ON KÂHLER-EINSTEIN METRIC

Öz

In this study, we were mentioned Kâhler-Einstein metrics and the first Chern class of the Chern classes which is the characteristic class of complex vector bundles with a orientation and used to distinguish the differences between two manifolds. Relation between Kâhler-Einstein metrics and first Chern class was mentioned. Positive case of the first Chern class which has positive, negative and zero was examined and for this case how to find Kâhler-Einstein metrics was explained.

Anahtar Kelimeler

• Kâhler-Einstein Metrics,First Chern Class,Futaki Invaryant

KÂHLER-EİNSTEİN METRİK ÜZERİNE

Öz

Bu çalışmada iki manifold arasındaki farklılıkları ayırt etmek için kullanılan ve bir yönlendirmeye sahip kompleks vektör demetlerinin karakteristik sınıfı olan Chern sınıflarından birinci Chern sınıfı ve Kâhler-Einstein metriklerine değinilmiş aynı zamanda da aralarındaki ilişkiden bahsedilmiştir. Birinci Chern sınıfının pozitif, negatif ve sıfır olduğu durumlardan pozitif olduğu durum incelenmiş ve bu durum için Kâhler-Einstein metriklerinin nasıl bulunabileceği açıklanmıştır.

 

Anahtar Kelimeler

• Kâhler-Einstein Metrik,Birinci Chern Sınıfı,Futaki İnvaryantı

Kaynakça

1. [1] Pragacz, P., (2012), Characteristic Classes with Applications yo Geometry, Topology and Number Theory, Instytut Matematyczny Polskiej Akademii Nauk, Poland.
2. [2] Ozan, Y., (2016), Türevlenebilir Manifoldlara Giriş, ODTÜ, Ankara.
3. [3] Hatcher, A., (2002), Algebraic Topology, Cambridge University Press, England.
4. [4] Szekelyhidi, G., (2013), Introduction to Extremal metrics Preliminary Version.
5. [5] Faulk, M., (2016), First Chern Classes of Kahler Manifolds, Columbia University, New York.
6. [6] Fine, J., (2012), A rapid introduction to Kähler geometry, Chapter 2: Holomorphic line bundles, preprints, Université Libre de Bruxelles, Belgique.
7. [7] Santoro, B., (2009), Introduction to Evolution Equations in Geometry, Nacional de Matematica Pura e Aplicada-IMPA, Rio de Janeiro, Brazil.
8. [8] Li, C., (2012), Kahler-Einstein Metrics and K-Stability, PhD thesis, Princeton University.

9. [9] Moroianu, A., (2007), Lectures on Kahler Geometry, Cambridge University Press, Paris.
10. [10] Tian, G., (2000), Kahler-Einstein Manifolds of Positive Scalar Curvature, İnternational Press.
11. [11] Wang, X. J., Zhu, X., (2004), Kahler-Ricci solitons on toric manifolds with positive first Chern class, Advances in Mathematics 188, 87-103.
12. [12] Spotti, C., (2016), Compact moduli spaces of Kahler-Einstein Fano varieties, Cambridge of University, England.
13. [13] Bando, S., Mabuchi, T., (1987), Uniqueness of Einstein Kahler metrics modulo connected group actions, Algebraic Geometry, Adv. Studies in Pure Math., 10.
14. [14] Tian, G., (2014), Kahler-Einstein metrics on Fano manifolds, Japanese Journal of Mathematics, Volume 10, Issue 1, pp 1–41.
15. [15] Tian, G., (1990), Kahler-Einstein on algebraic manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II ,587-598, Math. Soc. Japan, Tokyo, 1991.
16. [16] Tsuboi, K., (2009), On the existence of Kahler metrics of constant scalar curvature, Tohoku Math. J., vol. 61(2), 241-252.
17. [17] Tian, G., (1991), Lectures on Einstein Manifolds: Kahler-Einstein Manifolds and Positive Scalar Curvature, Mathematics Subject Classification. Primary 53C20, İnternational Press.
18. [18] Tian, G., (1997), Kahler-Einstein metrics with positive scalar curvature. Invent. Math., 130, 1-39.
19. [19] Matsushima, Y., (1957), Sur la structure du group homeomorphismes analytiques d’une certain variete, Kaehlerienne, Nagoya Math. J., 11, 145-150.
20. [20] Futaki, A., (1983), An obstruction to the existence of Kahler-Einstein metrics, Inv. Math., 73, 437-443.
21. [21] Stoppa, J., (2009), K-stability of constant scalar curvature Kahler manifolds. Adv. in Math., 221, 1397-1408.
22. [22] Yetim, C., (2018), Kahler-Einstein Metrik Üzerine, Yüksek Lisans Tezi, Dumlupınar Üniversitesi Fen Bilimleri Enstitüsü, Kütahya, 70s.