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Zbl 1094.14005
de Cataldo, Mark Andrea A.; Migliorini, Luca
The Hodge theory of algebraic maps. (English)
[J] Ann. Sci. Éc. Norm. Supér. (4) 38, No. 5, 693-750 (2005). ISSN 0012-9593

Let $f:X\to Y$ be a proper map of complex algebraic varieties. When $f$ is smooth and $X,Y$ are projective, the classical Hard Lefschetz Theorem applied to the fibers of $f$ gives rise to a direct sum decomposition $Rf_*\Bbb Q_X \cong\bigoplus_iR^if_*\Bbb Q_x[-i]$ in the derived category of the category of sheaves on $Y$. In this paper the authors give a geometric proof of the decomposition theorem of {\it A. A. Beilinson}, {\it J. Bernstein} and {\it P. Deligne} [Faisceaux pervers, Astérisque 100 (Soc. Math. France, Paris) (1982; Zbl 0536.14011)], which is a vast generalization of the classical one mentioned above. For the proof the authors employ a double induction on the defect of semismallness $r(f)=\max\{2i+\dim Y^i-\dim X;Y^i\ne\varphi\}$, where $Y^i=\{y\in Y;\dim f^{-1}(y)=i\}$, and on the dimension of the target of the map $f$. They make great efforts for the readability of this important paper by collecting results on the theory of stratifications, constructible sheaves and perverse sheaves and by providing us with several results in a form that is less general but sharper than what one could find in the literature.
[Fumio Hazama (Hatoyama)]

MSC 2000:: *14C30 Transcendental methods
14F43 Other algebro-geometric (co)homologies
14D06 Fibrations, degenerations

Keywords: intersection cohomology; perverse sheaves

Citations: Zbl 0536.14011

Cited in: Zbl 1124.14002

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