Adjoint linear series on weakly 1-complete Kähler manifolds. I: Global projective embedding (Q1264324)

A complex manifold \(X\) is said to be weakly 1-complete if there exists a smooth function \(\Phi:X\to \mathbb R\) which is plurisubharmonic and exhaustive. For each point \(x\) on \(X\), put \(d(x)=\max\{\dim V: V\) is a compact subvariety of \(X\) passing through \(x\}\). The main results are the following. Theorem 1. Let \(X\) be an \(n\)-dimensional weakly 1-complete manifold with a positive line bundle \(L\). Then \(K_X\otimes L^{\otimes m}\) is ample for every \(m>n(n+1)/2\). Theorem 2. Let \(x_1,\ldots,x_r\) be \(r\) distinct points on a sublevel set \(X_c\) and let \(d_x=\max\{d(x_i):i=1,\ldots,r\}\). Then for every positive integer \(m>{1\over 2}d_x(d_x+2r-1)\), the restriction map \(H^0(X_c,K_X\otimes L^{\otimes m})\to \oplus^r_{i=1} {\mathcal O}_X/{\mathcal M}_{X,x_i}\) is surjective. Theorem 3. Let \(X\) be a weakly 1-complete manifold. Then the following three statements are equivalent. (1) \(X\) is holomorphically embeddable into a projective space. (2) \(X\) admits a positive line bundle. (3) There exists an integral Kähler form on \(X\). Theorem 4. Every holomorphically convex complex manifold with a positive line bundle admits a proper holomorphic embedding into a product space of a projective space and a complex Euclidean space. Theorem 5. A weakly 1-complete manifold with a negative canonical bundle is Stein if and only if it has no compact subvarieties of positive dimension.