Bochner-Hartogs type extension theorem for roots and logarithms of holomorphic line bundles

We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton's equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, Vol. 18-I (Academic Press, Boston, MA, 1990), pp. 327–337]. Our examples can be viewed a generalization of previous examples by Cao [Existense of gradient Kähler–Ricci solitons, in Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), pp. 1–16], Chave and Valent [On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B 478 (1996) 758–778], Pedersen, Tønnesen-Friedman, and Valent [Quasi-Einstein Kähler metrics, Lett. Math. Phys. 50(3) (1999) 229–241], and Feldman, Ilmanen and Knopf [Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169–209]. We also prove a uniformization result on complete steady gradient Kähler–Ricci solitons with non-negative Ricci curvature under additional assumptions.

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A Note on Cohomology Groups of Holomorphic Line Bundles over a Complex Torus

Manifolds and Lie Groups ◽

10.1007/978-1-4612-5987-9_15 ◽

1981 ◽

pp. 301-313 ◽

Cited By ~ 2

Author(s):

Shingo Murakami

Keyword(s):

Line Bundles ◽

Cohomology Groups ◽

Complex Torus ◽

Holomorphic Line Bundles

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THE PICARD GROUP OF THE LOOP SPACE OF THE RIEMANN SPHERE

International Journal of Mathematics ◽

10.1142/s0129167x10006471 ◽

2010 ◽

Vol 21 (11) ◽

pp. 1387-1399

Author(s):

NING ZHANG

Keyword(s):

Lie Algebra ◽

Lie Group ◽

Loop Space ◽

Riemann Sphere ◽

Loop Group ◽

Picard Group ◽

Line Bundles ◽

Projective Embedding ◽

Infinite Dimensional ◽

Holomorphic Line Bundles

The loop space Lℙ1 of the Riemann sphere consisting of all Ck or Sobolev Wk, p maps S1 → ℙ1 is an infinite dimensional complex manifold. We compute the Picard group pic(Lℙ1) of holomorphic line bundles on Lℙ1 as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H0, 1(Lℙ1). The group G of Möbius transformations and its loop group LG act on Lℙ1. We prove that an element of pic(Lℙ1) is LG-fixed if it is G-fixed, thus completely answering the question of Millson and Zombro about the G-equivariant projective embedding of Lℙ1.

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