Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture

Albano, Alberto; Katz, Sheldon

doi:10.1090/S0002-9947-1991-1024767-6

Lines on the Fermat quintic threefold and the infinitesimal generalized Hodge conjecture
HTML articles powered by AMS MathViewer

by Alberto Albano and Sheldon Katz

Trans. Amer. Math. Soc. 324 (1991), 353-368

DOI: https://doi.org/10.1090/S0002-9947-1991-1024767-6

PDF | Request permission

Abstract:

We study the deformation theory of lines on the Fermat quintic threefold. We formulate an infinitesimal version of the generalized Hodge conjecture, and use our analysis of lines to prove it in a special case.

References

Infinite generation of the Griffiths group: a local proof

M. Artin, On the solutions of analytic equations, Invent. Math. 5 (1968), 277–291. MR 232018, DOI 10.1007/BF01389777

Homological modulo algebraic equivalence is not finitely generated

58

-modules and subcanonical deformations

William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460–495; 90 (1969), 496–541. MR 0260733, DOI 10.2307/1970746

Periods of integrals on algebraic manifolds

90

Joe Harris, Galois groups of enumerative problems, Duke Math. J. 46 (1979), no. 4, 685–724. MR 552521
Sheldon Katz, On the finiteness of rational curves on quintic threefolds, Compositio Math. 60 (1986), no. 2, 151–162. MR 868135
Sheldon Katz, Degenerations of quintic threefolds and their lines, Duke Math. J. 50 (1983), no. 4, 1127–1135. MR 726321, DOI 10.1215/S0012-7094-83-05048-2
Sheldon Katz, Tangents to a multiple plane curve, Pacific J. Math. 124 (1986), no. 2, 321–331. MR 856166
K. Kodaira, L. Nirenberg, and D. C. Spencer, On the existence of deformations of complex analytic structures, Ann. of Math. (2) 68 (1958), 450–459. MR 112157, DOI 10.2307/1970256
K. Kodaira, On stability of compact submanifolds of complex manifolds, Amer. J. Math. 85 (1963), 79–94. MR 153033, DOI 10.2307/2373187
A. Ogus, Griffiths transversality in crystalline cohomology, Ann. of Math. (2) 108 (1978), no. 2, 395–419. MR 506993, DOI 10.2307/1971182
Ziv Ran, Cycles on Fermat hypersurfaces, Compositio Math. 42 (1980/81), no. 1, 121–142. MR 594486
Miles Reid, The moduli space of $3$-folds with $K=0$ may nevertheless be irreducible, Math. Ann. 278 (1987), no. 1-4, 329–334. MR 909231, DOI 10.1007/BF01458074
Tetsuji Shioda, The Hodge conjecture for Fermat varieties, Math. Ann. 245 (1979), no. 2, 175–184. MR 552586, DOI 10.1007/BF01428804
Tetsuji Shioda, Algebraic cycles on hypersurfaces in $\textbf {P}^N$, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 717–732. MR 946254, DOI 10.2969/aspm/01010717
Gang Tian, Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric, Mathematical aspects of string theory (San Diego, Calif., 1986) Adv. Ser. Math. Phys., vol. 1, World Sci. Publishing, Singapore, 1987, pp. 629–646. MR 915841