Algebraicity of formal varieties and positivity of vector bundles
The aim of the article under review is to propose a positivity condition for vector bundles on a projective variety and use this positivity condition to present an algebraicity criterion for formal schemes. Then the author applies the algebraicity criterion to the study of formal principle in algebraic geometry. On the technical side, let \(X\) be an integral projective scheme over any field \(k\) and let \(Y\) be an integral closed subscheme of \(X\) which defines a formal closed subscheme \(\widehat{V}\) of \(\widehat{X}_Y\). The author actually proves that under the condition \(P_3(Y,N)\), there exist an ample line bundle \(L\) on \(Y\) and \(\lambda>0\) such that for all integers \(n\) and \(D\) with \(n/\lambda>D>\lambda\), one has \(\mathrm{H}^0(Y,S^n(N^\vee)\otimes L^{\otimes D})=0\), the formal scheme \(\widehat{V}\) is algebraic. Here \(N\) is the normal bundle \(N_Y\widehat{V}\). This result relies on the slope method coming from Arakelov geometry and it largely weakens the ampleness condition of \(N_Y\widehat{V}\) in usual algebraicity criterion for \(\widehat{V}\). Concerning the application of the main result in the study of formal principle, the author proves that: let \((W,A)\) be a pair of smooth projective varieties where \(A\) has dimension \(\geq1\), if the normal bundle \(N_AW\) verifies the condition \(P_3\), then the formal principle holds for the pair \((W,A)\).
algebraicity criterion of formal varieties
scientific article; zbMATH DE number 6083999
scientific article