Discriminants, resultants and a conjecture of S. Halperin. (English) Zbl 1018.55008

The Halperin conjecture states that for every orientable fibration \(X\to Y\to B\) with fibre \(X\) a space of type \(F_0\) (i.e. \(\dim \pi_* (X)\otimes Q<\infty\), \(H^{\text{odd}}(X,\mathbb{Q})=0\), and \(\dim H^*(X,\mathbb{Q}) <\infty)\) the Serre spectral sequence degenerates at the term \(E_2\). The conjecture has been proved for a large family of spaces of type \(F_0\), for example for homogeneous spaces \(G/H\) of rank zero by Shiga and Tezuka, and in the case of \(H^*(X,\mathbb{Q})\) being generated by \(\geq 3\) elements by Hao Chen. The paper under review contains a very nice survey on the subject. The author also introduces differential calculus and algebraic geometry around the conjecture. Consider all the algebras of the form \(\mathbb{Q}[a_1, \dots,a_n]/(f_1, \dots,f_n)\) where \(n\), the degrees of the \(a_i\) and the degrees of the homogeneous polynomials \(f_i\) are fixed. Each sequence \((f_1,\dots,f_n)\) corresponds to a point in some \(\mathbb{Q}^N\). The sequence is regular if some resultant does not vanish. Therefore regular sequences form a Zariski open set \(V\). The author proves that the sequences \((f_1,\dots,f_n)\) that satisfy the Halperin conjecture form an open subset of \(W\).


55P62 Rational homotopy theory
55T10 Serre spectral sequences
13D10 Deformations and infinitesimal methods in commutative ring theory
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
14B07 Deformations of singularities
14B12 Local deformation theory, Artin approximation, etc.
14M10 Complete intersections