Hodge spectral sequence on compact Kähler spaces. (English) Zbl 0626.32029
The purpose of this paper is to prove the following
Theorem 1. Let X be a compact Kähler space of pure dimension n whose singular points are isolated, and let \(X^*\) be the complement of the singular points. Then \[ E_ 1^{p,q}=E_{\infty}^{p,q},\quad E_ 1^{p,q}\cong E_ 1^{p,q}\quad for\quad any\quad p,q \] (E\({}_ r^{p,q}\) denotes the Hodge spectral sequence on X) holds on \(X^*\) for the range \(p+q<n-1.\)
The author also gives a partial answer to a question of J. Cheeger, M. Goresky and R. MacPherson [Ann. Math. Stud. 102, 303-340 (1982; Zbl 0503.14008)] by proving the following:
Theorem 2. Under the situation of Theorem 1, \[ H^ r(X^*)\cong H^ r_{(2)}(X^*)\;if\;r<n-1,\quad H^{p,q}(X^*)\cong H^{p,q}_{(2)}(X^*)\;if\;p+q<n-1 \] and \[ H^ r_ 0(X^*)\cong H^ r_{(2)}(X^*)\;if\;r>n+1,\quad H_ 0^{p,q}(X^*)\cong H^{p,q}_{(2)}(X^*)\;if\;p+q>n+1. \] Here, H, \(H_ 0\) and \(H_{(2)}\) denote respectively the ordinary cohomology, the cohomology with compact support, and the \(L^ 2\) cohomology.
Theorem 2 implies the following
Corollary. \(IH^ r(X)\cong H^ r_{(2)}(X^*)\) if \(r\neq n\), \(n\pm 1\). \((IH^ r(X)\) is the intersection cohomology).
Cheeger-Goreski-MacPherson conjectured that the above isomorphism is valid for any degree, and in some special case it has been verified [cf. L.Saper, Invent. Math. 82, 207-255 (1985; Zbl 0611.14018)].
Theorem 1. Let X be a compact Kähler space of pure dimension n whose singular points are isolated, and let \(X^*\) be the complement of the singular points. Then \[ E_ 1^{p,q}=E_{\infty}^{p,q},\quad E_ 1^{p,q}\cong E_ 1^{p,q}\quad for\quad any\quad p,q \] (E\({}_ r^{p,q}\) denotes the Hodge spectral sequence on X) holds on \(X^*\) for the range \(p+q<n-1.\)
The author also gives a partial answer to a question of J. Cheeger, M. Goresky and R. MacPherson [Ann. Math. Stud. 102, 303-340 (1982; Zbl 0503.14008)] by proving the following:
Theorem 2. Under the situation of Theorem 1, \[ H^ r(X^*)\cong H^ r_{(2)}(X^*)\;if\;r<n-1,\quad H^{p,q}(X^*)\cong H^{p,q}_{(2)}(X^*)\;if\;p+q<n-1 \] and \[ H^ r_ 0(X^*)\cong H^ r_{(2)}(X^*)\;if\;r>n+1,\quad H_ 0^{p,q}(X^*)\cong H^{p,q}_{(2)}(X^*)\;if\;p+q>n+1. \] Here, H, \(H_ 0\) and \(H_{(2)}\) denote respectively the ordinary cohomology, the cohomology with compact support, and the \(L^ 2\) cohomology.
Theorem 2 implies the following
Corollary. \(IH^ r(X)\cong H^ r_{(2)}(X^*)\) if \(r\neq n\), \(n\pm 1\). \((IH^ r(X)\) is the intersection cohomology).
Cheeger-Goreski-MacPherson conjectured that the above isomorphism is valid for any degree, and in some special case it has been verified [cf. L.Saper, Invent. Math. 82, 207-255 (1985; Zbl 0611.14018)].
Reviewer: T.Rassias
MSC:
| 32J25 | Transcendental methods of algebraic geometry (complex-analytic aspects) |
| 55N10 | Singular homology and cohomology theory |
| 14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |
| 53C55 | Global differential geometry of Hermitian and Kählerian manifolds |
| 32C15 | Complex spaces |
| 55N35 | Other homology theories in algebraic topology |
