Abstract.
I will discuss positive and negative results on the Hodge conjecture. The negative aspects come on one side from the study of the Hodge conjecture for integral Hodge classes, and on the other side from the study of possible extensions of the conjecture to the general Kähler setting. The positive aspects come from algebraic geometry. They concern the structure of the so-called locus of Hodge classes, and of the Hodge loci.
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Y. André, Déformation et spécialisation de cycles motivés, J. Inst. Math. Jussieu, 5 (2006), 563–603.
Y. André, Pour une théorie inconditionnelle des motifs, Inst. Hautes Études Sci. Publ. Math., 83 (1996), 5–49.
M. Artin and D. Mumford, Some elementary examples of unirational varieties which are not rational, Proc. London Math. Soc. (3), 25 (1972), 75–95.
M. F. Atiyah and F. Hirzebruch, Analytic cycles on complex manifolds, Topology, 1 (1962), 25–45.
S. Bando and Y.-T. Siu, Stable sheaves and Einstein–Hermitian metrics, In: Geometry and Analysis on Complex Manifolds, (eds. T. Mabuchi et al.), World Sci. Publ., New Jersey, 1994, pp. 39–50.
S. Bloch and H. Esnault, The coniveau filtration and non-divisibility for algebraic cycles, Math. Ann., 304 (1996), 303–314.
S. Bloch and V. Srinivas, Remarks on correspondences and algebraic cycles, Amer. J. Math., 105 (1983), 1235–1253.
A. Borel and J.-P. Serre, Le théorème de Riemann–Roch, Bull. Soc. Math. France, 86 (1958), 97–136.
J. Carlson and P. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli theorem, In: Géométrie Algébrique, (ed. A. Beauville), Sijthoff and Noordhoff, Angers, 1980, pp. 51–76.
E. Cattani, P. Deligne and A. Kaplan, On the locus of Hodge classes, J. Amer. Math. Soc., 8 (1995), 483–506.
H. Clemens and P. Griffiths, The intermediate Jacobian of the cubic threefold, Ann. of Math. (2), 95 (1972), 281–356.
A. Conte and J. Murre, The Hodge conjecture for fourfolds admitting a covering by rational curves, Math. Ann., 238 (1978), 79–88.
P. Deligne, Théorie de Hodge. II, Inst. Hautes Études Sci. Publ. Math., 40 (1971), 5–57.
P. Deligne, Hodge Cycles on Abelian Varieties (notes by J. S. Milne), Lecture Notes in Math., 900, Springer-Verlag, 1982, pp. 9–100.
P. Deligne, P. Griffiths, J. Morgan and D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math., 29 (1975), 245–274.
S. Druel, Espaces des modules des faisceaux semi-stables de rang 2 et de classes de Chern c 1 = 0, c 2 = 2 et c 3 = 0 sur une hypersurface cubique lisse de \({\mathbb{P}}^4\), Int. Math. Res. Not., 19 (2000), 985–1004.
H. Esnault and K. Paranjape, Remarks on absolute de Rham and absolute Hodge cycles, C. R. Acad. Sci. Paris Sér. I Math., 319 (1994), 67–72.
W. Fulton, Intersection Theory, Ergeb. Math. Grenzgeb. (3), 2, Springer-Verlag, 1984.
Ph. Griffiths, Periods of integrals on algebraic manifolds. I, II, Amer. J. Math., 90 (1968), 568–626, 805–865.
Ph. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2), 90 (1969), 460–541.
A. Grothendieck, Hodge’s general conjecture is false for trivial reasons, Topology, 8 (1969), 299–303.
F. Hirzebruch, Topological Methods in Algebraic Geometry, 3rd ed., Springer-Verlag, 1966.
W. V. D. Hodge, The topological invariants of algebraic varieties, In: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., Aug. 30-Sept. 6, 1950, 1, Amer. Math. Soc., 1952, pp. 182–192.
V. Iskovskikh and Y. Manin, Three dimensional quartics and counterexamples to the Lüroth problem, Math. USSR-Sb., 15 (1971), 141–166.
S. Kleiman, Algebraic cycles and the Weil conjectures, In: Dix esposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968, pp. 359–386.
K. Kodaira, On Kähler varieties of restricted type (an intrinsic characterization of algebraic varieties), Ann. of Math. (2), 60 (1954), 28–48.
J. Kollár, Lemma p. 134, In: Classification of Irregular Vvarieties, (eds. E. Ballico, F. Catanese and C. Ciliberto), Lecture Notes in Math., 1515, Springer-Verlag, 1990.
J. Kollár, Rational Curves on Algebraic Varieties, Ergeb. Math. Grenzgeb. (3), 32, Springer-Verlag, 1996.
J. Kollár, Y. Miyaoka and S. Mori, Rationally connected varieties, J. Algebraic Geom., 1 (1992), 429–448.
Yu. Manin, Correspondences,motifs and monoidal transformations. (Russian), Mat. Sb. (N. S.), 77 (1968), 475–507.
D. Markushevich and A. Tikhomirov, The Abel–Jacobi map of a moduli component of vector bundles on the cubic threefold, J. Algebraic Geom., 10 (2001), 37–62.
H.-W. Schuster, Locally free resolutions of coherent sheaves on surfaces, J. Reine Angew. Math., 337 (1982), 159–165.
J.-P. Serre, Faisceaux algébriques cohérents, Ann. of Math. (2), 61 (1955), 197–278.
J.-P. Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble), 6 (1956), 1–42.
C. Soulé and C. Voisin, Torsion cohomology classes and algebraic cycles on complex projective manifolds, Adv. Math., special volume in honor of Michael Artin, Part I, 198 (2005), 107–127.
B. Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc., 10 (1997), 467–493.
K. Uhlenbeck and S.-T. Yau, On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Comm. Pure Appl. Math., 39 (1986), 257–293.
C. Voisin, A counterexample to the Hodge conjecture extended to Kähler varieties, Int. Math. Res. Not., 2002 (2002), 1057–1075.
C. Voisin, Hodge Theory and Complex Algebraic Geometry. I, II, Cambridge Stud. Adv. Math., 76, 77, Cambridge Univ. Press, 2003.
C. Voisin, On integral Hodge classes on uniruled and Calabi–Yau threefolds, In: Moduli Spaces and Arithmetic Geometry, Adv. Stud. Pure Math., 45, 2006, pp. 43–73.
C. Voisin, Hodge loci and absolute Hodge classes, Compos. Math., 143 (2007), 945–958.
S. Zucker, The Hodge conjecture for cubic fourfolds, Compos. Math., 34 (1977), 199–209.
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Communicated by: Hiraku Nakajima
This article is based on the 1st Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on November 25 and 26, 2006.
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Voisin, C. Some aspects of the Hodge conjecture. Jpn. J. Math. 2, 261–296 (2007). https://doi.org/10.1007/s11537-007-0639-x
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DOI: https://doi.org/10.1007/s11537-007-0639-x