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A generalization of Griffiths’ theorem on rational integrals. II. (English) Zbl 1192.14009
Let $$Y$$ be a reduced hypersurface in $$X = \mathbb P^n$$ defined by a polynomial $$f$$ of degree $$d,$$ and let $$F$$ and $$P$$ denote the global Hodge and pole order filtration on $$H^n(X\setminus Y, \mathbb C),$$ respectively. The authors show that for any sufficient general hypersurface $$Y$$ whose singular locus $$\text{Sing}\, Y$$ consists of one ordinary double point and $$d=3,$$ $$n\geq 5,$$ or $$d=4,$$ $$n\geq 3,$$ one has $$F^p \neq P^p,$$ where $$1+ (n+1)/d \leq p \leq n -[n/2].$$ The proof is based on the construction of a series of examples. Under additional assumptions they then compute $$\text{Gr}_F^p H^n(X\setminus Y, \mathbb C),$$ $$p< n - [n/2],$$ for a hypersurface $$Y$$ with some ordinary double singular points in terms of graded pieces of the Jacobian ideal of $$f$$ and of powers of the homogeneous ideal of $$\text{Sing}\, Y.$$
In fact, their computations partially support a Conjecture by L. Wotzlaw [Intersection cohomology of hypersurfaces (Dissertation), Humboldt Universität zu Berlin (2007)] which is a generalization of the Griffiths’ theorem on rational integrals [P. A. Griffiths, Ann. Math. (2) 90, 460–495, 496–541 (1969; Zbl 0215.08103)].
[For part I, cf. Duke Math. J. 135, No. 2, 303–326 (2006; Zbl 1117.14012)].

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 32S35 Mixed Hodge theory of singular varieties (complex-analytic aspects) 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects) 14F40 de Rham cohomology and algebraic geometry 14J70 Hypersurfaces and algebraic geometry 32S20 Global theory of complex singularities; cohomological properties 55R55 Fiberings with singularities in algebraic topology 57R60 Homotopy spheres, Poincaré conjecture 57R70 Critical points and critical submanifolds in differential topology
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