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Wolfgang Ebeling

Publications and source records attributed to Wolfgang Ebeling.

51 records · Page 3Linked to original sources
arXiv2008

Poincaré series and monodromy of the simple and unimodal boundary singularities

A boundary singularity is a singularity of a function on a manifold with boundary. The simple and unimodal boundary singularities were classified by V. I. Arnold and V. I. Matov. The McKay correspondence can be generalized to the simple boundary singularities. We consider the monodromy of the simple, parabolic, and exceptional unimodal boundary singularities. We show that the characteristic polynomial of the monodromy is related to the Poincaré series of the coordinate algebra of the ambient singularity.

math.AG↗
arXiv2006

Low-dimensional Singularities with Free Divisors as Discriminants

We present versal complex analytic families, over a smooth base and of fibre dimension zero, one, or two, where the discriminant constitutes a free divisor. These families include finite flat maps, versal deformations of reduced curve singularities, and versal deformations of Gorenstein surface singularities in C^5. It is shown that such free divisors often admit a "fast normalization", obtained by a single application of the Grauert-Remmert normalization algorithm. For a particular Gorenstein surface singularity in C^5, namely the simple elliptic singularity of type \tilde A_4, we exhibit an explicit discriminant matrix and show that the slice of the discriminant for a fixed j-invariant is the cone over the dual variety of an elliptic curve.

math.AG↗
arXiv2006

Indices of vector fields and 1-forms on singular varieties

We discuss different generalizations of the classical notion of the index of a singular point of a vector field to the case of vector fields or 1-forms on singular varieties, describe relations between them and formulae for their computation.

math.AG↗
arXiv2005

Monodromy

Let $(X,x)$ be an isolated complete intersection singularity and let $f : (X,x) \to (\CC,0)$ be the germ of an analytic function with an isolated singularity at $x$. An important topological invariant in this situation is the Picard-Lefschetz monodromy operator associated to $f$. We give a survey on what is known about this operator. In particular, we review methods of computation of the monodromy and its eigenvalues (zeta function), results on the Jordan normal form of it, definition and properties of the spectrum, and the relation between the monodromy and the topology of the singularity.

math.AG↗
arXiv2005

Mirror symmetry, Kobayashi's duality, and Saito's duality

M. Kobayashi introduced a notion of duality of weight systems. We tone this notion slightly down to a notion called coupling. We show that coupling induces a relation between the reduced zeta functions of the monodromy operators of the corresponding singularities generalizing an observation of K. Saito concerning Arnold's strange duality. We show that the weight systems of the mirror symmetric pairs of M. Reid's list of 95 families of Gorenstein K3 surfaces in weighted projective 3-spaces are strongly coupled. This includes Arnold's strange duality where the corresponding weight systems are strongly dual in Kobayashi's original sense. We show that the same is true for the extension of Arnold's strange duality found by the author and C. T. C. Wall.

math.AG↗
arXiv2005

Quadratic forms for a 1-form on an isolated complete intersection singularity

We consider a holomorphic 1-form $ω$ with an isolated zero on an isolated complete intersection singularity $(V,0)$. We construct quadratic forms on an algebra of functions and on a module of differential forms associated to the pair $(V,ω)$. They generalize the Eisenbud-Levine-Khimshiashvili quadratic form defined for a smooth $V$.

math.AG↗
arXiv2003

A filtration defined by arcs on a variety

We define a natural filtration on the ring ${\cal O}_{V,0}$ of germs of functions on a germ of a complex analytic variety $(V,0)$ related with the geometry of arcs on the variety and describe some properties of it.

math.AG↗
arXiv2001

The Poincaré series of some special quasihomogeneous surface singularities

In the author's paper ''Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity'' a relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. We study this relation for Fuchsian singularities and show that it is connected with the mirror symmetry of K3 surfaces and with automorphisms of the Leech lattice. We also indicate relations between other singularities and Conway's group.

math.AG↗
arXiv2001

Poincaré series and monodromy of a two-dimensional quasihomogeneous hypersurface singularity

A relation is proved between the Poincaré series of the coordinate algebra of a two-dimensional quasihomogeneous isolated hypersurface singularity and the characteristic polynomial of its monodromy operator. For a Kleinian singularity not of type $A_{2n}$, this amounts to the statement that the Poincaré series is the quotient of the characteristic polynomial of the Coxeter element by the characteristic polynomial of the affine Coxeter element of the corresponding root system. We show that this result also follows from the McKay correspondence.

math.AG↗
arXiv2001

Indices of 1-forms on an isolated complete intersection singularity

There are some generalizations of the classical Eisenbud-Levine-Khimshashvili formula for the index of a singular point of an analytic vector field on $R^n$ for vector fields on singular varieties. We offer an alternative approach based on the study of indices of 1-forms instead of vector fields. When the variety under consideration is a real isolated complete intersection singularity (icis), we define an index of a (real) 1-form on it. In the complex setting we define an index of a holomorphic 1-form on a complex icis and express it as the dimension of a certain algebra. In the real setting, for an icis $V=f^{-1}(0)$, $f:(C^n, 0) \to (C^k, 0)$, $f$ is real, we define a complex analytic family of quadratic forms parameterized by the points $ε$ of the image $(C^k, 0)$ of the map $f$, which become real for real $ε$ and in this case their signatures defer from the "real" index by $χ(V_ε)-1$, where $χ(V_ε)$ is the Euler characteristic of the corresponding smoothing $V_ε=f^{-1}(ε)\cap B_δ$ of the icis $V$.

math.AG↗
arXiv1998

Strange duality and polar duality

We describe a relation between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M.~Kobayashi. We show that this relation continues to hold for the extension of Arnold's strange duality found by C.~T.~C.~Wall and the author. By a method of Ehlers-Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. We generalize this method to the isolated complete intersection singularities embraced in the extended duality. We use this to explain the duality of characteristic polynomials of the monodromy discovered by K.~Saito for Arnold's original strange duality and extended by the author to the other cases.

math.AG↗
arXiv1998

Strange duality, mirror symmetry, and the Leech lattice

We give a survey on old and new results concerning Arnold's strange duality. We show that most of the features of this duality continue to hold for the extension of it discovered by C. T. C. Wall and the author. The results include relations to mirror symmetry and the Leech lattice.

alg-geom↗
arXiv1998

On the index of a vector field at an isolated singularity

We consider manifolds with isolated singularities, i.e., topological spaces which are manifolds (say, $C^\infty$--) outside discrete subsets (sets of singular points). For (germs of) manifolds with, so called, cone--like singularities, a notion of the index of an isolated singular point of a vector field is introduced. There is given a formula for the index of a gradient vector field on a (real) isolated complete intersection singularity. The formula is in terms of signatures of certain quadratic forms on the corresponding spaces of thimbles.

alg-geom↗