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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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