Abstract
We begin by introducing the concept of a Hodge structure and give some of its basic properties, including the Hodge and Lefschetz decompositions. We then define the period map, which relates families of Kähler manifolds to the families of Hodge structures defined on their cohomology, and discuss its properties. This will lead us to the more general definition of a variation of Hodge structure and the Gauss-Manin connection. We then review the basics about mixed Hodge structures with a view towards degenerations of Hodge structures; including the canonical extension of a vector bundle with connection, Schmid’s limiting mixed Hodge structure and Steenbrink’s work in the geometric setting. Finally, we give an outlook about Hodge theory in the Gross-Siebert program.
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Notes
- 1.
A third characterization of Hodge structures is given in terms of certain representations of \(\mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{C}^{{\ast}}\) (see, for example, [31]). More precisely, a rational Hodge structure of weight n on a \(\mathbb{Q}\)-vector space H can be identified with an algebraic representation \(\rho: \mathbb{C}^{{\ast}}\rightarrow GL(H_{\mathbb{R}})\), where \(H_{\mathbb{R}}:= H \otimes _{\mathbb{Q}}\mathbb{R}\), such that the restriction of ρ to \(\mathbb{R}^{{\ast}}\) is given by \(\rho (\lambda ) =\lambda ^{n}\). From this point of view, it is clearly completely natural to use constructions from multi-linear algebra to produce new Hodge structures.
- 2.
References
Barth, W.P., Hulek, K., Peters, C.A.M., van de Ven, A.: Compact Complex Surfaces. Volume 4 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Fogle, A Series of Modern Surveys in Mathematics, 2nd edn. Springer, Berlin/New York (2004)
Carlson, J., Müller-Stach, S., Peters, C.A.M.: Period Mappings and Period Domains. Volume 85 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2003)
Clemens, C.H.: Degeneration of Kähler manifolds. Duke Math. J. 44(2), 215–290 (1977)
Deligne, P.: Théorie de Hodge II. Inst. Hautes Études Sci. Publ. Math. 40, 5–57 (1971)
Deligne, P.: Comparaison avec la théorie transcendante. In: Groupes de Monodromie en Géométrie Algébrique. Volume 340 of Lecture Notes in Mathamatics, pp. 116–164. Springer, Berlin/Heidelberg (1973)
Deligne, P.: Le formalisme des cycles évanescents. In: Groupes de Monodromie en Géométrie Algébrique. Volume 340 of Lecture Notes in Mathamatics, pp. 82–115. Springer, Berlin/Heidelberg (1973)
Deligne, P.: Théorie de Hodge III. Inst. Hautes Études Sci. Publ. Math. 44, 5–77 (1974)
Deligne, P.: Local behavior of Hodge structures at infinity. In: Mirror Symmetry, II. Volume 1 of AMS/IP Studies in Advanced Mathematics, pp. 683–699. American Mathematical Society, Providence (1997)
Griffiths, P.: Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties. Am. J. Math. 90, 568–626 (1968)
Griffiths, P.: Periods of integrals on algebraic manifolds. II. Local study of the period mapping. Am. J. Math. 90, 805–865 (1968)
Griffiths, P.: On the periods of certain rational integrals. I, II. Ann. Math. (2) 90, 460–495, 496–541 (1969)
Griffiths, P.: Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping. Inst. Hautes Études Sci. Publ. Math. 38, 125–180 (1970)
Griffiths, P.: Periods of integrals on algebraic manifolds: summary of main results and discussion of open problems. Bull. Am. Math. Soc. 76, 228–296 (1970)
Gross, M.: Mirror symmetry and the Strominger-Yau-Zaslow conjecture. In: Current Developments in Mathematics 2012, pp. 133–191. International Press, Somerville (2013)
Gross, M., Katzarkov, L., Ruddat, H.: Towards mirror symmetry for varieties of general type (February 2012, preprint). arXiv:1202.4042
Gross, M., Siebert, B.: Affine manifolds, log structures, and mirror symmetry. Turkish J. Math. 27(1), 33–60 (2003)
Gross, M., Siebert, B.: Mirror symmetry via logarithmic degeneration data I. J. Differ. Geom. 72(2), 169–338 (2006)
Gross, M., Siebert, B.: Mirror symmetry via logarithmic degeneration data II. J. Algebraic Geom. 19(4), 679–780 (2010)
Gross, M., Siebert, B.: An invitation to toric degenerations. In: Geometry of Special Holonomy and Related Topics. Volume 16 of Surveys in Differential Geometry, pp. 43–78. International Press, Somerville (2011)
Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. Volume 339 of Lecture Notes in Mathematics. Springer, Berlin/Heidelberg (1973)
Kulikov, V.: Mixed Hodge Structures and Singularities. Volume 132 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge/New York (1998)
Malgrange, B.: Intégrales asymptotiques et monodromie. Ann. Sci. École Norm. Sup. (4) 7, 405–430 (1974)
Morrison, D.: The Clemens-Schmid exact sequence and applications. In: Griffiths, P. (ed.) Topics in Transcendental Algebraic Geometry (Princeton, 1981/1982). Volume 106 of Annals of mathematics studies, pp. 101–119. Princeton University Press, Princeton (1984)
Morrison, D.: Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians. J. Am. Math. Soc. 6(1), 223–247 (1993)
Peters, C.A.M., Steenbrink, J.H.M.: Mixed Hodge Structures. Volume 52 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Fogle, A Series of Modern Surveys in Mathematics. Springer, Berlin (2008)
Ruddat, H., Siebert, B.: Canonical coordinates in toric degenerations. (September 2014, preprint). arXiv:1409.4750
Ruddat, H.: Log Hodge groups on a toric Calabi-Yau degeneration. In: Mirror Symmetry and Tropical Geometry. Volume 527 of Contemporary Mathematics, pp. 113–164. AMS, Providence (2010)
Ruddat, H., Sibilla, N., Treumann, D., Zaslow, E.: Skeleta of affine hypersurfaces. Geom. Topol. 18(3), 1343–1395 (2014)
Schmid, W.: Variation of Hodge structure: the singularities of the period mapping. Invent. Math. 22, 211–319 (1973)
Steenbrink, J.H.M.: Limits of Hodge structures. Invent. Math. 31(3), 229–257 (1975)
van Geemen, B.: Kuga-Satake varieties and the Hodge conjecture. In: Gordon, B.B., Lewis, J.D., Müller-Stach, S., Saito, S., Yui, N. (eds.) The Arithmetic and Geometry of Algebraic Cycles (Proceedings of the NATO Advanced Study Institute held as part of the 1998 CRM Summer School at Banff, AB, 7–19 June 1998). Volume 548 of NATO Science Series C: Mathematical and Physical Sciences, pp. 51–82. Kluwer, Dordrecht (2000)
Varčenko, A.N.: Asymptotic behaviour of holomorphic forms determines a mixed Hodge structure. Dokl. Akad. Nauk SSSR 255(5), 1035–1038 (1980)
Voisin, C.: Hodge Theory and Complex Algebraic Geometry. I. Volume 76 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2007)
Acknowledgements
A part of these notes was written while the authors were in residence at the Fields Institute Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry and Physics; we would like to thank the Fields Institute for their support and hospitality. A. Thompson was supported by a Fields-Ontario-PIMS postdoctoral fellowship with funding provided by NSERC, the Ontario Ministry of Training, Colleges and Universities, and an Alberta Advanced Education and Technology Grant.
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Filippini, S.A., Ruddat, H., Thompson, A. (2015). An Introduction to Hodge Structures. In: Laza, R., Schütt, M., Yui, N. (eds) Calabi-Yau Varieties: Arithmetic, Geometry and Physics. Fields Institute Monographs, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2830-9_4
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