Skip to main content

2015 | OriginalPaper | Buchkapitel

An Introduction to Hodge Structures

verfasst von : Sara Angela Filippini, Helge Ruddat, Alan Thompson

Erschienen in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Verlag: Springer New York

Aktivieren Sie unsere intelligente Suche um passende Fachinhalte oder Patente zu finden.

search-config
loading …

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Fußnoten
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
Note that here we use the original notation by Steenbrink [30]; the two indices p, q are swapped in [25].
 
Literatur
1.
Zurück zum Zitat 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) 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)
2.
Zurück zum Zitat 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) 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)
5.
Zurück zum Zitat 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.: 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)
6.
Zurück zum Zitat 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.: 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)
8.
Zurück zum Zitat 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) 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)
9.
Zurück zum Zitat Griffiths, P.: Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties. Am. J. Math. 90, 568–626 (1968)MATH Griffiths, P.: Periods of integrals on algebraic manifolds. I. Construction and properties of the modular varieties. Am. J. Math. 90, 568–626 (1968)MATH
10.
Zurück zum Zitat Griffiths, P.: Periods of integrals on algebraic manifolds. II. Local study of the period mapping. Am. J. Math. 90, 805–865 (1968)MATH Griffiths, P.: Periods of integrals on algebraic manifolds. II. Local study of the period mapping. Am. J. Math. 90, 805–865 (1968)MATH
11.
Zurück zum Zitat Griffiths, P.: On the periods of certain rational integrals. I, II. Ann. Math. (2) 90, 460–495, 496–541 (1969) Griffiths, P.: On the periods of certain rational integrals. I, II. Ann. Math. (2) 90, 460–495, 496–541 (1969)
12.
Zurück zum Zitat 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)CrossRefMATH 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)CrossRefMATH
13.
Zurück zum Zitat 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)CrossRefMathSciNetMATH 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)CrossRefMathSciNetMATH
14.
Zurück zum Zitat 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.: Mirror symmetry and the Strominger-Yau-Zaslow conjecture. In: Current Developments in Mathematics 2012, pp. 133–191. International Press, Somerville (2013)
15.
Zurück zum Zitat Gross, M., Katzarkov, L., Ruddat, H.: Towards mirror symmetry for varieties of general type (February 2012, preprint). arXiv:1202.4042 Gross, M., Katzarkov, L., Ruddat, H.: Towards mirror symmetry for varieties of general type (February 2012, preprint). arXiv:1202.4042
16.
Zurück zum Zitat Gross, M., Siebert, B.: Affine manifolds, log structures, and mirror symmetry. Turkish J. Math. 27(1), 33–60 (2003)MathSciNetMATH Gross, M., Siebert, B.: Affine manifolds, log structures, and mirror symmetry. Turkish J. Math. 27(1), 33–60 (2003)MathSciNetMATH
17.
Zurück zum Zitat Gross, M., Siebert, B.: Mirror symmetry via logarithmic degeneration data I. J. Differ. Geom. 72(2), 169–338 (2006)MathSciNetMATH Gross, M., Siebert, B.: Mirror symmetry via logarithmic degeneration data I. J. Differ. Geom. 72(2), 169–338 (2006)MathSciNetMATH
18.
19.
Zurück zum Zitat 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) 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)
20.
Zurück zum Zitat Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. Volume 339 of Lecture Notes in Mathematics. Springer, Berlin/Heidelberg (1973) Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. Volume 339 of Lecture Notes in Mathematics. Springer, Berlin/Heidelberg (1973)
21.
Zurück zum Zitat Kulikov, V.: Mixed Hodge Structures and Singularities. Volume 132 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge/New York (1998) Kulikov, V.: Mixed Hodge Structures and Singularities. Volume 132 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge/New York (1998)
22.
Zurück zum Zitat Malgrange, B.: Intégrales asymptotiques et monodromie. Ann. Sci. École Norm. Sup. (4) 7, 405–430 (1974) Malgrange, B.: Intégrales asymptotiques et monodromie. Ann. Sci. École Norm. Sup. (4) 7, 405–430 (1974)
23.
Zurück zum Zitat 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.: 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)
24.
Zurück zum Zitat Morrison, D.: Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians. J. Am. Math. Soc. 6(1), 223–247 (1993)CrossRefMathSciNetMATH Morrison, D.: Mirror symmetry and rational curves on quintic threefolds: a guide for mathematicians. J. Am. Math. Soc. 6(1), 223–247 (1993)CrossRefMathSciNetMATH
25.
Zurück zum Zitat 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) 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)
26.
Zurück zum Zitat Ruddat, H., Siebert, B.: Canonical coordinates in toric degenerations. (September 2014, preprint). arXiv:1409.4750 Ruddat, H., Siebert, B.: Canonical coordinates in toric degenerations. (September 2014, preprint). arXiv:1409.4750
27.
Zurück zum Zitat 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.: 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)
28.
29.
31.
Zurück zum Zitat 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) 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)
32.
Zurück zum Zitat Varčenko, A.N.: Asymptotic behaviour of holomorphic forms determines a mixed Hodge structure. Dokl. Akad. Nauk SSSR 255(5), 1035–1038 (1980)MathSciNet Varčenko, A.N.: Asymptotic behaviour of holomorphic forms determines a mixed Hodge structure. Dokl. Akad. Nauk SSSR 255(5), 1035–1038 (1980)MathSciNet
33.
Zurück zum Zitat Voisin, C.: Hodge Theory and Complex Algebraic Geometry. I. Volume 76 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2007) Voisin, C.: Hodge Theory and Complex Algebraic Geometry. I. Volume 76 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2007)
Metadaten
Titel
An Introduction to Hodge Structures
verfasst von
Sara Angela Filippini
Helge Ruddat
Alan Thompson
Copyright-Jahr
2015
Verlag
Springer New York
DOI
https://doi.org/10.1007/978-1-4939-2830-9_4