Infinitesimal variations of Hodge structure at infinity. (English) Zbl 1168.14008
In the fundamental paper [J. Carlson, M. Green, P. Griffiths and J. Harris [Compos. Math. 50, 109–205 (1983; Zbl 0531.14006)], Carlson et al. introduced the idea of infinitesimal variation of Hodge structure (IVHS) as a way to associate to a polarized variation of Hodge structure (PVHS) an integral element of the corresponding Griffiths’ exterior differential system. Similarly, in the paper under review, to any degenerating PVHS it is associated an infinitesimal variation of Hodge structure at infinity (IVI). Using the local description of PVHS near infinity [W. Schmid, Invent. Math. 22, 211–319 (1973; Zbl 0278.14003)], it is proved that every IVI integrates to a PVHS and that every IVI is a limit of IVHS at infinity. It is also showed that IVIs encode somehow more refined information of a PVHS than an IVHS does for finite points, in the sense that, even though the upper bounds for the maximal dimension of an IVHS remain valid and sharp for IVIs in a given period domain, considering only IVIs with underlying particular mixed Hodge structures or nilpotent cones leads to lower maximal dimensions.
Reviewer: Davide Franco (Napoli)
MSC:
| 14D07 | Variation of Hodge structures (algebro-geometric aspects) |
| 32G20 | Period matrices, variation of Hodge structure; degenerations |
Keywords:
polarized variations of Hodge structures; infinitesimal variations of Hodge strucrures; mixed Hodge structures; nilpotent conesReferences:
| [1] | Carlson J.: Bounds on the dimension of variations of Hodge structure. Trans. Am. Math. Soc. 294(1), 45–64 (1986) · Zbl 0593.14006 · doi:10.1090/S0002-9947-1986-0819934-6 |
| [2] | Carlson J., Toledo T.: Rigidity of harmonic maps of maximum rank. J. Geom. Anal. 3(2), 99–140 (1993) · Zbl 0771.58009 |
| [3] | Carlson, J., Cattani, E., Kaplan, A.: Mixed Hodge structures and compactifications of Siegel’s space (preliminary report), Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn, 1980, pp. 77–105 |
| [4] | Carlson J., Green M., Griffiths P., Harris J.: Infinitesimal variations of Hodge structure. I. Compos. Math. 50(2–3), 109–205 (1983) · Zbl 0531.14006 |
| [5] | Carlson J., Donagi R.: Hypersurface variations are maximal. I. Invent. Math. 89(2), 371–374 (1987) · Zbl 0639.14003 · doi:10.1007/BF01389084 |
| [6] | Cattani, E., Fernandez, J.: Asymptotic Hodge theory and quantum products. Advances in algebraic geometry motivated by physics, Lowell, MA, 2000. Contemporary Mathematics, vol. 276, pp. 115–136. Ame. Math. Soc., Providence, RI (2001). Also, arXiv:math.AG/0011137 |
| [7] | Cattani, E., Kaplan, A.: Degenerating variations of Hodge structure. Astérisque, vol. 9(179–180), pp. 67–96, 1989. Actes du Colloque de Théorie de Hodge, Luminy, (1987) |
| [8] | Cattani E., Kaplan A., Schmid W.: Degeneration of Hodge structures. Ann. Math. 123(3), 457–535 (1986) · Zbl 0617.14005 · doi:10.2307/1971333 |
| [9] | Griffiths, P. (eds): Topics in transcendental algebraic geometry. Annals of Mathematics Studies Vol. 106. Princeton University Press, Princeton, NJ (1984 · Zbl 0528.00004 |
| [10] | Kato, K, Usui, S.: Logarithmic Hodge structures and classifying spaces. The arithmetic and geometry of algebraic cycles, Banff, AB, 1998. Amer. Math. Soc., pp. 115–130. Providence, RI, (2000) · Zbl 0981.14006 |
| [11] | Mayer R.: Coupled contact systems and rigidity of maximal dimensional variations of Hodge structure. Trans. AMS. 352(5), 2121–2144 (1999) Also, arXiv:alg-geom/9712001 · Zbl 0947.37033 · doi:10.1090/S0002-9947-99-02395-8 |
| [12] | Schmid W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973) · Zbl 0278.14003 · doi:10.1007/BF01389674 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
