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Théorèmes d’ annulation. (Vanishing theorems). (French) Zbl 0698.14010

Hyperrésolutions cubiques et descente cohomologique, Expo. Sémin. Théor. Hodge-Deligne, Barcelona/Spain 1982, Lect. Notes Math. 1335, 133-160 (1988).
[For the entire collection see Zbl 0638.00011.]
In the following a \({\mathbb{C}}\)-scheme means a separated scheme of finite type over the complex numbers \(\mathbb{C}\). Let \(X\) be a \(\mathbb{C}\)-scheme and \(a: X_{\bullet} \to X\) a cubic hyperresolution of \(X\). [For the definition, see, for example, I.3 of the present conference: F. Guillén, Lect. Notes Math. 1335, 1-42 (1988; Zbl 0673.14013).] Let \((\Omega_{X_\bullet},F)\) be the de Rham complex of \(X_\bullet\) with stupid filtration \(F\). Then \(\mathbb{R} a_*(\Omega^*_{X_\bullet},F)\) is an element of the derived category \(D^ b_{\text{diff,coh}}(X)\) of filtered bounded complexes with differential operators of degree \(\leq 1\). Moreover, \(\mathbb{R}^ a_* (\Omega^ *_{X_\bullet},F)\) is independent of the choice of resolutions. By \((\bar\Omega_ X,F)\) we denote the filtered complex \(\mathbb{R} ^ a_*(\Omega^*_{X_\bullet},F)\) and call it the filtered de Rham complex of \(X\). The filtered de Rham complex was introduced by P. Du Bois [Bull. Soc. Math. Fr. 109, 41-81 (1981; Zbl 0465.14009)]. Put \(\bar\Omega^ p_ X = \text{Gr}^ p_ F \bar\Omega_ X[p]\).
One of the main results of the present paper is the following theorem, which is a generalization of the vanishing theorem of Kodaira-Akizuki- Nakano to a singular variety:
Let \(X\) be a projective \(\mathbb{C}\)-scheme and \(L\) an ample line bundle on \(X\). Then we have \(H^ q(X,\bar\Omega^ n_ X\otimes L)=0\) for \(p+q>\dim(X)\).
A relative version of the theorem is the following theorem:
Let \(f:X\to Y\) be a proper morphism of \(\mathbb{C}\)-schemes and \(L\) be an \(f\)-ample line bundle on \(X\). Then, we have \(\mathbb{R}^ q f_*(\bar\Omega^ p_ X\otimes L) =0\) for \(p+q>\dim(X)\).
As a corollary the author obtains the following generalization of the vanishing theorem of Grauert-Riemenschneider:
Let \(X\) be a proper \(\mathbb{C}\)-scheme. Then, we have \(H^ q(X,\bar\Omega^ p_ X)= 0\) for \(p+q>\dim(X).\)
For a proof of the first theorem, the author uses the method of C. P. Ramanujan, Indian Math. Soc., New Ser. 36, 41-51 (1972; Zbl 0276.32018)] combined with the weak Lefschetz theorem and the Hodge-Deligne theory. The second theorem is an easy consequence of the first one.
Reviewer: K.Ueno

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32L20 Vanishing theorems