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 In the present paper, continuing some very recent (unpublished) work of C. Faber and R. Pandharipande, the authors study a canonically defined $$0$$-cycle $$z_K$$ on the product $$X= Y\times Y$$ of a curve $$Y$$ of genus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

##### 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
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