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Szegö kernels and a theorem of Tian. (English) Zbl 0922.58082

Let \(M\) be a compact complex manifold of dimension \(n\), \((L,h)\to M\) a positive Hermitian holomorphic line bundle, and \(g\) the Kähler metric on \(M\) corresponding to the Kähler form \(\omega_g= Ric(h)\), where \( Ric(h)\) is the Ricci curvature of \(h\). For each \(N\in \mathbb{N}\), \(h\) induces a Hermitian metric \(h_N\) on \(L^{\otimes N}\), and let \(\{S^N_0,\ldots,S^N_{d_N}\}\) be any orthonormal basis of \(H^0(M,L^{\otimes N})\) with respect to the inner product \(\langle s_1,s_2\rangle_{h_N}=\int_M h_N(s_1(z),s_2(z)) dV_g\), \(dV_g\) being the volume form of \(g\). By using the Boutet de Monvel-Sjöstrand parametrix for the Szegö kernel, the author proves that, for any \(k\), \(\|\sum_{i=0}^{d_N}\| S^N_i(z) \|^2_{h_N}-\sum_{j<R} a_j(z) N^{n-j}\|_{C^k} \leq C_{R,k} N^{n-R}\), for certain smooth coefficients \(a_j(z)\) with \(a_0=1\).
This result implies that for sufficiently large \(N\), one can define the holomorphic map \(\phi_N\colon M\to\mathbb{C}\roman P^{d_N}\) taking \(z\) to the line through \((S^N_0(z),\ldots,S^N_{d_N}(z))\). Now, if \(\omega_FS\) denotes the Fubini-Study form on \(\mathbb{C}\roman P^{d_N}\) then \(\|(1/N)\phi^*_N(\omega_FS)-\omega_g\|_{C^k}=O(1/N)\), for any \(k\), which proves that the convergence of \((1/N)\phi^*_N(\omega_FS) \rightarrow \omega_g\) takes place in the \(C^\infty\)-topology. This was conjectured by G. Tian in J. Differ. Geom. 32, No. 1, 99-130 (1990; Zbl 0706.53036).

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
32Q15 Kähler manifolds

Citations:

Zbl 0706.53036
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