## 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

Zbl 0706.53036
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