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The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components. 相似文献
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LetV ⊂ ℙℝ
n
be an algebraic variety, such that its complexificationV
ℂ ⊂ ℙ
n
is irreducible of codimensionm ≥ 1. We use a sufficient condition on a linear spaceL ⊂ ℙℝ
n
of dimensionm + 2r to have a nonempty intersection withV, to show that any six dimensional subspace of 5 × 5 real symmetric matrices contains a nonzero matrix of rank at most 3. 相似文献
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Anatoly Libgober 《manuscripta mathematica》2002,107(2):251-269
Multivariable Alexander invariants of algebraic links calculated in terms of algebro-geometric invariants (polytopes and
ideals of quasiadjunction). The relations with log-canonical divisors, the multiplier ideals and a semicontinuity property
of polytopes of quasiadjunction are discussed.
Received: 8 February 2001 / Revised version: 1 December 2001 相似文献
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Anatoly Libgober 《manuscripta mathematica》2009,128(1):1-31
We show that closures of families of unitary local systems on quasiprojective varieties for which the dimension of a graded
component of Hodge filtration has a constant value can be identified with a finite union of polytopes. We also present a local
version of this theorem. This yields the “Hodge decomposition” of the set of unitary local systems with a non-vanishing cohomology
extending Hodge decomposition of characteristic varieties of links of plane curves studied by the author earlier. We consider
a twisted version of the characteristic varieties generalizing the twisted Alexander polynomials. Several explicit calculations
for complements to arrangements are made.
A. Libgober was supported by National Science Foundation grant. 相似文献