Resonance varieties and Dwyer-Fried invariants (Q2906438)

The paper under review is illuminating and illustrates how to use the technique of resonance varieties and Dwyer-Fried invariants for studying broad class of important object, such as hyperplane arrangements, toric complexes, Kähler and quasi Kähler manifolds, etc. The exposition of the paper is excellent and familiar even to the non-specialists in the topic. The various application and connections with algebraic topology, algebraic geometry, combinatorics, groups, etc. make this article attractive to wide spectra of researchers.NEWLINENEWLINEThere are two parts of the paper. In the first part the technique used in the latter part is presented. The Aomoto complex and universal Aomoto complex of the cohomology algebra have been \(H^\ast (X, \mathbb{C})\) defined at the beginning, where \(X\) is a connected \(CW\) complex with finite \(k\)-skeleton. These complexes are studied in the previous papers of Papadima and Suciu. They are especially interesting when \(X\) is minimal cell complex. From the cohomology algebra of the Aomoto complex the resonance varieties sets of \(X\) are defined. They are homogeneous algebraic subvarieties of the affine space \(H^1 (X, \mathbb{C})\) and provide its a descending filtration. The case of depth one resonance varieties is carefully treated, because they could be calculated. The tangent cone and the exponential tangent cone of a Zariski closed subset \(W\) of algebraic torus are defined and related to each other by Lemma 4.4. In Section 5, a nice exposition about the characteristic varieties of a connected CW-complex \(X\) with finite \(k\)-skeleton is given. Their relation with other topological invariant such as the Alexander polynomial of the link and varieties is also discussed. The notions of straightness, \(k\)-straightness and locally straightness are then developed and explained on concrete examples. It is shown that they behave well with respect to topological operations (products and wedges of spaces, etc.). Also, an alternative approach is explained and in Theorem 6.16 and Corollary 6.17 the criteria in terms of the resonance and the characteristic varieties for locally \(k\)-straightness and \(k\)-straightness are given. Dwyer-Fried invariants are defined as certain subsets of a rational Grassmannian \(\mathrm{Gr}_r (H^1 (X, \mathbb{Q})\), and it is proved that for \(k\)-straight spaces they are complements to the special Schubert variety. Finally, the formality notion and its connection with straightness is studied in the Section 8.NEWLINENEWLINEThe rest of the paper is devoted to applications. Toric complexes are spaces naturally related to the simplicial complex and it is shown that they are straight spaces. Also their Dwyer-Fried invariants are determined. In Section 10 Kähler and quasi-Kähler manifolds are studied. They are a broad class of manifolds that includes Riemann surfaces. Their varieties and Dwyer-Fried invariants are treated. Finally, all the technique is applied to the case of hyperplane arrangements.NEWLINENEWLINEThis paper gives to the reader fresh ideas and it could be interesting to wide spectra of researchers. The theory of the resonance varieties for the first time look very abstract, it is connected with algebraic topology, algebraic and differential geometry, combinatorics, etc. and it is exposed in a reader friendly way.NEWLINENEWLINEFor the entire collection see [Zbl 1242.14003].