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Depth and perversity. (English) Zbl 0777.14005
Algebraic geometry and analytic geometry, Proc. Conf., Tokyo/Jap. 1990, ICM-90 Satell. Conf. Proc., 111-125 (1991).
[For the entire collection see Zbl 0744.00034.]
This paper continues and completes some results on: (1) H. A. Hamm and the author [in The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck, Vol. II, Prog. Math. 87, 311-351 (1990; Zbl 0725.14016)], and (2) the author [J. Algebr. Geom. 1, No. 1, 83-99 (1992)].
The main results are: (a) An improvement of the homotopy version of the classical Lefschetz theorem: namely, that the level of homotopy comparison in this theorem is maximal for the spaces satisfying Milnor’s property (see (2)). – (b) A theorem of Lefschetz type for constructible sheaves satisfying some depth condition. In fact, the author defines the “rectified homological depth” of a constructible complex \(\mathbb{K}^*\) on a complex analytic space, noted rHd\((\mathbb{K}^*)\), and proves that if \(V\) is a complex projective \(d\)-dimensional variety and \(V\cap H\) is a hyperplane section, then for every \(\mathbb{K}^*\) as above such that the rHd\((\mathbb{K}^*|_{V\backslash V\cap H})\geq n\), the morphism \(\mathbb{H}^ i(V,\mathbb{K}^*)\to\mathbb{H}^ i(V\cap H,\mathbb{K}^*)\) is an isomorphism for \(i\leq n-d-2\) and is an injection for \(i=n-d-1\). The result generalizes another result of M. Goresky and R. MacPherson [Invent. Math. 72, 77-129 (1983; Zbl 0529.55007)] for the intersection complex.
To finish, let us mention that the notion of rectified homological depth of a constructible complex can be easily expressed in terms of the \(t\)- structure with respect to the autodual perversity, in the sense of A. A. Beilinson, J. Bernstein and P. Deligne, Astérisque 100 (1982; Zbl 0536.14011), and the proof of the above result follows from a theorem of M. Artin and A. Grothendieck on the direct images by affine maps.
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
32S60 Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects)