On the Hodge spectral sequence for some classes of non-complete algebraic manifolds
The following two theorems concerning the degeneration of the Hodge spectral sequence for non-complete algebraic manifolds are proved via Deligne's and Illusie's purely algebraic proof of the \(E_ 1\)- degeneration [\textit{P. Deligne} and \textit{L. Illusie}, Invent. Math. 89, 247-270 (1987; Zbl 0632.14017)]: Theorem A. Let K be a field of characteristic zero, X a scheme, integral and proper over Spec(K) and \(Y\subset X\) a closed subscheme such that \(U:=X\setminus Y\) is smooth. Then (1) the K-vector spaces \(H^ n_{DR}(U/K)\) and \(H^ j(U,\Omega^ i_{U/K})\) are finite dimensional for n, j\(<co\dim (Y,X)-1\) and any i, (2) the differentials \(d_ n^{i,j}\) of the Hodge spectral sequence \(E_ 1^{i,j}=H^ j(U,\Omega^ i_{U/K})\Rightarrow H_{DR}^{i+j}(U/K)\) are zero for \(i+j<co\dim (Y,X)-1\) and \(r\geq 1.\) Theorem B. Let K be a field of characteristic zero, X an integral scheme, smooth and proper over Spec(K), \(Y\subset X\) a divisor and \(U:=X\setminus Y\). If the normal bundle \(N_{Y/X}\) of Y in X is k-ample, then (1) the K-vector spaces \(H^ n_{DR}(U/K)\) and \(H^ j(U,\Omega^ i_{U/K})\) are finite dimensional for \(n\geq \dim (X)+k+1\) and \(j\geq k+1\), respectively, (2) the differentials \(d_ r^{i,j}\) of the Hodge spectral sequence \(E_ 1^{i,j}=H^ j(U,\Omega^ i_{U/K})\Rightarrow H_{DR}^{i+j}(U/K)\) are zero for \(i+j\geq \dim (X)+k+1\) and \(r\geq 1.\)
degeneration of the Hodge spectral sequence for non- complete algebraic manifolds
scientific article; zbMATH DE number 4160908
scientific article