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Scalar curvature and projective embeddings. I. (English) Zbl 1052.32017
Let $$X$$ be a compact complex manifold of complex dimension $$n$$.
The author investigates the relations between the existence of a Kähler metric of constant scalar curvature on $$X$$ and the existence of certain sequence of projective embeddings of $$X$$. For $$n=1$$, as is well-known, both existences above hold. Let $$L\to X$$ be a positive line bundle. The Kodaira embedding theorem asserts that for large enough $$k$$ the sections $$H^0(L^k)$$ of $$L^k$$ define a projective embedding $$\iota_k: X\to P(H^0(L^k)^*)$$. Let $$[z_0,z_1,\dots, z_N]$$ be the standard homogeneous coordinates on $$\mathbb{C} P^N$$ and define $$b_{\alpha\beta}= z_\alpha z_\beta/| z|^2$$, where $$| z|^2= \sum| z_\alpha|^2$$. If $$V\subset \mathbb{C} P^N$$ is any projective variety we define $$M(V)$$ to be the $$(N+ 1)\times(N+ 1)$$ matrix with entires $(M(V))_{\alpha\beta}= i \int_V b_{\alpha\beta} \,d\mu_V,$ where $$d\mu_V$$ is the standard measure on $$V$$ induced by the Fubini-Study metric. If $$M(V)$$ is a multiple of the identity matrix we call $$V$$ a balanced variety in $$\mathbb{C} P^N$$. We also say that $$(X,L^k)$$ is balanced if we can choose a basis of $$H^0(L^k)$$ such that $$\iota_k(X)$$ is a balanced variety in $$\mathbb{C} P^N$$.
In this case we have a well defined Kähler metric $$\omega_k$$ on $$X$$ defined by $$\omega_k= (2\pi/k)\iota^*_k(\omega_{FS})$$, where $$\omega_{FS}$$ is defined by the Fubini-Study metric, the cohomology class $$[\omega_k]= 2\pi\, c_1(L)\in H^2(X)$$ being independent of $$k$$. Now, suppose that the group $$\operatorname{Aut}(X,L)$$ of holomorphic automorphisms of the pair $$(X,L)$$ is discrete. The author proves that if $$(X,L^k)$$ is balanced for all sufficiently large $$k$$, and if $$\omega_k$$ converges in $$C^\infty$$ to some limit $$\omega_\infty$$ as $$k\to\infty$$, then $$\omega_\infty$$ has constant scalar curvature.
The author proves also the converse: Suppose $$\omega_\infty$$ is a Kähler metric in the class $$2\pi\,c_1(L)$$ with constant scalar curvature. Then $$(X,L^k)$$ is balanced for large enough $$k$$ and the sequence of metrics $$\omega_k$$ converges in $$C^\infty$$ to $$\omega_\infty$$ as $$k\to\infty$$. As a corollary we obtain the following: Suppose $$\operatorname{Aut}(X,L)$$ is discrete. Then there is at most one Kähler metric of constant scalar curvature in the cohomology class $$2\pi\,c_1(L)$$.

##### MSC:
 32Q15 Kähler manifolds 32Q40 Embedding theorems for complex manifolds 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
##### Keywords:
constant scalar curvature; balanced variety
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