Skip to main content

2015 | OriginalPaper | Buchkapitel

Some Remarks About Chow, Hilbert and K-stability of Ruled Threefolds

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

search-config
loading …

Abstract

Given a rank 2 holomorphic vector bundle E over a projective surface, we explain some relationships between the Gieseker stability of E and the Chow, Hilbert and K-stability of the polarized ruled manifold \(\mathbb{P}E\) with respect to polarizations that make fibres sufficiently small.

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!

Literatur
1.
Zurück zum Zitat V. Apostolov, D.M.J. Calderbank, P. Gauduchon, C.W. Tønnesen-Friedman, Extremal Kähler metrics on ruled manifolds and stability, in Géométrie Différentielle, Physique Mathématique, Mathématiques et Société. II. Astérisque, vol. 322 (2008), pp. 93–150 V. Apostolov, D.M.J. Calderbank, P. Gauduchon, C.W. Tønnesen-Friedman, Extremal Kähler metrics on ruled manifolds and stability, in Géométrie Différentielle, Physique Mathématique, Mathématiques et Société. II. Astérisque, vol. 322 (2008), pp. 93–150
2.
3.
Zurück zum Zitat A. Della Vedova, F. Zuddas, Scalar curvature and asymptotic Chow stability of projective bundles and blowups. Trans. Am. Math. Soc. 364(12), 6495–6511 (2012)CrossRefMATH A. Della Vedova, F. Zuddas, Scalar curvature and asymptotic Chow stability of projective bundles and blowups. Trans. Am. Math. Soc. 364(12), 6495–6511 (2012)CrossRefMATH
4.
Zurück zum Zitat S.K. Donaldson, Scalar curvature and projective embeddings. I. J. Differ. Geom. 59(3), 479–522 (2001)MATHMathSciNet S.K. Donaldson, Scalar curvature and projective embeddings. I. J. Differ. Geom. 59(3), 479–522 (2001)MATHMathSciNet
5.
Zurück zum Zitat S.K. Donaldson, Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)MATHMathSciNet S.K. Donaldson, Scalar curvature and stability of toric varieties. J. Differ. Geom. 62(2), 289–349 (2002)MATHMathSciNet
6.
Zurück zum Zitat S.K. Donaldson, Lower bounds on the Calabi functional. J. Differ. Geom. 70(3), 453–472 (2005)MATHMathSciNet S.K. Donaldson, Lower bounds on the Calabi functional. J. Differ. Geom. 70(3), 453–472 (2005)MATHMathSciNet
7.
Zurück zum Zitat A. Futaki, Asymptotic Chow polystability in Kähler geometry, in Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt. 1, vol. 2 (Am. Math. Soc., Providence, 2012), pp. 139–153 A. Futaki, Asymptotic Chow polystability in Kähler geometry, in Fifth International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 51, pt. 1, vol. 2 (Am. Math. Soc., Providence, 2012), pp. 139–153
8.
Zurück zum Zitat Y.-J. Hong, Constant Hermitian scalar curvature equations on ruled manifolds. J. Differ. Geom. 53(3), 465–516 (1999)MATH Y.-J. Hong, Constant Hermitian scalar curvature equations on ruled manifolds. J. Differ. Geom. 53(3), 465–516 (1999)MATH
9.
Zurück zum Zitat Y.-J. Hong, Gauge-fixing constant scalar curvature equations on ruled manifolds and the futaki invariants. J. Differ. Geom. 60, 389–453 (2002) Y.-J. Hong, Gauge-fixing constant scalar curvature equations on ruled manifolds and the futaki invariants. J. Differ. Geom. 60, 389–453 (2002)
10.
Zurück zum Zitat Y.-J. Hong, Stability and existence of critical Kaehler metrics on ruled manifolds. J. Math. Soc. Jpn. 60(1), 265–290 (2008)CrossRefMATH Y.-J. Hong, Stability and existence of critical Kaehler metrics on ruled manifolds. J. Math. Soc. Jpn. 60(1), 265–290 (2008)CrossRefMATH
12.
Zurück zum Zitat T. Mabuchi, An energy-theoretic approach to the Hitchin–Kobayashi correspondence for manifolds. I. Invent. Math. 159(2), 225–243 (2005)CrossRefMATHMathSciNet T. Mabuchi, An energy-theoretic approach to the Hitchin–Kobayashi correspondence for manifolds. I. Invent. Math. 159(2), 225–243 (2005)CrossRefMATHMathSciNet
13.
Zurück zum Zitat T. Mabuchi, Chow-stability and Hilbert-stability in Mumford’s geometric invariant theory. Osaka J. Math. 45(3), 833–846 (2008)MATHMathSciNet T. Mabuchi, Chow-stability and Hilbert-stability in Mumford’s geometric invariant theory. Osaka J. Math. 45(3), 833–846 (2008)MATHMathSciNet
15.
Zurück zum Zitat S. Mukai, Semi-homogeneous vector bundles on an Abelian variety. J. Math. Kyoto Univ. 18(2), 239–272 (1978)MATHMathSciNet S. Mukai, Semi-homogeneous vector bundles on an Abelian variety. J. Math. Kyoto Univ. 18(2), 239–272 (1978)MATHMathSciNet
16.
Zurück zum Zitat D. Mumford, Stability of projective varieties, in L’Enseignement Mathématique, Lectures given at the “Institut des Hautes Études Scientifiques”, Bures-sur-Yvette, Geneva March–April 1976, Monographie de l’Enseignement Mathématique, vol. 24 (1977) D. Mumford, Stability of projective varieties, in L’Enseignement Mathématique, Lectures given at the “Institut des Hautes Études Scientifiques”, Bures-sur-Yvette, Geneva March–April 1976, Monographie de l’Enseignement Mathématique, vol. 24 (1977)
17.
Zurück zum Zitat J. Ross, R. Thomas, An obstruction to the existence of constant scalar curvature Kähler metrics. J. Differ. Geom. 72(3), 429–466 (2006)MATHMathSciNet J. Ross, R. Thomas, An obstruction to the existence of constant scalar curvature Kähler metrics. J. Differ. Geom. 72(3), 429–466 (2006)MATHMathSciNet
18.
Zurück zum Zitat J. Ross, R. Thomas, A study of the Hilbert–Mumford criterion for the stability of projective varieties. J. Algeb. Geom. 16(2), 201–255 (2007)CrossRefMATHMathSciNet J. Ross, R. Thomas, A study of the Hilbert–Mumford criterion for the stability of projective varieties. J. Algeb. Geom. 16(2), 201–255 (2007)CrossRefMATHMathSciNet
21.
Zurück zum Zitat R.P. Thomas, Notes on GIT and symplectic reduction for bundles and varieties, in Surveys in Differential Geometry. Vol. X. Surv. Differ. Geom., vol. 10 (International Press, Somerville, 2006), pp. 221–273 R.P. Thomas, Notes on GIT and symplectic reduction for bundles and varieties, in Surveys in Differential Geometry. Vol. X. Surv. Differ. Geom., vol. 10 (International Press, Somerville, 2006), pp. 221–273
22.
Zurück zum Zitat X. Wang, Moment map, Futaki invariant and stability of projective manifolds. Commun. Anal. Geom. 12(5), 1009–1037 (2004)CrossRefMATH X. Wang, Moment map, Futaki invariant and stability of projective manifolds. Commun. Anal. Geom. 12(5), 1009–1037 (2004)CrossRefMATH
Metadaten
Titel
Some Remarks About Chow, Hilbert and K-stability of Ruled Threefolds
verfasst von
Julien Keller
Copyright-Jahr
2015
DOI
https://doi.org/10.1007/978-3-319-12577-0_42