. SG ] 2 3 Ju n 20 06 Semiclassical almost isometry
نویسنده
چکیده
Let M be an irreducible n-dimensional complex projective manifold, and A → M an ample line bundle on it. Then there exists an Hermitian metric h on A such that the curvature of the unique compatible covariant derivative is −2πiω, where ω is a Kähler form on M . As is well-known, for k ≫ 0 the full linear series of global holomorphic sections of A⊗k determines a projective embedding φk : M → P ( H0(M,A⊗k)∗ ) ; the asymptotic metric properties of φk as k → +∞ are governed by the Tian-Zelditch almost isometry theorem [T], [Z] which is crucial in many recent advances in complex and symplectic geometry. Explicitly, the choice of h and ω endows every space of global holomorphic sections H0(M,A⊗k) with a natural Hermitian structure; let us choose for every k ≫ 0 an orthonormal basis of H0(M,A⊗k), so as to view φk as a map φk : M → Pk . Then, when the projective space Pk is endowed with the Fubini-Study metric ωFS, for every l ≥ 0 one has ∥∥∥∥ω − 1 k φk(ωFS) ∥∥∥∥ Cl = O(k−1), (1)
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تاریخ انتشار 2006