41 research outputs found
On an equivariant analogue of the monodromy zeta function
We offer an equivariant analogue of the monodromy zeta function of a germ
invariant with respect to an action of finite group G as an element of the
Grothendieck ring of finite (Z x G)-sets. We formulate equivariant analogues of
the Sebastiani-Thom theorem and of the A'Campo formula
Indices of vector fields or 1-forms and characteristic numbers
We define an index of a collection of 1-forms on a complex isolated complete
intersection singularity corresponding to a Chern number and, in the case when
the 1-forms are complex analytic, express it as the dimension of a certain
algebra
On a Newton filtration for functions on a curve singularity
In a previous paper, there was defined a multi-index filtration on the ring
of functions on a hypersurface singularity corresponding to its Newton diagram
generalizing (for a curve singularity) the divisorial one. Its Poincar\'e
series was computed for plane curve singularities non-degenerate with respect
to their Newton diagrams. Here we use another technique to compute the
Poincar\'e series for plane curve singularities without the assumption that
they are non-degenerate with respect to their Newton diagrams. We show that the
Poincar\'e series only depends on the Newton diagram and not on the defining
equation.Comment: 11 page
Saito duality between Burnside rings for invertible polynomials
We give an equivariant version of the Saito duality which can be regarded as
a Fourier transformation on Burnside rings. We show that (appropriately
defined) reduced equivariant monodromy zeta functions of Berglund-H\"ubsch dual
invertible polynomials are Saito dual to each other with respect to their
groups of diagonal symmetries. Moreover we show that the relation between
"geometric roots" of the monodromy zeta functions for some pairs of
Berglund-H\"ubsch dual invertible polynomials described in a previous paper is
a particular case of this duality.Comment: 12 pages; the main result has been improve
An algebraic formula for the index of a 1-form on a real quotient singularity
Let a finite abelian group act (linearly) on the space and
thus on its complexification . Let be the real part of the
quotient (in general ). We give an
algebraic formula for the radial index of a 1-form on the real quotient . It
is shown that this index is equal to the signature of the restriction of the
residue pairing to the -invariant part of . For a
-invariant function , one has the so-called quantum cohomology group
defined in the quantum singularity theory (FJRW-theory). We show that, for a
real function , the signature of the residue pairing on the real part of the
quantum cohomology group is equal to the orbifold index of the 1-form on
the preimage of under the natural quotient map.Comment: 19 page
Quadratic forms for a 1-form on an isolated complete intersection singularity
We consider a holomorphic 1-form with an isolated zero on an
isolated complete intersection singularity . We construct quadratic
forms on an algebra of functions and on a module of differential forms
associated to the pair . They generalize the
Eisenbud-Levine-Khimshiashvili quadratic form defined for a smooth
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, --) 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.Comment: AMS-LaTeX, 11 p. with 1 fig.; remarks, definition, and references
added to Section
A version of the Berglund-H\"ubsch-Henningson duality with non-abelian groups
A. Takahashi suggested a conjectural method to find mirror symmetric pairs
consisting of invertible polynomials and symmetry groups generated by some
diagonal symmetries and some permutations of variables. Here we generalize the
Saito duality between Burnside rings to a case of non-abelian groups and prove
a "non-abelian" generalization of the statement about the equivariant Saito
duality property for invertible polynomials. It turns out that the statement
holds only under a special condition on the action of the subgroup of the
permutation group called here PC ("parity condition"). An inspection of data on
Calabi-Yau threefolds obtained from quotients by non-abelian groups shows that
the pairs found on the basis of the method of Takahashi have symmetric pairs of
Hodge numbers if and only if they satisfy PC.Comment: 22 pages, to appear in Int. Math. Res. Not. IMR
A filtration defined by arcs on a variety
We define a natural filtration on the ring of germs of
functions on a germ of a complex analytic variety related with the
geometry of arcs on the variety and describe some properties of it
Homological indices of collections of 1-forms
Homological index of a holomorphic 1-form on a complex analytic variety with
an isolated singular point is an analogue of the usual index of a 1-form on a
non-singular manifold. One can say that it corresponds to the top Chern number
of a manifold. We offer a definition of homological indices for collections of
1-forms on a (purely dimensional) complex analytic variety with an isolated
singular point corresponding to other Chern numbers. We also define new
invariants of germs of complex analytic varieties with isolated singular points
related to "vanishing Chern numbers" at them.Comment: 12 page