Lectures on topology of complements and fundamental groups

Libgober, Anatoly

doi:10.1142/9789812707499_0003

Cited by 9 publications

(13 citation statements)

References 71 publications

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“…The following consequence of Theorem 3.2, Remark 3.3, and of Example 2.8 is similar to some results in [22], [36], [37]. …”

Section: Theorem 31 For All I and Allsupporting

confidence: 83%

“…The first part of the example below corresponds to the germ of a normal crossing divisor. The second part of the example below corresponds to isolated non-normal crossing divisors (for short INNC); see [22], [36], [37]. Similarly, instead of localizing at a point, one may localize along the hyperplane H at infinity, i.e.…”

Section: Proposition 25 For Any Point λ ∈ Tmentioning

confidence: 99%

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On the Alexander invariants of hypersurface complements

Maxim

2007

Singularity Theory

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Abstract. We start with a discussion on Alexander invariants, and then prove some general results concerning the divisibility of the Alexander polynomials and the supports of the Alexander modules, via Artin's vanishing theorem for perverse sheaves. We conclude with explicit computations of twisted cohomology following an idea already exploited in the hyperplane arrangement case, which combines the degeneration of the Hodge to de Rham spectral sequence with the purity of some cohomology groups.

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“…The following consequence of Theorem 3.2, Remark 3.3, and of Example 2.8 is similar to some results in [22], [36], [37]. …”

Section: Theorem 31 For All I and Allsupporting

confidence: 83%

Section: Proposition 25 For Any Point λ ∈ Tmentioning

confidence: 99%

On the Alexander invariants of hypersurface complements

Maxim

2007

Singularity Theory

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“…In [28], under the same assumptions, the sets V q i were related to the homotopy groups π q (U ). See [25,28,29] for properties and applications of local and global polytopes of quasiadjunction. Even if the end result is the same as ours in this special case, the methods are different (see Example 6.6).…”

Section: Introductionmentioning

confidence: 99%

Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

Budur

2009

Advances in Mathematics

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The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green-Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers h q,0 of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak-Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers.

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“…An answer to Problem 5.5.1 would shed a new light on the general problem of realizability of finite groups as fundamental groups of plane curves, see [54], [55]. Certainly, this approach has its limitations; for example, all groups admit a presentation with at most three generators.…”

Section: Topology Of Trigonal Curves the Class Realized By A Trigonamentioning

confidence: 99%

“…In the last 25 years, topology of singular plane algebraic curves has been an area of active research; the modern state of affairs and important open problems are thoroughly presented in recent surveys [7], [54], [55], and [71]. Analyzing recent achievements, one cannot help noticing that, with relatively few exceptions, algebraic objects are studied by purely topological means, making very little use of the analytic structure.…”

Section: Introductionmentioning

confidence: 99%