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Vanishing theorems on singular spaces. (English) Zbl 0582.32039
Systèmes différentiels et singularités, Colloq. Luminy/France 1983, Astérisque 130, 330-341 (1985).
[For the entire collection see Zbl 0559.00004.]
Generalizing a classical vanishing theorem of Kodaira-Akizuki-Nakano [see Y. Akizuki and Sh. Nakano, Proc. Japan Acad. 30, 266-272 (1954; Zbl 0059.147)], H. Grauert and O. Riemenschneider [Invent. Math. 11, 263-292 (1970; Zbl 0202.076)] proved, for \(p=n,\) a) \(H^ q(X,\pi_*\Omega^ p_{\tilde X}\otimes L)=0\) for \(q>0,\)
b) \(R^ q\pi_*\Omega^ p_{\tilde X}=0\) for \(q>0,\)
where X is a compact complex space of dimension n, L an ample line bundle on X, and \(\pi: \tilde X\to X\) a proper birational morphism with \(\tilde X\) smooth. It is known that a) may fail for \(p\neq n\). Guillen, Navarro Aznar, and Puerta [Barcelona Notes (1982)] showed that, for X and L as above, \[ a)\quad H^ m(X,Gr^ p_ FK_{\dot X}\otimes L)=0\text{ for } m>n, \]
\[ b)\quad {\mathcal H}^ m(Gr^ p_ FK_{\dot X})=0\;text{ for } m<p \text{ or } m>n, \] where \((K^._{X},F)\) is the filtered de Rham complex of X, and H denotes hypercohomology and \({\mathcal H}\) denotes cohomology sheaf.
The author proves the following vanishing theorem, from which he derives a’) and b’): Let X be an n-dimensional complex projective variety, \(\Sigma\) \(\subset X\) such that \(X\setminus \Sigma\) is nonsingular, L an ample line bundle on X and \(\pi: \tilde X\to X\) a proper birational mapping such that \(\tilde X\) is nonsingular, \(E=\pi^{-1}(\Sigma)\) is a divisor with normal crossings on \(\tilde X\) and \(\pi\) maps \(X\setminus E\) isomorphically to \(X\setminus \Sigma\). Then \[ a)\quad H^ q(\tilde X,J_ E \Omega^ P_{\tilde X}(\log E)\otimes \pi L)=0\quad for\quad p+q>n,\quad b)\quad R^ q\pi_*J_ E\Omega^ P_{\tilde X}(\log E)=0\quad for\quad p+q>n. \] Here \(\Omega^._{\tilde X}(\log E)\) is the logarithmic de Rham complex and \(J_ E\) is the ideal sheaf of the divisor E.
Reviewer: M.Röhrl

32L20 Vanishing theorems
32C35 Analytic sheaves and cohomology groups
14F40 de Rham cohomology and algebraic geometry
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
32J99 Compact analytic spaces