The one-motif of an algebraic surface. (English) Zbl 0629.14027

Carlson, James A.

Compos. Math. 56, 271-314 (1985).

Let X be a singular projective algebraic complex variety, and let \(X_{\bullet}\) be a smooth projective semisimplicial resolution of X. The author defines (by geometrical methods) the trace homomorphism \(\tau_ X: NS(X_{\bullet})\to P(X),\) where \(NS(X_{\bullet})\) is the Néron-Severi group and P(X) is a certain torus. For example, if X consists of two smooth surfaces A and B meeting transversely in the curve C, then P(X) specializes to \(Pic^ 0(C)\) and the trace is induced by the map which transforms the divisor \(Z=Z_ A+Z_ B\) on \(A\cup B\) into the zero-cycle \(C\cdot Z_ A-C\cdot Z_ B\). Then, by cohomological methods, the author defines the group \(L^ 1H^ 2(X_{\bullet})\), \(J^ 1H^ 2(X_{\bullet})\) and the motivic homomorphism \(\eta_ x: L^ 1H^ 1(X_{\bullet})\to J^ 1H^ 2(X_{\bullet}).\)
Main result: there is a natural isomorphism between the trace and motivic homomorphism modulo torsion in \(NS(X_{\bullet})\). - As typical application, the author proves the following Torelli theorem: Let X be the union of a smooth cubic surface and a plane, tangent in at most one point. Then X is determined up to isomorphism by the polarized mixed Hodge structure on the primitive cohomology of \(H^ 2(X)\).

Reviewer: A.M.Šermenev

MSC:

14C22	Picard groups
14A20	Generalizations (algebraic spaces, stacks)
14C30	Transcendental methods, Hodge theory (algebro-geometric aspects)

Keywords:

Néron-Severi group; motivic homomorphism; Torelli theorem; polarized mixed Hodge structure

References:

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