Convexity on the space of Kähler metrics

Bo Berndtsson

Annales de la faculté des sciences de Toulouse Mathématiques (2013)

  • Volume: 22, Issue: 4, page 713-746
  • ISSN: 0240-2963

Abstract

top
These are the lecture notes of a minicourse given at a winter school in Marseille 2011. The aim of the course was to give an introduction to recent work on the geometry of the space of Kähler metrics associated to an ample line bundle. The emphasis of the course was the role of convexity, both as a motivating example and as a tool.

How to cite

top

Berndtsson, Bo. "Convexity on the space of Kähler metrics." Annales de la faculté des sciences de Toulouse Mathématiques 22.4 (2013): 713-746. <http://eudml.org/doc/275376>.

@article{Berndtsson2013,
abstract = {These are the lecture notes of a minicourse given at a winter school in Marseille 2011. The aim of the course was to give an introduction to recent work on the geometry of the space of Kähler metrics associated to an ample line bundle. The emphasis of the course was the role of convexity, both as a motivating example and as a tool.},
author = {Berndtsson, Bo},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {6},
number = {4},
pages = {713-746},
publisher = {Université Paul Sabatier, Toulouse},
title = {Convexity on the space of Kähler metrics},
url = {http://eudml.org/doc/275376},
volume = {22},
year = {2013},
}

TY - JOUR
AU - Berndtsson, Bo
TI - Convexity on the space of Kähler metrics
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2013/6//
PB - Université Paul Sabatier, Toulouse
VL - 22
IS - 4
SP - 713
EP - 746
AB - These are the lecture notes of a minicourse given at a winter school in Marseille 2011. The aim of the course was to give an introduction to recent work on the geometry of the space of Kähler metrics associated to an ample line bundle. The emphasis of the course was the role of convexity, both as a motivating example and as a tool.
LA - eng
UR - http://eudml.org/doc/275376
ER -

References

top
  1. Bando (Sh.), Mabuchi (T.).— Uniqueness of Einstein Kähler metrics modulo connected group actions. Algebraic geometry, Sendai, 1985, 11-40, Adv. Stud. Pure Math., 10, North-Holland, Amsterdam, (1987). Zbl0641.53065MR946233
  2. Berman (R.), Berndtsson (B.) and Sjöstrand (J.).— Asymptotics of Bergman kernels Ark. Mat. 46, no. 2, p. 197-217 (2008). Zbl1161.32001MR2430724
  3. Berman (R.).— Analytic torsion, vortices and positive Ricci curvaturearXiv:1006.2988 
  4. Berndtsson (B.).— A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem arXiv:1103.0923 
  5. Berndtsson (B.).— Positivity of direct image bundles and convexity on the space of Kähler metrics, J. Differential geom. 81.3, p. 457-482 (2009). Zbl1187.53076MR2487599
  6. Berndtsson (B.).— Strict and non strict positivity of direct image bundles Math. Z. 269, no. 3-4, p. 1201-1218 (2011). Zbl1252.32027MR2860284
  7. Berndtsson (B.).— Positivity of direct image bundles and convexity on the space of Kähler metrics, J. Differential Geom. 81, p. 457-482 (2009). Zbl1187.53076MR2487599
  8. Brascamp (H. J.) and Lieb (E. H.).— On extensions of the Brunn-Minkowski and Prèkopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22, no. 4, p. 366-389 (1976). Zbl0334.26009MR450480
  9. Chen (X. X.).— The space of Käler metrics, J. Differential Geom. 56, no. 2, p. 189-234 (2000). Zbl1041.58003MR1863016
  10. Ding (W.) and Tian (G.).— The generalized Moser-Trudinger Inequality. Proceedings of Nankai International Conference on Nonlinear Analysis (1993). Zbl1049.53507
  11. Donaldson (S. K.).— Scalar curvature and projective embeddings. I. J. Differential Geom. 59, no. 3, p. 479-522 (2001). Zbl1052.32017MR1916953
  12. Donaldson (S. K.).— Scalar curvature and projective embeddings. II. Q. J. Math. 56, no. 3, p. 345-356 (2005). Zbl1159.32012MR2161248
  13. Donaldson (S. K.).— Symmetric spaces, Kähler geometry and Hamiltonian dynamics. Northern California Symplectic Geometry Seminar, p. 13-33. Zbl0972.53025
  14. Chen (X.X.), Donaldson (S.K.), and Sun (S.).— Kähler-Einstein metrics and stability. arXiv:1210.7494 
  15. Kobayashi (S.).— Differential geometry of complex vector bundles. Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo, 1987. xii+305 pp. ISBN: 0-691-08467-X Zbl0708.53002MR909698
  16. Lempert (L.) and Vivas (L.).— Geodesics in the space of Kähler metrics. arXiv:1105.2188 Duke Math. Journal, to appear. Zbl1275.32020
  17. Lu (Z.).— On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math. 122, no. 2, p. 235-273 (2000). Zbl0972.53042MR1749048
  18. Mabuchi (T.).— Some symplectic geometry on compact Kähler manifolds. Osaka J. Math. 24, p. 227-252 (1987). Zbl0645.53038MR909015
  19. Mabuchi (T.).— K -energy maps integrating Futaki invariants. Tohoku Math. J. (2) 38, no. 4, p. 575-593 (1986). Zbl0619.53040MR867064
  20. Phong (D.H.), Sturm (J.).— The Monge-Ampère operator and geodesics in the space of Kähler potentials, Inventiones Math. 166, p. 125-149 (2006). Zbl1120.32026MR2242635
  21. Phong (D.H.), Song (J.), Sturm (J.), Weinkove (B.).— The Moser-Trudinger inequality on Kahler-Einstein manifolds. Amer. J. Math. 130, no. 4, p. 1067-1085 (2008). Zbl1158.58005MR2427008
  22. Prekopa (A.).— On logarithmic concave measures and functions. Acad. Sci. Math. (Szeged) 34, p. 335-343 (1973). Zbl0264.90038MR404557
  23. Ruan (W.-D.).— Canonical coordinates and Bergman metrics. Comm. Anal. Geom. 6, p. 589-631 (1998). Zbl0917.53026MR1638878
  24. Semmes (S.).— Complex Monge-Ampère and symplectic manifolds. Amer. J. Math. 114, no. 3, p. 495-550 (1992). Zbl0790.32017MR1165352
  25. Tian (G.).— On a set of polarized metrics on Kähler manifolds. J. Differential Geom. 32 (1990). Zbl0706.53036MR1064867
  26. Tian (G.).— K-stability and Kähler-Einstein metrics. arXiv:1211.4669 
  27. Zelditch (S.).— Szegö kernels and a theorem of Tian. Internat. Math. Res. Notices, no. 6, p. 317-331 (1998). Zbl0922.58082MR1616718

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.