Abstract
We describe methods for calculation of polytopes of quasiadjunction for plane curve singularities which are invariants giving a Hodge theoretical refinements of the zero sets of multivariable Alexander polynomials. In particular we identify some hyperplanes on which all polynomials in multivariable Bernstein ideal vanish.
Highlights
The purpose of this paper is to study Hodge theoretical invariants of local systems on the complements to germs of plane curve singularities
These invariants, called the faces of quasiadjunction, yield a refinement for the multivariable Alexander polynomial of a link of isolated singularity or, more precisely, for the characteristic varieties associated with the homology of universal abelian covers of the complements to a germ of plane curve
While a multivariable Alexander polynomial is a product of factors of the form tm1 1 ...tmr r − ω where ω = e2π√−1α is a root of unity (i.e. α ∈ Q), the faces of quasiadjunction are subsets of the union of hyperplanes of the form mixi = α + k, k ∈ Z and are related to the zero set of the Alexander polynomial via the exponential map
Summary
The purpose of this paper is to study Hodge theoretical invariants of local systems on the complements to germs of plane curve singularities. Faces of quasiadjunction for links of plane curve singularities were defined in [9], [10] using adjunction conditions for the abelian covers of germs In these papers they were related to the cohomology of the finite order local systems of the complements. In particular the families of (unitary) local systems on the complement to a link with fixed Hodge data have a linear structure and form polytopes of various dimensions in the universal covering of the space of unitary local systems This description of the Mixed Hodge structure on the local systems in terms of log-resolution of the germ provides first approach to calculations of faces of quasiadjunction. The authors thank the referee for careful reading of the manuscript
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