Partial \(C^0\)-estimate for Kähler-Einstein metrics (Q376056)

Let \(M\) be a compact Kähler manifold with positive Chern class \(c_1(M)\) and let \(\mathcal{K}(M)\) be the set of all Kähler metrics with Kähler class \([\omega ]=c_1(M)\). Consider its subset \(\mathcal{K}(M, t_0)=\{\omega \in \mathcal{K}(M); \text{Ric}(\omega )\geq t_0\omega \}\). Hence, \(\mathcal{K}(M, t_0)\) is empty unless \(t_0\leq 1\) and \(\mathcal{K}(M, 1)\) is the set of Kähler-Einstein metrics on \(M\) with Kähler class \(c_1(M)\). The author stated some time ago the following conjecture: There are uniform constants \(c_k=c(k, n)>0\) for \(k\geq 1\) and \(l_i\rightarrow \infty \) such that for any \(\omega \in \mathcal{K}(M, t_0)\) and \(l=l_i\) for each \(i\), \(\rho _{\omega , l}\geq c_l>0\). Here \(\rho _{\omega , l}(x)=\sum _{i=0}^N\|\sigma _i\|^2_h(x)\) where \(M\) is embedded in \(\mathbb{C}\mathbb P^N\) and \(\{\sigma _i\}_{0\leq i\leq N}\) is an orthonormal basis of \(H^0(M, K_M^{-l})\). The main result of this paper is a proof of this conjecture for \(\mathcal{K}(M, 1)\). As a corollary, the Gromov-Hausdorff limits of Kähler-Einstein manifolds are proved to be projective varieties.