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**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.

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.

Reviewer: Cristian D. Gonzales-Aviles (La Serena)