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The Hodge- ${\mathcal D}$-conjecture for $\text{\rm K3}$ and Abelian surfaces

Author(s): Xi Chen; James D. Lewis
Journal: J. Algebraic Geom. 14 (2005), 213-240.
Posted: December 30, 2004
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Abstract | References | Additional information

Abstract: Let $X$ be a projective algebraic manifold, and $\text{CH}^{k}(X,1)$ the higher Chow group, with corresponding real regulator $\text{r}_{k,1}\otimes {{\mathbb R}}: \text{CH}^k(X, 1)\otimes {{\mathbb R}} \to H_{\mathcal D}^{2k-1}(X,{{\mathbb R}}(k))$. If $X$ is a general K3 surface or Abelian surface, and $k=2$, we prove the Hodge- ${\mathcal D}$-conjecture, i.e. the surjectivity of $\text{r}_{2,1}\otimes {{\mathbb R}}$. Since the Hodge- ${\mathcal D}$-conjecture is not true for general surfaces in ${\mathbb P}^{3}$ of degree $\geq 5$, the results in this paper provide an effective bound for when this conjecture is true.

References:

[Bei1]
A. Beilinson, Higher regulators and values of $L$-functions, J. Soviet math. 30, 1985, 2036-2070. MR 0862627 (88f:11060)

[Bei2]
-, Notes on absolute Hodge cohomology, In: Contemp. Math. 55, Part I, AMS, pp. 35-68 (1985). MR 0862628 (87m:14019)

[Blo1]
S. Bloch, Algebraic cycles and higher $K$-theory, Adv. Math. 61, 1986, 267-304. MR 0852815 (88f:18010)

[Blo2]
-, Lectures on Algebraic Cycles, Duke University Mathematics Series IV, Duke University, Mathematics Department, Durham, N.C., 1980. 182 pp. (not consecutively paged). MR 0558224 (82e:14012)

[Blo3]
-, Algebraic cycles and the Beilinson conjectures, Cont. Math. 58 (1) (1986), 65-79. MR 0860404 (88e:14006)

[B-L]
J. Bryan and N.C. Leung, The Enumerative Geometry of K3 surfaces and Modular Forms, J. Amer. Math. Soc. 13 (2000), no. 2, 371-410 (electronic). Also preprint alg-geom/9711031. MR 1750955 (2001i:14071)

[C1]
X. Chen, Rational Curves on K3 Surfaces, J. Alg. Geom. 8 (1999), 245-278. Also preprint math.AG/9804075. MR 1675158 (2000d:14057)

[C2]
-, A simple proof that rational curves on K3 are nodal, Math. Ann. 324 (2002), no. 1, 71-104. Also preprint math.AG/0011190. MR 1931759 (2003k:14047)

[Co1]
A. Collino, Griffiths' infinitesimal invariant and higher $K$-theory on hyperelliptic jacobians, J. Alg. Geom. 6, 1997, 393-415. MR 1487220 (98m:14010)

[Co2]
-, Indecomposable motivic cohomology classes on quartic surfaces and on cubic fourfolds. Algebraic $K$-theory and its applications (Trieste, 1997), 370-402, World Sci. Publishing, River Edge, NJ, 1999. MR 1715883 (2000i:14013)

[EV]
H. Esnault and E. Viehweg, Deligne-Beilinson cohomology, in Beilinson's Conjectures on Special Values of $L$-Functions, (Rapoport, Schappacher, Schneider, eds.), Perspect. Math. 4, Academic Press, 1988, 43-91. MR 0944991 (89k:14008)

[GL]
B. Gordon and J. Lewis, Indecomposable higher Chow cycles on products of elliptic curves, J. Alg. Geometry 8, (1999), 543-567. MR 1689357 (2000d:14003)

[GH]
P. Griffiths and J. Harris, Principles of Algebraic Geometry, John Wiley & Sons, New York, 1978. MR 1288523 (95d:14001)

[G-S]
M. Green and S. Müller-Stach, Algebraic cycles on a general complete intersection of high multi-degree of a smooth projective variety, Comp. Math. 100 (3), 305-309 (1996). MR 1387668 (97b:14008)

[Gr]
P. Griffiths, Periods of integrals on algebraic manifolds, III. Pub. Math. I.H.E.S. 38, 125-180 (1970). MR 0282990 (44:224)

[Ja]
U. Jannsen, Deligne cohomology, Hodge- ${\mathcal D}$-conjecture, and motives, in Beilinson's Conjectures on Special Values of $L$-Functions, (Rapoport, Schappacher, Schneider, eds.), Perspect. Math. 4, Academic Press, 1988, 305-372. MR 0944998 (89h:14016)

[La]
S. Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159-205. MR 0828820 (87h:32051)

[Lev]
M. Levine, Localization on singular varieties, Invent. Math. 91, 1988, 423-464. MR 0928491 (89c:14015a)

[Lw1]
J. D. Lewis, Regulators of Chow cycles on Calabi-Yau varieties, Proceedings of the Field's Institute Workshop, ``Arithmetic, Geometry and Physics around Calabi-Yau Varieties and Mirror Symmetry'', July 22-29, 2001, Fields Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003. MR 2019148

[Md]
D. Mumford, Rational equivalence of $0$-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195-204. MR 0249428 (40:2673)

[MS1]
S. Müller-Stach, Constructing indecomposable motivic cohomology classes on algebraic surfaces, J. Alg. Geom. 6, 1997, 513-543. MR 1487225 (99k:14016)

[MS2]
-, Algebraic cycle complexes, in Proceedings of the NATO Advanced Study Institute on the Arithmetic and Geometry of Algebraic Cycles Vol. 548, (Lewis, Yui, Gordon, Müller-Stach, S. Saito, eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands, (2000), 285-305. MR 1746324 (2001b:14003)

[Mo]
S. Mori and S. Mukai, The uniruledness of the moduli space of curves of genus $11$, Algebraic Geometry (Tokyo/Kyoto, 1982) Lecture Notes in Math.1016, (1993), 334-353, Springer, Berlin, 1983. MR 0726433 (85b:14033)

[No]
M. Nori, Algebraic cycles and Hodge theoretic connectivity, Invent. math. 111, (1993), 349-373. MR 1198814 (94b:14007)

[Sa]
S. Saito, Motives and filtrations on Chow groups, Invent. math. 125, (1996), 149-196. MR 1389964 (97i:14002)

[S-S]
I. I. Pjatecki{\u{\i}}\kern.15em-Sapiro and I. R. Safarevic, A Torelli theorem for surfaces of type K3, Math. USSR Izvestija, Vol. 5, No. 3 (1971), 547-588. MR 0284440 (44:1666)

[RS]
A. Rosenschon and M. Saito, Cycle map for strictly decomposable cycles, Amer. J. Math 125 (2003), 773-790. MR 1993741 (2004g:14013)

[Y-Z]
Yau S.T. and Zaslow E., BPS States, String Duality, and Nodal Curves on K3, Nuclear Physics B 471(3), (1996) 503-512. Also preprint hep-th/9512121. MR 1398633 (97e:14066)

Additional Information:

Xi Chen
Affiliation: 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
Email: xichen@math.ualberta.ca

James D. Lewis
Affiliation: 632 Central Academic Building, University of Alberta, Edmonton, Alberta T6G 2G1, CANADA
Email: lewisjd@gpu.srv.ualberta.ca

PII: S 1056-3911(04)00390-X
Received by editor(s): April 11, 2003
Received by editor(s) in revised form: November 2, 2003
Posted: December 30, 2004
Additional Notes: Both authors were partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada

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