Let \(X\) be a projective variety of dimension \(n\) over an algebraically closed field \(k\), i.e. a reduced projective scheme of finite type over \(k\), not necessarily irreducible or equidimensional. It is well known that, if \(X\) is irreducible and smooth, then there exists an abelian variety \({\text{Alb}}(X)\), the Albanese variety of \(X\), together with a morphism \(\alpha:X\to{\text{Alb}}(X)\) (depending on the choice of a point on \(X\)), which is universal among the morphisms to abelian varieties. The Albanese morphism \(\alpha\) factorizes through \(\text{CH}^{n}(X)_{\text{deg } 0}\), the Chow group of \(0\)-cycles of degree \(0\) on \(X\) modulo rational equivalence. Precisely, \(\alpha=\phi\circ\gamma\), where \(\gamma\) is the natural cycle map from \(X\) to \(\text{CH}^{n}(X)_{\text{deg } 0}\) and \(\phi\) is a regular homomorphism. So \({\text{Alb}}(X)\) can be seen as a universal regular quotient of \(\text{CH}^{n}(X)_{\text{deg } 0}\).
If \(X\) is a curve, \(\phi\) turns out to be an isomorphism and \(\text{CH}^{1}(X)_{\text{deg } 0}\) is also isomorphic to the Picard variety of \(X\): This fact allows to define the generalized Albanese variety of \(X\), \(A^1(X)\), also for singular curves. In this important paper, the authors give the definition of the generalized Albanese variety \(A^n(X)\) for all singular projective varieties \(X\). More precisely, they study both the algebraic case, in which \(k\) is any algebraically closed field, and the case \(k=\mathbb{C}\), in which case they give a construction using transcendental arguments.
In the first case, to introduce \(A^n(X)\), the existence is proved of a smooth connected commutative algebraic group, with a regular homomorphism \(\phi: \text{CH}^{n}(X)_{\text{deg } 0}\to A^n(X)\), such that \(\phi\) is universal among the regular homomorphisms to smooth commutative algebraic groups. In the complex case, \(A^n(X)\) is characterized as the kernel of the natural map from \(D^n(X)\) to \(H^{2n}(X, {\mathbb{Z}}(n))\): Here \(D^n(X)\) denotes the cohomology group \({\mathbb{H}}^{2n}(X,{\mathcal D}(n)_X)\) of the Deligne complex: \[ {\mathcal D}(n)_X=(0 \to {\mathbb{Z}}_X(n) \to {\mathcal O}_X \to \Omega^1_{X|\mathbb{C}} \to \dots \to \Omega^{n-1}_{X|\mathbb{C}} \to 0). \] Moreover the authors prove that, if \(k\) is a universal domain, then \(\phi\) is an isomorphism if and only if the Chow group \(\text{CH}^{n}(X)_{\text{deg } 0}\) is finite dimensional, which means that there exists a positive integer \(m\) such that the natural map from \(S^m(X_{\text{reg}})\) to the Chow group is surjective.