cao
Theorem: Let $X$ be a projective manifold with nef anti-canonical bundle. Then the Albanese map $α_X : X \to Y = Alb(X)$ is locally isotrivial, i.e., for any small open set $U ⊂ Y $, $α^{−1}_X(U) $ is biholomorphic to the product $U × F$, where $F$ is the generic fiber of $α_X $. Moreover $−K_F$ is again nef
The following result proved by Campana et al.
Theorem: Let $X$ be a compact Kahler manifold of dimension three or four such that $−K_X$ is nef. Then the Albanese map $f : X → A(X)$ is surjective and has connected fibers
We have the following result of Paun by using Cheeger-Colding-Tian theory, see
Theorem: Let $(X,\omega)$ be a compact complex manifold and let $(\omega_k)_k$ be a sequence of Kahler metrics on $X$ with the following properties:
i) for each $k > 0;$ the metric $\omega_k$ belongs to the cohomology class $\{\omega\}$
ii) the Ricci curvature of $\omega_k$ is bounded from below by $-1/k$.
iii) the diameters $d_k := diam(X, \omega_k)$ satisfy $d_k/\sqrt{k}\to 0$ as $k $ goes to infinity. Then the Albanese map of $X$ is surjective.
We say a morphism $f : X → Y$ from $X$ to a smooth projective variety $Y$ an algebraic fiber space, if $f$ is surjective and $f_∗\mathcal O_X = \mathcal O_Y $.
We have the following result due to Y. Kawamata
Let $X$ be a smooth projective variety of Kodaira dimension $0$. Then the Albanese morphism $a_X : X → Alb(X)$ is an algebraic fiber space
We have the following result due to Hacon-Pardini-Kollar
Let $X$ be a smooth projective variety. If $0 < P_m(X) ≤ 2m-3$ for some $m ≥ 2$, $a_X : X → Alb(X)$ is surjective. Hear $P_m(X) = h^0(X, mK_X)$ is the $m$-th plurigenera of $X$,
See this thesis for more results
-Let $X$ be an Abelian variety. Then $X ≃ Alb(X)$.
-The Albanese variety is dual to the onnected component of zero of Picard variety
$$\operatorname {Alb}\,X=(\operatorname {Pic}_{0}\,X)^{\vee }.$$
The Albanese variety $\operatorname {Alb}\,(X)$ is of the dimension $q(X) := h^1(X, \mathcal O_X),$
To explain more: Let $J$ be the Picard variety of a normal projective variety $X$ in Matsusaka's sense, and $Ext(J,G_m)$ the group of all isomorphism classes of extensions of $J$ by the multiplicative group $G_m$. There is a natural isomorphism $Ext(J,G_m)→P(J)$, as an extension can be thought of as a principal $G_m$-bundle on $J$. Now, let $φ:X→A$ be the Albanese map. Then $A$ can be identified with the Picard variety $P(J)$ of $J$ by duality. This is due to J.-P. Serre
Fujiki studied the relative Albanese map. It will be nice to extend the Paun result in the relative setting.
Let me explain the Fujiki theorem here.
Fujiki's theorem: Let $f:X\to Y$ be a fiber space of compact complex manifolds, let $U⊂Y$ be a Zariski open subset such that $X_U\to U$ is smooth and $XU:=f^{-1}(U)$. For each $y\in U$ there is the Albanese map $ψ_y:X_y\to Alb(X_y)$, where $X_y:=f^{-1}(y)$. If $X_y$ is in the Fujiki class $\mathcal C$, then $\{AlbX_y:y\in U\}$ is a complex manifold $Alb(X_U)/U$ over $U$
Ueno conjecture: The Albanese map $α$ is birationally equivalent to an étale fiber bundle with fiber $F$ of Kodaira dimension zero. If $dimX=2$, this conjecture is, in fact, a theorem which follows from the complete classification of surfaces. As for the case $dimX≥3$, Conjecture is known to be true for varieties X satisfying the inequality $q(X)≥dimX-2$, where $q(X)$ denotes the irregularity of $X$.
-Viehweg showed the integer $\text{var}(α)$, i.e., the variation of the Albanese map $α$, is zero. Hacon showed that the Kodaira dimension of a general fibre of $a_X$ is zero.
-The existence of the Albanese map is rather remarkable in itself.
Let $X\to Y$ be an algebraic fibre space, in the sense that it is a surjective morphism between smooth complex projective varieties which has connected general fibre $F$. If $dimX=n$ and $dimY=m$, then the fundamental conjecture $C_{n,m}$ of Iitaka states that $κ(X)\geq κ(Y)+κ(F)$. The variety $X$ is said to be of maximal Albanese dimension if $dimAlb(X)=dimX$. Iitaka's conjecture when $Y$ is of maximal Albanese dimension holds true due to recent result of Hacon et al.
Campana and Fujiki introduced and studied the relative version of the Albanese map.
A relative Albanese map for $f : X\to Y$ is a commutative diagram of compact complex manifolds and holomorphic maps
$$ \matrix{ X & \xrightarrow{g} & Alb(X/Y)\\ & f \searrow & \downarrow{ A} \\ & & Y } $$
where $A : Alb(X/Y ) \to Y$ is a smooth submersion with fiber $A^{-1}(y)\cong Alb(X_y)$ the Albanese torus of $X_y = f^{−1}(y)\cong F$, and with $g|_{X_y}:X_y\to A^{-1}(y)$ isomorphic to the Albanese map $A_y : X_y \to Alb(X_y)$ (for all $y\in Y$ )
Now we explain quasi-Albanese map on log pair $(X,D)$.
Let $V$ be an algebraic variety. By Nagata, there is a complete algebraic variety $\tilde V$ containing $V$ as a Zariski-dense open subvariety. Then by Hironaka, there exists a log smooth pair $(X, D)$, where $X$ is a smooth projective variety and $D$ is a reduced effective divisor with only simple normal crossing singularities, and a projective birational morphism $\pi: X\to\tilde V$ such that $D = \pi^{-1}(\bar V \setminus V )$ and $X\setminus D=\pi^{-1}(V)$. Such pair is called a log smooth completion of $V$. We then define the logarithmic irregularity $\bar q(V ) := h^0(X;\Omega_X(\log D))$ where $\Omega_X(\log D)$ is the logarithmic differential sheaf. Also we define the logarithmic Kodaira dimension $\bar\kappaκ(V ) := \kappa (X, K_X + D)$.
(Quasi-abelian varieties). Let $G$ be a connected algebraic group. Then we have the Chevalley decomposition $$1 \to \mathcal G \to G \to \mathcal A \to 1$$ in which $\mathcal G$ is the maximal affine algebraic subgroup of $G$ and $\mathcal A$ is an abelian variety. If $\mathcal G$ is an algebraic torus $\mathbb G_m^d$ of dimension $d$, then $G$ is called a quasi-abelian variety. For the quasi-Albanese map we denote by $\alpha_V:V\to \mathcal A_V$ where $ \mathcal A_V$ is the quasi-Albanese variety which is the quasi-Abelian variety.
Let $V$ be a smooth algebraic variety with some log smooth completion $(X, D)$ obtained by blowing up subvarieties of the boundary such that $V = X\setminus D$. Then the Albanese varieties of $V$ and $X$ are isomorphic to each other and the Albanese morphism $alb_V$ of $V$ is just the restriction of the Albanese morphism $alb_X$ of $X$ . We have $\dim \mathcal A_V = \bar q(X)$ and $\dim Alb(V ) = q(X) := h^1(X, \mathcal O_X )$. If we assume further that $V$ is projective, then the quasi-Albanese morphism $\alpha_V$ of $V$ is just the original Albanese morphism $alb_V$ of $V$
We quickly recall the important properties of the Albanese map.
Some properties of Albanese map:
The Albanese variety is functorial in nature. If $f : X \to Y$ is a morphism of smooth projective varieties, there is a unique morphism $F : Alb(X) \to Alb(Y )$ such that the diagram commute.
$\require{AMScd}$ \begin{CD} X @>f>> Y \\ @V a_X VV= @VV a_{Y} V\\ Alb(X) @>>F> Alb(Y) \end{CD}
Note that, If $X$ is an abelian variety then $Alb(X) = X$.
For a projective variety $X$ , we take a resolution $\tilde X\dashrightarrow X$ and define the Albanese map $$alb_X : X \to Alb(X ) := Alb(\tilde X )$$ as the natural composition $X\dashrightarrow \tilde X\to Alb(\tilde X)$. It is known that $alb_X$ is a well-defined morphism when $X$ has at worst rational singularities. Ueno showed that for any smooth algebraic variety $X$, the Albanese map $\alpha: X\to Alb(X)$ is surjective if and only if, $\kappa(\alpha(X))=0$. Let $X$ be an irreducible complex algebraic manifold of Kodaira dimension 0; then Kawamata showed that the Albanese map $\alpha:X\to Alb(X)$ is surjective and has connected fibres. For example, let $X$ be a Calabi-Yau manifold, then Albanese map is surjective. Let $X$ be a compact K\"ahler manifold with nef tangent bundle. Then Demailly et al. showed that there exists an etale finite cover $\tilde X$ such that the Albanese mapping $\alpha:\tilde X\to Alb(\tilde X)$ is a surjective, smooth morphism, every fibre of which is a Fano manifold with nef tangent bundle. Zhang showed that if $X$ is a projective variety and $\Delta$ is an effective $\mathbb Q$-divisor such that the pair $(X, \Delta)$ is log canonical and $-(K_X +\Delta)$ is nef, then the Albanese morphism from any smooth model of $X$ is an algebraic fiber space. However, in the cases of compact K\"ahler manifolds, a new level of sophistication is involved and things become much more difficult. Recently Mihai Paun solved it in non-logarithmic setting by using positivity theory of twisted relative canonical bundle. We have the surjectivity of the quasi-Albanese map when $-(K_X +\Delta)$ is nef and $X$ is K\"ahler manifold. We can solve it by extending Paun method on variational Kahler-Ricci flow on pairs $(X_s,\Delta_s)$ by using positivity theory of twisted relative canonical bundle
Let $X$ be a compact K\"ahler manifold with nef Ricci class. and $D$ be a snc divisor on it. Then the quasi-Albanese map of $X\setminus D$ is surjective.
Let $k$ be an algebraically closed field, $char(k)≥0$, and let $X$ be a smooth projective connected variety over $k$. Let $CH_0(X)_0$ be the Chow group of zero cycles of degree zero on $X$, let $Alb(X)$ be the Albanese variety of $X$ and let $a_X:CH_0(X)_0\to Alb(X)$ be the Albanese map for $X$. Roitman's theorem asserts that $a_X$ induces an isomorphism on prime-to-$p$ torsion subgroups.
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:
- 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.
- 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.