Albanese varieties with modulus over a perfect field. (English) Zbl 1282.14078

Let \(X\) be a smooth proper variety over a perfect field \(k\) and let \(D\) be an effective divisor on \(X\) with multiplicity. In this paper the author introduces an Albanese variety \(\text{Alb}^{(1)}(X,D)\) associated to the pair \((X,D)\) as a higher-dimensional analog of the generalized Jacobian variety with modulus introduced by Rosenlicht and Serre for smooth proper curves. If \(P\) is a torsor under a commutative algebraic \(k\)-group \(G\) and \(\varphi: X\dashrightarrow P\) is a rational map, the author defines a certain effective divisor \(\text{mod}(\varphi)\) on \(X\), called the modulus of \(\varphi\), which coincides with the classical definition in the curve case. Then \(\text{Alb}^{(1)}(X,D)\) and the corresponding Albanese map \(\text{alb}_{X,D}^{(1)}: X\dashrightarrow \text{Alb}^{(1)}(X,D)\) are defined by the following universal property: for every torsor \(P\) under a commutative algebraic \(k\)-group \(G\) and every rational map \(\varphi\) from \(X\) to \(P\) of modulus \(\leq D\), there exists a unique homomorphism of torsors \(h: \text{Alb}^{(1)}(X,D)\to P\) such that \(\varphi=h\circ \text{alb}_{X,D}^{(1)}\). To establish the existence of \(\text{alb}_{X,D}^{(1)}\), the author works with a broader notion of generalized Albanese varieties defined by a universal mapping property for categories of rational maps from \(X\) to torsors for commutative algebraic groups. To construct the latter, the author develops a notion of duality for smooth connected commutative algebraic groups over a perfect field via 1-motives with unipotent part, which generalize both Deligne’s and Laumon’s 1-motives. A 1-motive with unipotent part is roughly a homomorphism \((\mathcal F\to G)\), where \(G\) is an extension of an abelian variety \(A\) by a commutative linear group \(L\) and \(\mathcal F\) is a dual-algebraic commutative formal group, i.e., the Cartier dual of \(\mathcal F\) is algebraic. These 1-motives admit duality, e.g., the dual of \((0\to A)\) is \((L^{\vee}\to A^{\vee})\), where \(L^{\vee}\) is the Cartier dual of \(L\) and \(A^{\vee}\) is the abelian variety dual to \(A\). Using these 1-motives, the author obtains explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. The following theorem, from which the existence of \(\text{Alb}^{(1)}(X,D)\) can be deduced, is one of the main results of the paper:
Theorem. Let \(\underline{\text{Div}}^{0,\text{red}}_{X}\) be the (sheaf) pullback of the Picard variety \(\text{Pic}^{0,\text{red}}_{X}\) of \(X\) under the cycle class map and let \(\mathcal F\) be a dual-algebraic formal \(k\)-subgroup of \(\underline{\text{Div}}^{0,\text{red}}_{X}\). By base change to an algebraic closure of \(k\), a rational map \(\varphi: X\dashrightarrow P\) induces a rational map \(\varphi: \overline{X}\dashrightarrow \overline{G}\) which in turn induces a natural transformation \(\tau_{\overline{\varphi}}: \overline{L}^{\vee}\to \underline{\text{Div}}^{0,\text{red}}_{\overline{X}}\). Let \(M_{\mathcal F}\) be the category of rational maps \(\varphi: X\dashrightarrow P\) such that the image of \(\tau_{\overline{\varphi}}\) lies in \(\overline{\mathcal F}\). Then \(M_{\mathcal F}\) admits a universal object \(\text{alb}_{\mathcal F}^{(1)}: X\dashrightarrow \text{Alb}^{(1)}_{_{\mathcal F}}(X)\), where \(\text{Alb}^{(1)}_{_{\mathcal F}}(X)\) is a torsor for an algebraic group \(\text{Alb}^{(0)}_{_{\mathcal F}}(X)\) which is an extension of the classical Albanese variety \(\text{Alb}(X)\) by \(\mathcal F^{\vee}\) and is dual to the 1-motive \((\mathcal F\to \text{Pic}^{0,\text{red}}_{X})\).
The author further defines a relative Chow group of zero cycles \(\text{CH}_{0}(X,D)\) of modulus \(D\) and shows that \(\text{Alb}^{(1)}(X,D)\) can be viewed as a universal quotient of \(\text{CH}_{0}(X,D)^{\text{deg}\,0}\). Finally, an application is given which rephrases Lang’s class field theory of function fields of varieties over finite fields in explicit terms.


14L10 Group varieties
11G45 Geometric class field theory
14C15 (Equivariant) Chow groups and rings; motives
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