47 research outputs found
Regular Functions Transversal at Infinity
We generalize and complete some of Maxim's recent results on Alexander
invariants of a polynomial transversal to the hyperplane at infinity. Roughly
speaking, and surprisingly, such a polynomial behaves both topologically and
algebraically (e.g. in terms of the variation of MHS on the cohomology of its
smooth fibers), like a homogeneous polynomial.Comment: This is a substantial improvement of the paper "Alexander Invariants
and Transversality" by the first author, see math.AG/0411329. Both the
topology and the associated mixed Hodge structures (not touched in the
previous paper) are clearly describe
Sequences of LCT-polytopes
To r ideals on a germ of smooth variety X one attaches a rational polytope in
the r-dimensional Euclidean space (the LCT-polytope) that generalizes the
notion of log canonical threshold in the case of one ideal. We study these
polytopes, and prove a strong form of the Ascending Chain Condition in this
setting: we show that if a sequence P_m of such LCT-polytopes converges to a
compact subset Q in the Hausdorff metric, then Q is equal to the intersection
of all but finitely many of the P_m. Furthermore, Q is an LCT-polytope.Comment: 16 pages; v3: minor corrections, to appear in Mathematical Research
Letter
Multivariable Hodge theoretical invariants of germs of plane curves
We describe methods for calculation of polytopes of quasiadjunction for plane
curve singularities which are invariants giving a Hodge theoretical refinement
of the zero sets of multivariable Alexander polynomials. In particular we
identify some hyperplanes on which all polynomials in multivariable Bernstein
ideal vanish
Motivic infinite cyclic covers
We associate with an infinite cyclic cover of a punctured neighborhood of a
simple normal crossing divisor on a complex quasi-projective manifold (assuming
certain finiteness conditions are satisfied) an element in the Grothendieck
ring , which we call {\it motivic
infinite cyclic cover}, and show its birational invariance. Our construction
provides a unifying approach for the Denef-Loeser motivic Milnor fibre of a
complex hypersurface singularity germ, and the motivic Milnor fiber of a
rational function, respectively.Comment: published versio