Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2015 | OriginalPaper | Buchkapitel

The Geometry and Moduli of K3 Surfaces

verfasst von : Andrew Harder, Alan Thompson

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

Verlag: Springer New York

Einloggen

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

search-config
loading …

Abstract

These notes will give an introduction to the theory of K3 surfaces. We begin with some general results on K3 surfaces, including the construction of their moduli space and some of its properties. We then move on to focus on the theory of polarized K3 surfaces, studying their moduli, degenerations and the compactification problem. This theory is then further enhanced to a discussion of lattice polarized K3 surfaces, which provide a rich source of explicit examples, including a large class of lattice polarizations coming from elliptic fibrations. Finally, we conclude by discussing the ample and Kähler cones of K3 surfaces, and give some of their applications.

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!

Jetzt informieren

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!

Jetzt informieren

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!

Jetzt informieren
Nächstes Kapitel Picard Ranks of K3 Surfaces of BHK Type
Anhänge
Nur mit Berechtigung zugänglich
Literatur
1.
Alexeev, V.: Log canonical singularities and complete moduli of stable pairs. Preprint, August 1996. arXiv:alg-geom/9608013
2.
Alexeev, V.: Moduli spaces M g, n (W) for surfaces. In: Higher-Dimensional Complex Varieties (Trento, 1994), pp. 1–22. de Gruyter, Berlin (1996)
3.
Alexeev, V.: Higher-dimensional analogues of stable curves. In: International Congress of Mathematicians, vol. II, pp. 515–536. European Mathematical Society, Zürich (2006)
4.
Artebani, M., Sarti, A., Taki, S.: K3 surfaces with non-symplectic automorphisms of prime order. Math. Z. 268(1–2), 507–533 (2011). With an appendix by Shigeyuki Kondō
5.
Ash, A., Mumford, D., Rapoport, M., Tai, Y.-S.: Smooth Compactification of Locally Symmetric Varieties. Lie Groups: History, Frontiers and Applications, vol. IV. Mathematical Science Press, Brookline (1975)MATH
6.
Baily, W.L., Jr. Borel, A.: Compactification of arithmetic quotients of bounded symmetric domains. Ann. Math. (2) 84:442–528 (1966)
7.
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)
8.
Beauville, A.: Fano threefolds and K3 surfaces. In: The Fano Conference, pp. 175–184. Univ. Torino, Turin (2004)
9.
10.
11.
Clingher, A., Doran, C.F., Lewis, J., Whitcher, U.: Normal forms, K3 surface moduli and modular parametrizations. In: Groups and Symmetries. Volume 47 of CRM Proceedings and Lecture Notes, pp. 81–98. American Mathematical Society, Providence (2009)
12.
Cossec, F.R., Dolgachev, I.V.: Enriques Surfaces. I. Volume 76 of Progress in Mathematics. Birkhäuser, Boston (1989)
13.
Dolgachev, I.V.: Integral quadratic forms: applications to algebraic geometry (after V. Nikulin). In: Bourbaki Seminar, Vol. 1982/83. Volume 105 of Astérisque, pp. 251–278. Société mathématique de France, Paris (1983)
14.
15.
Elkies, N.: Shimura curve computations via K3 surfaces of Néron-Severi rank at least 19. In: Algorithmic Number Theory. Volume 5011 of Lecture Notes in Computer Science, pp. 196–211. Springer, Berlin (2008)
16.
Elkies, N., Kumar, A.: K3 surfaces and equations for Hilbert modular surfaces. Algebra Number Theory 8(10), 2297–2411 (2014)MathSciNetCrossRefMATH
17.
Friedman, R.: A new proof of the global Torelli theorem for K3 surfaces. Ann. Math. (2) 120(2), 237–269 (1984)
18.
Friedman, R.: The period map at the boundary of moduli. In: Griffiths, P. (ed.) Topics in Transcendental Algebraic Geometry (Princeton, N.J., 1981/1982). Volume 106 of Annals of Mathematics Studies, pp. 183–208. Princeton University Press, Princeton (1984)
19.
Friedman, R., Morgan, J.W.: Smooth Four-Manifolds and Complex Surfaces. Volume 27 of Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Fogle, A Series of Modern Surveys in Mathematics. Springer, Berlin/New York (1994)
20.
Friedman, R., Morrison, D.: The birational geometry of degenerations: an overview. In: Friedman, R., Morrison, D. (eds.) The Birational Geometry of Degenerations. Number 29 in Progress in Mathematics, pp. 1–32. Birkhäuser, Boston (1983)
21.
Galluzzi, F., Lombardo, G.: Correspondences between K3 surfaces. Michigan Math. J. 52(2), 267–277 (2004). With an appendix by I. V. Dolgachev
22.
Galluzzi, F., Lombardo, G., Peters, C.: Automorphs of indefinite binary quadratic forms and K3-surfaces with Picard number 2. Rend. Semin. Mat. Univ. Politec. Torino 68(1), 57–77 (2010)MathSciNetMATH
23.
Gritsenko, V., Hulek, K., Sankaran, G.K.: Moduli of K3 surfaces and irreducible symplectic manifolds. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli, Vol. I. Number 24 in Advanced Lectures in Mathematics, pp. 459–526. International Press, Somerville (2013)
24.
Hartshorne, R.: Algebraic Geometry. Volume 52 of Graduate Texts in Mathematics. Springer, New York (1977)
25.
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces. Volume 80 of Pure and Applied Mathematics. Academic, New York (1978)
26.
27.
Iano-Fletcher, A.R.: Working with weighted complete intersections. In: Explicit Birational Geometry of 3-Folds. Volume 281 of London Mathematical Society Lecture Note Series, pp. 101–173. Cambridge University Press, Cambridge/New York (2000)
28.
Inose, H.: Defining equations of singular K3 surfaces and a notion of isogeny. In: Proceedings of the International Symposium on Algebraic Geometry (Kyoto University, Kyoto, 1977), pp. 495–502 (1978). Kinokuniya Book Store, Tokyo
29.
Kempf, G., Knudsen, F., Mumford, D., Saint-Donat, B.: Toroidal Embeddings I. Volume 339 of Lecture Notes in Mathematics Springer, Berlin/Heidelberg (1973)
30.
Kodaira, K.: On compact complex analytic surfaces, I. Ann. Math. (2) 71, 111–152 (1960)
31.
Kodaira, K.: On compact analytic surfaces: II. Ann. Math. (2) 77, 563–626 (1963)
32.
Kodaira, K.: On compact analytic surfaces, III. Ann. Math. (2) 78, 1–40 (1963)
33.
Kollár, J., Shepherd-Barron, N.: Threefolds and deformations of surface singularities. Invent. Math. 91(2), 299–338 (1988)MathSciNetCrossRefMATH
34.
Kondō, S.: Algebraic K3 surfaces with finite automorphism groups. Nagoya Math. J. 116, 1–15 (1989)MathSciNetMATH
35.
36.
Kulikov, V.: Degenerations of K3 surfaces and Enriques surfaces. Math. USSR Izv. 11(5), 957–989 (1977)CrossRefMATH
37.
Kulikov, V.: On modifications of degenerations of surfaces with \(\kappa = 0\). Math. USSR Izv. 17(2), 339–342 (1981)CrossRefMATH
38.
39.
Laza, R.: The KSBA compactification for the moduli space of degree two K3 pairs (2012, preprint). arXiv:1205.3144
40.
Ma, X., Marinescu, G.: Characterization of Moishezon manifolds. In: Holomorphic Morse Inequalities and Bergman Kernels. Number 254 in Progress in Mathematics, pp. 69–126. Birkhäuser, Basel (2007)
41.
42.
Miranda, R.: The Basic Theory of Elliptic Surfaces. Dottorato di Ricerca in Matematica [Doctorate in Mathematical Research]. ETS Editrice, Pisa (1989)
43.
Morrison, D.: Some remarks on the moduli of K3 surfaces. In: Classification of Algebraic and Analytic Manifolds (Katata, 1982). Volume 39 of Progress in Mathematics, pp. 303–332. Birkhäuser, Basel (1983)
44.
45.
Morrow, J., Kodaira, K.: Complex Manifolds. Holt, Rinehart and Winston, New York/Montreal/London (1971)MATH
46.
47.
Namikawa, Y.: Toroidal Compactification of Siegel Spaces. Volume 812 of Lecture Notes in Mathematics Springer, Berlin/New York (1980)
48.
Nikulin, V.V.: Finite automorphism groups of Kähler K3 surfaces. Trans. Moscow Math. Soc. 38(2), 71–135 (1980)
49.
50.
Nikulin, V.V.: Factor groups of groups of automorphisms of hyperbolic forms with respect to subgroups generated by 2-reflections. Algebrogeometric applications. J. Soviet Math. 22(4), 1401–1475 (1983)CrossRefMATH
51.
Nikulin, V.V.: Surfaces of type K3 with finite group of automorphisms and Picard group of rank three. Proc. Steklov Inst. Math. 165, 131–155 (1985)
52.
Persson, U.: On degenerations of algebraic surfaces. Mem. Am. Math. Soc. 11(189) (1977)
53.
Persson, U., Pinkham, H.: Degenerations of surfaces with trivial canonical bundle. Ann. Math. (2) 113(1), 45–66 (1981)
54.
Pjateckiĭ-Šapiro, I.I., Šafarevič, I.R.: A Torelli theorem for algebraic surfaces of type K3. Math. USSR Izv. 5(3), 547–588 (1971)CrossRef
55.
Reid, M.: Canonical 3-folds. In: Beauville, A. (ed.) Journées de Géométrie Algébrique d’Angers, Juillet 1979, pp. 273–310. Sijthoff & Noordhoff, Alphen aan den Rijn (1980)
56.
Reid, M.: Chapters on algebraic surfaces. In: Complex Algebraic Geometry (Park City, UT, 1993). Volume 3 of IAS/Park City Mathematics Series, pp. 3–159. American Mathematical Society, Providence (1997)
57.
Rohsiepe, F.: Lattice polarized toric K3 surfaces (2004, preprint). arXiv:hep-th/0409290
58.
Rohsiepe, F.: Calabi-Yau-Hyperflächen in Torischen Varietäten, Faserungen und Dualitäten. PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn (2005)
59.
Scattone, F.: On the compactification of moduli spaces for algebraic K3 surfaces. Mem. Am. Math. Soc. 70(374) (1987)
60.
Schütt, M., Shioda, T.: Elliptic surfaces. In: Algebraic Geometry in East Asia – Seoul 2008. Volume 60 of Advanced Studies in Pure Mathematics, pp. 51–160. Mathematical Society of Japan, Tokyo (2010)
61.
Serre, J.-P.: A Course in Arithmetic. Volume 7 of Graduate Texts in Mathematics. Springer, New York (1973)
62.
Shah, J.: A complete moduli space for K3 surfaces of degree 2. Ann. Math. (2) 112(3), 485–510 (1980)
63.
Shah, J.: Degenerations of K3 surfaces of degree 4. Trans. Am. Math. Soc. 263(2), 271–308 (1981)MATH
64.
Silverman, J.H.: Advanced Topics in the Arithmetic of Elliptic Curves. Volume 151 of Graduate Texts in Mathematics. Springer, New York (1994)
65.
66.
67.
Tate, J.: Algorithm for determining the type of a singular fiber in an elliptic pencil. In: Modular Functions of One Variable IV. Volume 476 of Lecture Notes in Mathematics, pp. 33–52. Springer, Berlin/Heidelberg (1975)
68.
van Geemen, B.: Kuga-Satake varieties and the Hodge conjecture. In: The Arithmetic and Geometry of Algebraic Cycles (Banff, AB, 1998). Volume 548 of Nato Science Series C: Mathematical and Physical Sciences, pp. 51–82. Kluwer, Dordrecht (2000)
69.
Xiao, G.: Galois covers between K3 surfaces. Ann. Inst. Fourier (Grenoble) 46(1), 73–88 (1996)
Metadaten
Titel
The Geometry and Moduli of K3 Surfaces
verfasst von
Andrew Harder
Alan Thompson
Copyright-Jahr
2015
Verlag
Springer New York
Buch
Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Print ISBN: 978-1-4939-2829-3

Electronic ISBN: 978-1-4939-2830-9

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-1-4939-2830-9_1