Real Regulators on Milnor Complexes

Let X/_C be a projective algebraic manifold. In the first part of this paper, we construct a natural real regulator on the cohomology of a Gersten–Milnor complex, and where appropriate, compare it to the Beilinson regulator into real Deligne cohomology. The second part is devoted to a calculation on curves. In particular, we arrive at an elementary new proof of the known nontriviality of the regulator on CH²(X,2), for a general elliptic curve X. Further, if X P ² is a general curve of degree 3, then for any open Riemann surface U contained in the complement , we arrive at a nontrivial regulator calculation on CH²(^U,2). We conclude this paper with an application to points of finite order on a curve X.     

Real Milnor Fibrations and (C)-Regularity

In this paper we study Milnor fibrations associated to real isolated singularities defined by map-germs f: (^m,0)(²,0). The main result relates the existence of the Milnor fibration with the (C)-regularity of the family of hypersurfaces with isolated singularity obtained by projecting f into the family L_– of all lines through the origin in the plane ².     

On Milnor Fibrations of Arrangements

We use covering space theory and homology with local coefficientsto study the Milnor fiber of a homogeneous polynomial. Thesetechniques are applied in the context of hyperplane arrangements,yielding an explicit algorithm for computing the Betti numbersof the Milnor fiber of an arbitrary real central arrangementin C³, as well as the dimensions of the eigenspaces of the algebraicmonodromy. We also obtain combinatorial formulas for these invariantsof the Milnor fiber of a generic arrangement of arbitrary dimensionusing these methods.     

On the Milnor Number of an Equivariant Singularity

Mathematical Notes -     

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

Functional Analysis and Its Applications - In this paper we study relationships between some topological and analytic invariants of zero-dimensional germs, or multiple points. Among other things,...     

The motivic real Milnor fibres

Given a polynomial with real coefficients, we produce a motivic analog of a simple identity that relates the complex conjugation and the monodromy of the Milnor fibre of its complexification. To that purpose, we introduce motivic zeta functions that take into account complex conjugation and monodromy.     

On the monodromy action on Milnor fibers of graphic arrangements

We analyze the monodromy action, over the rationals, on the first homology group of the Milnor fiber, for arbitrary subarrangements of Coxeter arrangements. We propose a combinatorial formula for the monodromy action, involving Aomoto complexes in positive characteristic. We verify the formula, in cases A, B and D.     

On the mixed Hodge structure on the cohomology of the Milnor fibre

Mathematische Annalen -     

The Milnor fiber of a generic arrangement

In memoriam Deane Montgomery     

On the Structures of the Milnor K-Groups of Some Complete Discrete Valuation Fields

For a complete discrete valuation field K, the unit group of K has a natural decreasing filtration with respect to the valuation, and the graded quotients of this filtration are written in terms of the residue field. The Milnor K-group of a field is a generalization of the unit group. The Milnor K-group of K has a natural decreasing filtration of the same kind. However, if K is of mixed characteristics and has an absolute ramification index greater than one, the structure is not yet known. The aim of this paper is to determine the structure of the Milnor K-group of some special K, which are of mixed characteristics (0,p), whose residue fields are allowed to be imperfect, and which are of absolute ramification index p(p–1).     

On the Milnor K-theory of the function fields of quasihomogeneous cones

Let V be a quasihomogeneous normal variety. The aim of this paper is to describe the Milnor K-theory of the function field of V in terms of the second residue homomorphisms associated with subvarieties and resolution data of V.Supported by KBN, 2 P301 010 06.     

On the Semicontinuity of Curvature Integrals

Let H_j(K, ·) be the j – th elementary symmetric function of the principal curvatures of a convex body K in Euclidean d – space. We show that the functionals ∫_bd f(H_j(K, x)) dℋ︁^d—1(x) depend upper semicontinuously on K, if f : [0, ∞) is concave, lim_t→0f(t) = 0, and lim_t→∞f(t)/t = 0. An analogous statement holds for integrals of elementary symmetric functions of the principal radii of curvature.     

子流形的测地曲率

该文研究了黎曼齐次空间G/H中子流形M~p和子流形N~q的交M~p∩gN~q的测地曲率,其中g∈G为等距群,并把M~p∩gN~q的第二基本形式表示成M~p和N~q的测地曲率以及它们之间的夹角的组合.     

On Milnor K-groups attached to semi-Abelian varieties

In this paper we define and study a Milnor K-group attached to a finite family of semi-Abelian varieties over a field, which is a generalization of the usual Milnor K-group. Using this group, we generalize the work of Bloch.     

On deformation with constant Milnor number and Newton polyhedron

We show that every \(\mu \)-constant family of isolated hypersurface singularities satisfying a nondegeneracy condition in the sense of Kouchnirenko, is topologically trivial, also is equimultiple.     

The boundary of the Milnor fibre of certain non-isolated singularities

Let \( \Phi : (\mathbb {C}^2, 0) \rightarrow ( \mathbb {C}^3, 0) \) be a finitely determined complex analytic germ and let \((\{f=0\},0)\) be the reduced equation of its image, a non-isolated hypersurface singularity. We provide the plumbing graph of the boundary of the Milnor fibre of f from the double-point-geometry of \(\Phi \).     

Jump of Milnor numbers

In this note we study a problem of A'Campo about the minimal non-zero difference between the Milnor numbers of a germ of plane curve and one of its deformation.