Mixed Hodge structures on the homotopy of links
This paper uses mixed Hodge theory to study the homotopy of a complex algebraic variety in the neighborhood of a singular point, or more generally, in the neighborhood of a subvariety. An important application is finding topological restrictions on the links of isolated singular points. One such application is the following theorem: Let L be the link of an isolated singularity of an n-dimensional variety. If s, t\(<n\) and \(s+t\geq n\), then the cup product \(H^ s(L;{\mathbb{Q}})\otimes H^ t(L;{\mathbb{Q}})\to H^{s+t}(L;{\mathbb{Q}})\quad vanishes\). (Recall that the link of an isolated singularity in an n-dimensional variety is a real (2n-1)-manifold.) This theorem is a direct consequence of the following three facts: (1) The cohomology \(H^ k(L)\) (which is isomorphic to \(H_ x^{k+1}(X))\) has a natural mixed Hodge structure. (2) The weights of \(H^ k(L)\) are less than or equal to k for \(k<n\), and greater than or equal to \(k+1\) for \(k\geq n.\) (3) The cup product in the cohomology of L is a morphism of mixed Hodge structures and therefore preserves weights. The first assertion is due to Deligne. The second is a well-known consequence of Gabber's purity theorem and the decomposition theorem of intersection homology, both deep theorems. This paper establishes fact 3. It is more convenient to work not just with isolated singularities, but in the following more general situation: Suppose that X is a projective variety and that Z and Z' are closed subvarieties. Set \(Y=Z\cup Z'\) and suppose that the singular locus of X is contained in Y. Let T be a sufficiently nice neighborhood of Z in X. Define the link L(X,Y,Z) of Z in X with Y removed by \(L(X,Y,Z)=T-Y\). This is well defined up to homotopy. A de Rham mixed Hodge complex is constructed for L(X,Y,Z). The main result of this paper then follows from work of \textit{R. M. Hain} [''The de Rham homotopy theory of complex algebraic varieties. I'', J. K-Theory (to appear)] or \textit{J. W. Morgan} [Publ. Math., Inst. Hautes Etud. Sci. 48, 137-204 (1978; Zbl 0401.14003)] This result may be summarized as follows: If \(L=L(X,Y,Z)\) is as above, then: \((i)\quad For\) all k, \(H^ k(L)\) has a real mixed Hodge structure. \((ii)\quad The\) cup product of H(L) is a morphism of mixed Hodge structures. \((iii)\quad The\) real homotopy type of L has a mixed Hodge structure. The main idea behind the proof is the use of the machinery developed in the paper of Hain cited above.
decomposition theorem of intersection homology
scientific article; zbMATH DE number 4000099
scientific article