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Complex structures on product of circle bundles over complex manifolds

Parameswaran Sankaran, Ajay Singh Thakur (2013)

Annales de l’institut Fourier

Let L ¯ i X i be a holomorphic line bundle over a compact complex manifold for i = 1 , 2 . Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S = S 1 × S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant ( * ) n i -bundles satisfying some additional conditions. The linear type complex structures...

Convergence of Bergman geodesics on CP 1

Jian Song, Steve Zelditch (2007)

Annales de l’institut Fourier

The space of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X is an infinite dimensional symmetric space whose geodesics ω t are solutions of a homogeneous complex Monge-Ampère equation in A × X , where A is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials ϕ ( t , z ) of ω t may be approximated in a weak C 0 sense by geodesics ϕ N ( t , z ) of the finite dimensional symmetric space of Bergman metrics of height N . In this article we prove that ϕ N ( t , z ) ϕ ( t , z ) in C 2 ( [ 0 , 1 ] × X ) in the case of...

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