An interesting 0-cycle. (English) Zbl 1058.14014

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 In the present paper, continuing some very recent (unpublished) work of C. Faber and R. Pandharipande, the authors study a canonically defined \(0\)-cycle \(z_K\) on the product \(X= Y\times Y\) of a curve \(Y\) of genus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.


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
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