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Chow groups are finite dimensional, in some sense. (English) Zbl 1067.14006
Let $$A_{*}(X)$$ denote the Chow group of $$*$$-dimensional cycles of an algebraic variety $$X$$ with rational coefficients and $$A_{*}(X)_{\mathbb Z}$$ be the corresponding Chow group with integer coefficients. For a curve $$C$$, the degree $$0$$ part of $$A_{0}(C)_{\mathbb Z}$$ is represented by the Jacobian variety which is a finite dimensional object. This is no longer the case for the surfaces. D. Mumford [J. Math. Kyoto Univ. 9, 195–204 (1968; Zbl 0184.46603)] proved that if $$S$$ is a surface with $$p_{g}(S)>0$$ then the group $$A_{0}(S)$$ cannot be represented by a finite dimensional variety. The author suggests a new definition of “finite dimensionality” of Chow groups. Let $$K$$ denote the kernel of the Albanese map $$\text{alb}: A_{0}(S)\rightarrow \mathbb Z\oplus \text{Alb}(S).$$ Let $$S$$ be the product of two smooth nonrational curves. Although $$K$$ is infinite dimensional in the sense of Mumford the author proves the following
Theorem 1.2. Assume that the base field is algebraically closed. Let $$c_{1},\dots c_{n}$$ be cycles in the Albanese kernel in $$A_{0}(X),$$ where $$X$$ is a product of two curves. When $$n$$ is larger than $$4p_{g}(x)$$, then $$c_{1}\wedge\cdots\wedge c_{n}=0.$$ Here $$c_{1}\wedge\cdots\wedge c_{n}=\sum_{\sigma\in S_n}\frac{{\text{sgn}}(\sigma)}{n!} c_{\sigma (1)}\times \dots \times c_{\sigma (n)}\in A_{*}(X\times \cdots \times X ).$$ The theorem shows that there are finitely many “linearly independent” cycles in $$K.$$ For a curve $$C$$ the author proves the following theorems.
Theorem 8.5. Let $$C$$ be a smooth projective curve. For any $$n>0$$, there exist $$0$$-cycles $$c_{1},\dots ,c_{n}$$ in $$A_{*}(C)$$ such that $$c_i$$’s have degree $$0$$ and $$c_{1}\wedge\cdots\wedge c_{n}\neq 0$$
Theorem 1.5. Let $$C$$ be a smooth projective curve and $$n>2g(C).$$ If $$c_{1},\dots ,c_{n}\in A_0 (C)$$ are $$0$$-cycles with degree $$0$$ then $$\text{Sym}( c_{1},\dots ,c_{n})=0.$$
For algebraic cycles $$c_{1},\dots ,c_{n}\in A_* (X)$$, $$\text{Sym}( c_{1},\dots ,c_{n})= \sum_{\sigma\in S_n}\frac{1}{n!} c_{\sigma (1)}\times \dots \times c_{\sigma (n)}$$ holds.

##### MSC:
 14C15 (Equivariant) Chow groups and rings; motives 14C25 Algebraic cycles
##### Keywords:
Chow group; Chow motive
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