Related papers: Equivariant indices of vector fields and 1-forms

Let a finite group $G$ act on the complex plane $({\Bbb C}^2, 0)$. We consider multi-index filtrations on the spaces of germs of holomorphic functions of two variables equivariant with respect to 1-dimensional representations of the group…

Let $G$ be a finite abelian group acting faithfully on ${\mathbb C}{\mathbb P}^1$ via holomorphic automorphisms. In \cite{DF2} the $G$--equivariant algebraic vector bundles on $G$--invariant affine open subsets of ${\mathbb C}{\mathbb P}^1$…

We give a concrete description of the category of G-equivariant vector bundles on certain affine G-varieties (where G is a reductive linear algebraic group over an algebraically closed field of characteristic 0) in terms of linear algebra…

We define a multi-index filtration on the ring of germs of functions on a hypersurface singularity associated with its Newton diagram and compute the multivariable Poincar\'e series of this filtration in some cases.

We offer a new approach to a definition of an equivariant version of the Poincar\'e series. This Poincar\'e series is defined not as a power series, but as an element of the Grothendieck ring of $G$-sets with an additional structure. We…

Let M be a foliated manifold and G a discrete group acting on M by diffeomorphisms mapping leaves to leaves. Then G naturally acts by automorphisms on the algebra of Heisenberg pseudodifferential operators on the foliation. Our main result…

In many relevant cases -- e.g., in hamiltonian dynamics -- a given vector field can be characterized by means of a variational principle based on a one-form. We discuss how a vector field on a manifold can also be characterized in a similar…

We construct the space of vector fields on a generic quantum group. Its elements are products of elements of the quantum group itself with left invariant vector fields. We study the duality between vector fields and 1-forms and generalize…

We study Ehrhart series with coefficients in Abelian group rings. This opens new enumeration applications and unifies earlier variants, in particular, polynomial weighted, $q$-weighted, and equivariant Ehrhart series.

We provide a complete characterization of the equivariant commutative ring structures of all the factors in the idempotent splitting of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite…

We consider one dimensional holomorphic foliations with isolated singularities that leave invariant a local complete intersection. We establish explicit formulas for the total GSV index of such foliations and obtain bounds for this index.…

For a proper, cocompact action by a locally compact group of the form $H \times G$, with $H$ compact, we define an $H \times G$-equivariant index of $H$-transversally elliptic operators, which takes values in $KK_*(C^*H, C^*G)$. This…

We use group representation theory to give algebraic formulae to compute complete transversals of singularities of vector fields, either in the nonsymmetric or in the reversible equivariant contexts. This computation produces normal forms…

In this work we introduce a GSV type index for varieties invariant by holomorphic Pfaff systems (possibly non locally decomposables) on projective manifolds. We prove a non-negativity property for the index. As an application, we prove that…

We give a general framework of equivariant model category theory. Our groups G, called Hopf groups, are suitably defined group objects in any well-behaved symmetric monoidal category V. For any V, a discrete group G gives a Hopf group,…

In this paper we study the relationship of the Brouwer degree of a vector field with the dynamics of the induced flow. Analogous relations are studied for the index of a vector field. We obtain new forms of the Poincar% \'{e}-Hopf theorem…

We study the structure of combinatorial Burnside groups, which receive equivariant birational invariants of actions of finite groups on algebraic varieties.

In this paper we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This…

The work proves that, for three-dimensional upper triangular groups over a field of odd characteristic with an abelian unipotent subgroup, the ring of invariants is polynomial if and only if the unipotent subgroup is generated by…

We apply the equivariant Burnside group formalism to distinguish linear actions of finite groups, up to equivariant birationality. Our approach is based on De Concini-Procesi models of subspace arrangements.