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Small codimension components of the Hodge locus containing the Fermat variety

R. Villaflor Loyola

Pontificia Universidad Católica de Chile, Ed. Rolando Chuaqui, Campus San Joaquín Av. Vicuña Mackenna 4860, 7820436, Macul Santiago, Chile

E-mail Address: roberto.villaflor@mat.uc.cl

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https://doi.org/10.1142/S021919972150053XCited by:2 (Source: Crossref)

Abstract

We characterize the smallest codimension components of the Hodge locus of smooth degree $d$ hypersurfaces of the projective space $ℙ^{n + 1}$ of even dimension $n$ , passing through the Fermat variety (with $d \neq 3, 4, 6$ ). They correspond to the locus of hypersurfaces containing a linear algebraic cycle of dimension $\frac{n}{2}$ . Furthermore, we prove that among all the local Hodge loci associated to a nonlinear cycle passing through Fermat, the ones associated to a complete intersection cycle of type $(1, 1, \dots, 1, 2)$ attain the minimal possible codimension of their Zariski tangent spaces. This answers a conjecture of Movasati, and generalizes a result of Voisin about the first gap between the codimension of the components of the Noether–Lefschetz locus to arbitrary dimension, provided that they contain the Fermat variety.

Keywords:

AMSC: 14C25, 14C30, 14D17, 13H10

References

1. E. Aljovin, H. Movasati and R. V. Loyola, Integral Hodge conjecture for Fermat varieties, J. Symbolic Comput. 95 (2019) 177–184. Crossref, Web of Science, Google Scholar
2. S. Bloch, Semi-regularity and de Rham cohomology, Invent. Math. 17 (1972) 51–66. Crossref, Web of Science, Google Scholar
3. U. Bruzzo and A. Grassi, The Noether–Lefschetz locus of surfaces in toric threefolds, Commun. Contemp. Math. 20(5) (2018) 1750070. Link, Web of Science, Google Scholar
4. U. Bruzzo, A. Grassi and A. F. Lopez, Existence and density of general components of the Noether–Lefschetz locus on normal threefolds, Int. Math. Res. Not. (2020). Web of Science, Google Scholar
5. J. Carlson, M. Green, P. Griffiths and J. Harris, Infinitesimal variations of Hodge structure. I, Compos. Math. 50(2–3) (1983) 109–205. Web of Science, Google Scholar
6. J. A. Carlson and P. A. Griffiths, Infinitesimal variations of Hodge structure and the global Torelli problem, in Journees de geometrie algebrique, Angers/France 1979 (Sijthoff & Noordhoff, 1980), pp. 51–76. Google Scholar
7. E. H. Cattani, P. Deligne and A. G. Kaplan, On the locus of Hodge classes, J. Amer. Math. Soc. 8(2) (1995) 483–506. Crossref, Web of Science, Google Scholar
8. C. Ciliberto, J. Harris and R. Miranda, General components of the Noether–Lefschetz locus and their density in the space of all surfaces, Math. Ann. 282(4) (1988) 667–680. Crossref, Web of Science, Google Scholar
9. D. A. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e, Undergraduate Texts in Mathematics (Springer-Verlag, Berlin, Heidelberg, 2007). Crossref, Google Scholar
10. A. Dan, Noether–Lefschetz locus and a special case of the variational Hodge conjecture: Using elementary techniques, in Analytic and Algebraic Geometry (Springer Singapore, Singapore, 2017), pp. 107–115. Crossref, Google Scholar
11. P. Deligne, J. S. Milne, A. Ogus and K.-Y. Shih, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, Vol. 900 (Springer-Verlag, Berlin, 1982). Philosophical Studies Series in Philosophy, 20. Crossref, Google Scholar
12. M. L. Green, Koszul cohomology and the geometry of projective varieties. II, J. Differential Geom. 20(1) (1984) 279–289. Crossref, Web of Science, Google Scholar
13. M. L. Green, A new proof of the explicit Noether–Lefschetz theorem, J. Differential Geom. 27(1) (1988) 155–159. Crossref, Web of Science, Google Scholar
14. M. L. Green, Components of maximal dimension in the Noether–Lefschetz locus, J. Differential Geom. 29(2) (1989) 295–302. Crossref, Web of Science, Google Scholar
15. P. A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969) 460–495; Ann. of Math. (2) 90 (1969) 496–541. Google Scholar
16. A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966) 95–103. Crossref, Google Scholar
17. S.-O. Kim, Noether–Lefschetz locus for surfaces, Trans. Amer. Math. Soc. 324(1) (1991) 369–384. Crossref, Web of Science, Google Scholar
18. R. Kloosterman, Higher Noether–Lefschetz loci of elliptic surfaces, J. Differential Geom. 76(2) (2007) 293–316. Crossref, Web of Science, Google Scholar
19. R. Villaflor Loyola, Periods of complete intersection algebraic cycles, Manuscripta Math. (2021), https://doi.org/10.1007/s00229-021-01290-x. Web of Science, Google Scholar
20. H. Movasati, A Course in Hodge theory: With emphasis on multiple integrals (International Press, 2017), Available at author’s webpage. Google Scholar
21. H. Movasati, Gauss–Manin connection in disguise: Noether–Lefschetz and Hodge loci, Asian J. Math. 21(3) (2017) 462–482. Crossref, Web of Science, Google Scholar
22. H. Movasati and R. Villaflor, Periods of linear algebraic cycles, Pure Appl. Math. Quart. 14 (2018) 563–577. Crossref, Web of Science, Google Scholar
23. A. Otwinowska, Sur la fonction de hilbert des algèbres graduées de dimension 0, J. Reine Angew. Math. 2002 (2002) 97–119. Crossref, Google Scholar
24. A. Otwinowska, Composantes de petite codimension du lieu de Noether–Lefschetz: Un argument asymptotique en faveur de la conjecture de Hodge pour les hypersurfaces, J. Algebraic Geom. 12(2) (2003) 307–320. Crossref, Web of Science, Google Scholar
25. Z. Ran, Cycles on Fermat hypersurfaces, Compos. Math. 42(1) (1980/1981) 121–142. Web of Science, Google Scholar
26. T. Shioda, The Hodge conjecture and the Tate conjecture for Fermat varieties, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979) 111–114. Crossref, Google Scholar
27. C. Voisin, Une précision concernant le théorème de Noether, Math. Ann. 280(4) (1988) 605–611. Crossref, Web of Science, Google Scholar
28. C. Voisin, Composantes de petite codimension du lieu de Noether–Lefschetz. Comment. Math. Helv. 64(4) (1989) 515–526. Crossref, Web of Science, Google Scholar
29. C. Voisin, Sur le lieu de Noether–Lefschetz en degrés $6$ et $7$ , Compos. Math. 75(1) (1990) 47–68. Google Scholar
30. C. Voisin, Contrexemple à une conjecture de J. Harris, C. R. Acad. Sci. Paris Sér. I Math. 313(10) (1991) 685–687. Google Scholar
31. C. Voisin, Hodge Theory and Complex Algebraic Geometry. II, Cambridge Studies in Advanced Mathematics, Vol. 77 (Cambridge University Press, Cambridge, 2003). Translated from the French by Leila Schneps. Crossref, Google Scholar