• DOI:10.4310/JDG/1214445039
• Corpus ID: 120882886

On a set of polarized Kähler metrics on algebraic manifolds

@article{Tian1990OnAS,
  title={On a set of polarized K{\"a}hler metrics on algebraic manifolds},
  author={Gang Tian},
  journal={Journal of Differential Geometry},
  year={1990},
  volume={32},
  pages={99-130}
}

• G. Tian
• Published 1 July 1990
• Mathematics
• Journal of Differential Geometry

A projective algebraic manifold M is a complex manifold in certain projective space CP, N > dim c M = n . The hyperplane line bundle of CP restricts to an ample line bundle L on M. This bundle L is a polarization on M. For the Kahler metric g on M, we can associate a positive, rf-closed (1, l)-form ωg . In any local coordinate system (Zj, , zn) of M , the metric g is expressed by a tensor (^/j)1</ 7-<fI, and ωg is defined to be ^^Σ" j=ι gqdzi Λ d~z.. We call this ω the Kahler form associated to the metric g . By a polarized Kahler metric with respect to L, we mean a Kahler metric with its associated Kahler form representing the Chern class CX(L) of L in H (M, Z). We denote by Ka(M) the set of all polarized Kahler metrics on M with respect to L. Given a Kahler metric # in Ka(Af), one can find a hermitian metric h on L with its Ricci curvature form equal to ω (cf. [7], or Lemma 1.1 in §1). For each positive integer m > 0, the hermitian metric h induces a hermitian metric h on L m . Choose an orthonormal basis {STM9 ,S n N}of the space H°{M, L ) of all holomorphic global sec-