An interesting 0-cycle. (English) Zbl 1058.14014

Green, Mark; Griffiths, Phillip

Duke Math. J. 119, No. 2, 261-313 (2003).

For a smooth complex algebraic variety \(X\), the higher Chow groups CH\(^p(X)\) are principal algebraic invariants. However, for \(p\geq 2\) these groups are non-classical in character and not yet completely understood. For instance, it is still generally difficult to decide whether a given higher-codimension cycle is or is not rationally equivalent to zero. In the present paper, continuing some very recent (unpublished) work of C. Faber and R. Pandharipande, the authors study a canonically defined \(0\)-cycle \(z_K\) on the product \(X= Y\times Y\) of a curve \(Y\) of genus \(g\geq 4\) with itself. Their main result yields the somewhat surprising fact that \(z_K\) is not rationally equivalent to zero when \(Y\) is sufficiently general. This is in contrast to the previously examined cases when \(g= 0,1,2,3\), or when \(Y\) is hyperelliptic, and therefore \(z_K\) is indeed a highly interesting \(0\)-cycle in the higher-genus case. The proof uses infinitesimal and variational methods, and for it the authors introduce a new, very subtle computational method using M. Schiffer’s classical variations. The condition \(g\geq 4\) enters via the properties of tangent lines to canonical curves.

Reviewer: Werner Kleinert (Berlin)

MSC:

14C25	Algebraic cycles
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)
14D07	Variation of Hodge structures (algebro-geometric aspects)
58A14	Hodge theory in global analysis
14C15	(Equivariant) Chow groups and rings; motives

Keywords:

algebraic cycles; zero-cycles; Abel-Jacobi map; Albanese variety; variation of Hodge structures; Schiffer variations

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