Vanishing theorems for the intersection homology of Stein spaces
The paper contains a proof of a general vanishing result for the intersection homology for complex spaces: For an (analytically) q- complete pure n-dimensional complex space X, fix \({\mathcal L}^ a \)locally constant sheaf of R-modules with finitely generated stalks on the maximal stratum of the Whitney stratification \({\mathfrak X}\) on X; for a perversity \({\mathfrak p}\), \(\Sigma_{{\mathfrak p}t}\) is the complex analytic subset of X such that \(X\setminus \Sigma_{{\mathfrak p}t}\) is the maximal open subset such that on it the Deligne complexes \({\mathcal P}\) \(._{{\mathfrak p}}{\mathcal L}\) and \({\mathcal P}\) \(._ t{\mathcal L}\) are quasiisomorphic; \(\Sigma_{{\mathfrak p}t}\) is s-complete for some \(s\leq q\). If \({\mathfrak m}\) denotes the middle perversity, \(\beta\) is an integer and \({\mathfrak p}^ a \)perversity such that \({\mathfrak p}\geq {\mathfrak m}-\beta\) then: The modules \(I_{{\mathfrak p}}H\) \(c_ j(X,{\mathcal L})={\mathbb{H}}_ c^{2n- j}(X,{\mathcal P}\) \(._{{\mathfrak p}}{\mathcal L})\) vanish for \(j\geq n+1+\max (q,s+\beta -1).\) If the sheaf \({\mathcal L}\) is torsion-free, then the module \(I_{{\mathfrak p}}H\) \(c_{n+\max (q,s+\beta)}(X,{\mathcal L})\) is torsion- free; it is even free if the ring R is completely reflexive. One of the main ideas consists in using the degree of topological completeness tc(X)\(\in {\mathbb{Z}}\) introduced in the authors' paper in Algebraic Geometry, Proc. Conf. Berlin 1985, Teubner-Texte Math. 92, 83- 104 (1986; Zbl 0625.32016). Application of Lefschetz-type results will be discussed in the authors' paper ``Theorems of Lefschetz-type in intersection homology'' (to appear).
scientific article; zbMATH DE number 4049990
scientific article