zbMATH — the first resource for mathematics

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 In the present paper, continuing some very recent (unpublished) work of C. Faber and R. Pandharipande, the authors study a canonically defined \(0\)-cycle \(z_K\) on the product \(X= Y\times Y\) of a curve \(Y\) of genus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
Full Text: DOI Euclid
[1] A. Ching, On the generalized Noether-Lefschetz locus , · Zbl 0776.90007
[2] H. Esnault and K. H. Paranjape, Remarks on absolute de Rham and absolute Hodge cycles , C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 67–72. · Zbl 0833.14013
[3] M. L. Green, Griffiths’ infinitesimal invariant and the Abel-Jacobi map , J. Differential Geom. 29 (1989), 545–555. · Zbl 0692.14003
[4] M. Green and P. Griffiths, Hodge-theoretic invariants for algebraic cycles , Internat. Math. Res. Notices 2003 , no. 9, 477–510. \CMP1 951 543 · Zbl 1049.14002
[5] M. Green, J. Murre, and C. Voisin, Algebraic Cycles and Hodge Theory (Torino, Italy, 1993) , Lecture Notes in Math. 1594 , Springer, Berlin, 1994.
[6] D. Mumford, Rational equivalence of \(0\)-cycles on surfaces , J. Math. Kyoto Univ. 9 (1968), 195–204. · Zbl 0184.46603
[7] M. V. Nori, Algebraic cycles and Hodge-theoretic connectivity , Invent. Math. 111 (1993), 349–373. · Zbl 0822.14008
[8] M. Saito, Mixed Hodge complexes on algebraic varieties , Math. Ann. 316 (2000), 283–331. · Zbl 0976.14011
[9] S. Saito, “Motives, algebraic cycles and Hodge theory” in The Arithmetic and Geometry of Algebraic Cycles (Banff, Alberta, 1998) , CRM Proc. Lecture Notes 24 , Amer. Math. Soc., Providence, 2000, 235–253. · Zbl 0976.14004
[10] V. Srinivas, Gysin maps and cycle classes for Hodge cohomology , Proc. Indian Acad. Sci. Math. Sci. 103 (1993), 209–247. · Zbl 0816.14003
[11] C. Voisin, “Transcendental methods in the study of algebraic cycles” in Algebraic Cycles and Hodge Theory (Torino, Italy, 1993) , Lecture Notes in Math. 1594 , Springer, Berlin, 1994, 153–222. · Zbl 0832.14004
[12] –. –. –. –., Variations de structure de Hodge et zéro-cycles sur les surfaces générales , Math. Ann. 299 (1994), 77–103. · Zbl 0837.14007
[13] –. –. –. –., Some results on Green’s higher Abel-Jacobi map , Ann. of Math. (2) 149 (1999), 451–473. JSTOR: · Zbl 1053.14006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.