Vanishing cycles of pencils of hypersurfaces

Tibăr, Mihai

doi:10.1016/j.top.2003.09.005

Cited by 8 publications

(21 citation statements)

References 24 publications

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“…Our approach is in the spirit of the Lefschetz method [Lef], as presented by Andreotti and Frankel in [AF]. Since the homology conterpart of our result (which is easier) is presented in [Ti3], we concentrate here on the more delicate constructions and ingredients of the proof which are needed in case of homotopy groups. We use homotopy excision (the Blackers-Massey theorem), more precisely the general version proved by Gray [Gr,Cor;16.27], and the comparison between homotopy and homology groups via the Hurewicz map.…”

Section: Then We Have the Following Two Conclusionmentioning

confidence: 99%

“…that one may have A ∩ σ(Sing S p) = ∅. This occurs for instance if the transversality of the axis to strata of Y fails at finitely many points: see Proposition 2.2 and [Ti2,§2], [Ti3,§2] for more details.…”

Section: Introductionmentioning

confidence: 99%

“…Such nongeneric pencils were introduced in our preprint [Ti1] and this viewpoint allowed us to prove connectivity theorems of Lefschetz type [Ti2], and a far reaching extension of the Second Lefschetz Hyperplane Theorem [Ti3]. In the literature, examples of nongeneric pencils occured sporadically; more recently they got into light, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…[Ok1,Ok2]. Nongenericity also proved to be a useful concept for treating the topology of polynomial functions (see [NN,Ti3]) and of complements of arrangements (e.g. [LT]).…”

Section: Introductionmentioning

confidence: 99%

“…[Ph, Mi, La, Lo, Ga, NN], [Va1,Va2]). We have introduced in [Ti3] a global variation map in homology which generalises the local variation map to pencils with isolated singularities on arbitrarily singular (open) underlying spaces.…”

Section: Introductionmentioning

confidence: 99%

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Polynomials and Vanishing Cycles

Tibăr¹

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Section: Then We Have the Following Two Conclusionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…[Ok1,Ok2]. Nongenericity also proved to be a useful concept for treating the topology of polynomial functions (see [NN,Ti3]) and of complements of arrangements (e.g. [LT]).…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Polynomials and Vanishing Cycles

Tibăr¹

2007

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Complements of Hypersurfaces, Variation Maps, and Minimal Models of Arrangements

Tibăr¹

2014

Bridging Algebra, Geometry, and Topology

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Abstract. We prove the minimality of the CW-complex structure for complements of hyperplane arrangements in C n by using the theory of Lefschetz pencils and results on the variation maps within a pencil of hyperplanes. This also provides a method to compute the Betti numbers of complements of arrangements via global polar invariants. IntroductionTo study the topology of the complement C n \ V of an affine hypersurface V ⊂ C n one employs Morse theory, see for instance Randell [Ra], or the Lefschetz method of scanning by pencils of hyperplanes, as done e.g. by Dimca and Papadima in [DP]. Both methods yield in particular a CW-complex model of the complement C n \ V . It was proved in the above two papers that whenever V is a union of hyperplanes, then there exists a CWcomplex model which is minimal, in the sense that the number of q-cells equals the Betti number b q (C n \ V ), for any q. This notion of minimality was introduced by Papadima and Suciu in [PS] for studying the higher homotopy groups of complements of hyperplane arrangements. We give here a new proof of the minimality by using another method. We first prove the following result: Theorem 1.1. Let A be an affine arrangement of hyperplanes, not necessarily central. Let V A ⊂ C n denote the union of hyperplanes in A and let H be a generic hyperplane with respect to A. Then, one has the isomorphisms of Z-modules:Our alternate proof uses the behaviour of the variation maps within a pencil of hyperplanes on C n \ V A . It is based on a particular case (see Theorem 3.1) of a general result on vanishing cycles of pencils, which involves variation maps, proved in [Ti3,Ti6]. We discuss in §2 some aspects of the topology of pencils on complements of affine hypersurfaces, extracted from a general theory of non-generic Lefschetz pencils of hypersurfaces, which we have developped in a series of papers [Ti3,Ti4,Ti6,Ti5,Ti7]. In this context, we also give a method to compute inductively the betti numbers of complements of arrangements by using global polar invariants [Ti2].2000 Mathematics Subject Classification. 32S22, 14N20.

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Density of monodromy actions on non-abelian cohomology

Katzarkov

Pantev

Simpson

2003

Advances in Mathematics

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In this paper we study the monodromy action on the first Betti and de Rham nonabelian cohomology arising from a family of smooth curves. We describe sufficient conditions for the existence of a Zariski dense monodromy orbit. In particular we show that for a Lefschetz pencil of sufficiently high degree the monodromy action is dense.(ii) In the case of M B (X o , n), considered with its natural structure of an affine algebraic variety, there exists a point x B ∈ M B (X o , n) so that the orbitis Zariski dense in M B (X o , n); or in the case of M DR (X/B, n) → B there is a leaf of the foliation defined by ∇ n DR which is Zariski dense in the algebraic Zariski topology.This theorem suggests that for families f : X → B with a "large enough" geometric monodromy one should expect Zariski dense monodromy actions on non-abelian cohomology. Geometrically families with large monodromy naturally arise from hyperplane sections and Lefschetz fibrations. In this context we prove the following Theorem B Let Z be a smooth projective surface with b 1 (Z) = 0. Let O Z (1) be an ample line bundle on Z and let n > 1 be a fixed odd integer. Then there exists a positive integer ℓ (depending only on Z and O Z (1)), such that for every k ≥ ℓ and for every Lefschetz fibration f : Z → P 1 in the linear system |O Z (k)| we have:(i) There is no meromorphic function on M B (Z o , n) an which is invariant under the action of mon n B (π 1 (P 1 \ {p 1 , . . . , p µ }, o)) (equivalently there is no meromorphic function on M DR ( Z/P 1 \ {p 1 , . . . , p µ }, n) an which is ∇ n DR -invariant);(ii) In the case of M B (Z o , n), there exist a point x B ∈ M B (Z o , n) so that the orbitis Zariski dense in M B (Z o , n); or in the case of the space M DR ( Z/P 1 \ {p 1 , . . . , p µ }, n) the foliation defined by ∇ n DR has a Zariski dense leaf.Here as usual Z is the blow-up of Z at the base points of the pencil and p 1 , . . . p µ ∈ P 1 are the points where the map Z → P 1 is not submersive. Preliminary reductionsWe start with some general results about linear group actions on algebraic varieties, which will allow us to localize at a point the Zariski density property of an action.

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Vanishing cycles of pencils of hypersurfaces

Abstract: We prove an extended Lefschetz principle for a large class of pencils of hypersurfaces having isolated singularities, possibly in the axis, and show that the module of vanishing cycles is generated by the images of certain variation maps. ?

Cited by 8 publications

References 24 publications

Polynomials and Vanishing Cycles

Polynomials and Vanishing Cycles

Complements of Hypersurfaces, Variation Maps, and Minimal Models of Arrangements

Density of monodromy actions on non-abelian cohomology

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