Mark Losik

CHOOSING ROOTS OF POLYNOMIALS SMOOTHLY

We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given.

Lifting smooth curves over invariants for representations of compact lie groups

We show that one can lift locally real analytic curves from the orbit space of a compact Lie group representation, and that one can lift smooth curves even globally, but under an assumption.

Diagonal cohomology of the Lie algebra of vector fields

Abstract Simple proofs of main theorems on the cohomology for the diagonal subcomplex of the complex of standard continuous cochains for the Lie algebra (M) of vector fields on a manifold M with coefficients in a tensor (M)-module and the trivial (M)-module R are given.

The cohomology of the complex ofG-invariant forms onG-manifolds

The complex ofG-invariant forms and its cohomology for arbitraryG-manifolds and especially for a certain class ofG-manifolds, which are locally trivial fiber bundles over the orbit space, are considered. The transgression in the differential graded algebra of basic elements for tensor product of two identical Weil algebras of a reductive Lie groupG is calculated. This is used to get two convenient differential graded algebras with the same minimal models as the differential algebra of differential forms on the cross product of two principalG-bundles overG and ofG-invariant forms onG-manifolds of the above class. In particular, for compactG the generalization of the Cartan theorem on the cohomology of a homogeneous space is proved.

Characteristic classes of transformation groups

Abstract For transformation group G of a topological space X a spectral sequence with the term Ep,q2 = Hp(G;Hq(X;A)), where A is a coefficient group, is introduced. The cohomology of th e complex of G-invariant cochains with coefficients in A determine the characteristic classes of G as elements of E∞; in particular cases they are elements of Hp(G;Hq(X;A)) ( or Hpc(G;Hq(X;A)) for continous transformation group G). The main applications concern the case when A = R and the de Rham complex is used. It is shown that the characteristic classes of Reiman, Semenov- Tian-Shansky and Fadeev for the automorphism group of a smooth principal fibre bundle and characteristic classes of Bott for the diffeomorphism group of a manifold are the partial cases of the above construction. The connection of the above characteristic classes with the functional of Atiyah-Patodi-Singer is indicated.

A GENERALIZATION OF PUISEUX'S THEOREM AND LIFTING CURVES OVER INVARIANTS

Let ρ:G→GL (V) be a rational representation of a reductive linear algebraic group G defined over ℂ on a finite dimensional complex vector space V. We show that, for any generic smooth (resp. CM) curve c:ℝ→V//G in the categorical quotient V//G (viewed as affine variety in some ℂn) and for any t0∈ℝ, there exists a positive integer N such that t↦c(t0±(t−t0)N) allows a smooth (resp. CM) lift to the representation space near t0. (CM denotes the Denjoy–Carleman class associated with M=(Mk), which is always assumed to be logarithmically convex and derivation closed). As an application we prove that any generic smooth curve in V//G admits locally absolutely continuous (not better!) lifts. Assume that G is finite. We characterize curves admitting differentiable lifts. We show that any germ of a C∞ curve which represents a lift of a germ of a quasianalytic CM curve in V//G is actually CM. There are applications to polar representations.

Invariant Tensor Fields and Orbit Varieties for Finite Algebraic Transformation Groups

Let X be a smooth algebraic variety endowed with an action of a finite group G such that there exists a geometric quotient Π X : X → X/G. We characterize rational tensor fields τ on X/G such that the pull back of τ is regular on X: these are precisely all τ such that \(di{v_{{R_{X/G}}}}(\tau ) \geqslant 0\) where RX/G is the reflection divisor of X/G and \(di{v_{{R_{X/G}}}}(\tau )\) is the RX/G-divisor of τ. We give some applications, in particular to a generalization of Solomon’s theorem. In the last section we show that if V is a finite dimensional vector space and G a finite subgroup of GL(V), then each automorphism Ψ of V/G admits a biregular lift φ : V → V provided that Ψ maps the regular stratum to itself and Ψ*(RX/G) = RX/G.

Cohomology for a Group of Diffeomorphisms of a Manifold Preserving an Exact Form

Let M be a G-manifold and ω a G-invariant exact m-form on M. We indicate when these data allow us to construct a cocycle on a group G with values in the trivial G-module ℝ, and when this cocycle is nontrivial.