Scalar curvature and projective embeddings. I.

*(English)*Zbl 1052.32017Let \(X\) 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 \(2\pi\,c_1(L)\).

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 \(2\pi\,c_1(L)\).

Reviewer: Akihiko Morimoto (Nagoya)