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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
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