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Mikhail Grinberg

Publications and source records attributed to Mikhail Grinberg.

9 recordsLinked to original sources
arXiv2020

Nearby Cycle Sheaves for Stable Polar Representations

Let G|V, G connected, reductive over C, be a stable polar representation in the sense of [DK], satisfying some mild additional hypotheses. Given a G-equivariant rank one local system L on the general fiber of the quotient map f : V --> V/G, we compute the Fourier transform of the corresponding nearby cycle sheaf P on the zero-fiber of f. This provides a partial generalization of the results of [Gr1] and [GVX1]. Our main intended application is to the theory of character sheaves for graded Lie algebras over C.

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arXiv2020

Errata and notes on the paper "A generalization of Springer theory using nearby cycles"

We provide some corrections and clarifications to the paper [Gr3] of the title. In particular, we clarify the "left/right" conventions on complex reflection groups and their braid groups. Most importantly, we fill in a gap related to the treatment of cuts in the Picard-Lefschetz theory part of the argument. The statements of the main results are not affected.

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arXiv2018

Nearby Cycle Sheaves for Symmetric Pairs

We present a nearby cycle sheaf construction in the context of symmetric spaces. This construction can be regarded as a replacement for the Grothendieck-Springer resolution in classical Springer theory.

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arXiv2010

Dimensions of the Ascending and Descending Sets in Complex Stratified Morse Theory

We present a new construction of gradient-like vector fields in the setting of Morse theory on a complex analytic stratification. We prove that the ascending and descending sets for these vector fields possess cell decompositions satisfying the dimension bounds conjectured by M. Goresky and R. MacPherson. Similar results by C.-H. Cho and G. Marelli have recently appeared in arXiv:0908.1862.

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arXiv2000

Gradient-like flows and self-indexing in stratified Morse theory

We develop the idea of self-indexing and the technology of gradient-like vector fields in the setting of Morse theory on a complex algebraic stratification. Our main result is the local existence, near a Morse critical point, of gradient-like vector fields satisfying certain ``stratified dimension bounds up to fuzz'' for the ascending and descending sets. As a global consequence of this, we derive the existence of self-indexing Morse functions.

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arXiv1998

Versal deformations of formal arcs

Let X be a complex algebraic variety, and L(X) be the scheme of formal arcs in X. Let f be an arc whose image is not contained in the singularities of X. We show that the formal neighborhood of f in L(X) admits a decomposition into a product of an infinite-dimensional smooth piece, and a piece isomorphic to the formal neighborhood of a closed point of a scheme of finite type.

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arXiv1998

A generalization of Springer theory using nearby cycles

Let g be a complex semisimple Lie algebra, and f : g --> g/G the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of f. We give a generalization of Springer theory to visible, polar representations. It is a class of rational representations of complex reductive groups, for which the invariant theory works by analogy with the adjoint representations. Let G|V be such a representation, f : V --> V/G the quotient map, and P the sheaf of nearby cycles of f. We show that the Fourier transform of P is an intersection homology sheaf on V*. Associated to G|V, there is a finite complex reflection group W, called the Weyl group of G|V. We describe the endomorphism ring of P as a deformation of the group algebra of W.

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arXiv1998

On the specialization to the asymptotic cone

Let X be a smooth, connected, closed subvariety of a complex vector space V. The asymptotic cone as(X) is naturally equipped with a nearby cycles sheaf P coming from the specialization of X to as(X). We show that if X is transverse to infinity in a suitable sense, then the Fourier transform of P is an intersection homology sheaf.

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