On Hodge spectrum and multiplier ideals

We describe a relation between two invariants which measure the complexity of a hypersurface singularity. One is the Hodge spectrum which is related to the monodromy and the Hodge filtration on the cohomology of the Milnor fiber. The other is the multiplier ideal, having to do with log resolutions. Mathematics Subject Classification (2000):14B05, 32S35     

Integral operators and integral cohomology classes of Hilbert schemes

The methods of integral operators on the cohomology of Hilbert schemes of points on surfaces are developed. They are used to establish integral bases for the cohomology groups of Hilbert schemes of points on a class of surfaces (and conjecturally, for all simply connected surfaces).Mathematics Subject Classification (2000): 14C05, 14F43, 17B69Partially supported by an NSF grant.     

Characteristic cycles of perverse sheaves and Milnor fibers

We show that the Milnor monodromy and the Milnor numbers naturally appear in the characteristic cycles of irreducible perverse sheaves having the singularities of a complex hypersurface. We find also a new numerical constraint on the Milnor monodromy for general hypersurface singularities.Mathematics Subject Classification (1991): 14B05, 32C38, 32S40, 35A27Dedicated to the 60th anniversary of Prof P. Schapira who taught us the theory of D-modules     

On the moduli space of deformations of bihamiltonian hierarchies of hydrodynamic type

We investigate the deformation theory of the simplest bihamiltonian structure of hydrodynamic type, that of the dispersionless KdV hierarchy. We prove that all of its deformations are quasi-trivial in the sense of B. Dubrovin and Y. Zhang, that is, trivial after allowing transformations where the first partial derivative ∂u of the field is inverted. We reformulate the question about deformations as a question about the cohomology of a certain double complex, and calculate the appropriate cohomology group.     

Lagrangian Floer theory on compact toric manifolds II: bulk deformations

This is a continuation of part I in the series of the papers on Lagrangian Floer theory on toric manifolds. Using the deformations of Floer cohomology by the ambient cycles, which we call bulk deformations, we find a continuum of non-displaceable Lagrangian fibers on some compact toric manifolds. We also provide a method of finding all fibers with non-vanishing Floer cohomology with bulk deformations in arbitrary compact toric manifolds, which we call bulk-balanced Lagrangian fibers.     

Three Power Series Techniques

The techniques and concepts we present are flags of regularschemes and their persistence under blow-up, the Gauss–Bruhatdecomposition of the group of formal automorphisms of affinespace, and coordinate-free initial ideals. All three are usedto construct and study invariants for resolution of singularities.2000 Mathematics Subject Classification 14B05, 14E15, 32S05,32S10, 32S45.     

Comparing Codimension and Absolute Length in Complex Reflection Groups

Reflection length and codimension of fixed point spaces induce partial orders on a complex reflection group. Motivated by connections to the algebraic structure of cohomology governing deformations of skew group algebras, we show that Coxeter groups and the infinite family G(m, 1, n) are the only irreducible complex reflection groups for which reflection length and codimension coincide. We then discuss implications for the degrees of generators of Hochschild cohomology. Along the way, we describe the codimension atoms for the infinite family G(m, p, n), give algorithms using character theory, and determine two-variable Poincaré polynomials recording reflection length and codimension.     

Deformations of Functions and F-Manifolds

We study deformations of functions on isolated singularities.A unified proof of the equality of Milnor and Tjurina numbersfor functions on isolated complete intersections singularitiesand space curves is given. As a consequence, the base spaceof their miniversal deformations is endowed with the structureof an F-manifold, and we can prove a conjecture of V. Goryunov,stating that the critical values of the miniversal unfoldingof a function on a space curve are generically local coordinateson the base space of the deformation. 2000 Mathematics SubjectClassification 32S05.     

Universal abelian covers of quotient-cusps

 The quotient-cusp singularities are isolated complex surface singularities that are double-covered by cusp singularities. We show that the universal abelian cover of such a singularity, branched only at the singular point, is a complete intersection cusp singularity of embedding dimension 4. This supports a general conjecture that we make about the universal abelian cover of a ℚ-Gorenstein singularity. Received: 3 February 2001 / Revised version: 8 March 2002 / Published online: 10 February 2003 Mathematics Subject Classification (2000): 14B05, 14J17, 32S25 This research was supported by grants from the Australian Research Council and the NSF (first author) and the the NSA (second author).     

On the first cohomology of arithmetic groups

We study the restriction to smaller subgroups, of cohomology classes on arithmetic groups (possibly after moving the class by Hecke correspondences), especially in the context of first cohomology of arithmetic groups. We obtain vanishing results for the first cohomology of cocompact arithmetic lattices in SU(n,1) which arise from hermitian forms over division algebras D of degree p ², p an odd prime, equipped with an involution of the second kind. We show that it is not possible for a ‘naive’ restriction of cohomology to be injective in general. We also establish that the restriction map is injective at the level of first cohomology for non co-compact lattices, extending a result of Raghunathan and Venkataramana for co-compact lattices. Received: 14 September 2000 / Accepted: 6 June 2001     

Local flat duality of abelian varieties

 Let K be a field complete for a discrete valuation and with algebraically closed residue field of positive characteristic p. We prove the existence of a non-degenerate pairing between the first (flat) cohomology group of an abelian variety A _K over K and the fundamental group of the Néron model of the dual abelian variety. This pairing extends to the p-primary components a pairing introduced by Shafarevich in [16]. We relate this pairing with Grothendieck's pairing. Received: 7 January 2002 / Revised version: 6 December 2002 Published online: 24 April 2003 Mathematics Subject Classification (2000): 14k05, 14F20, 14G22     

Local moduli spaces and Kuranishi maps

 We explain, in a non technical way, several general methods for constructing semi-universal deformations, especially by Kuranshi maps. Moreover, we give (standard) criteria of universality and smoothness of the semi-universal deformation, discuss the existence of operations on the base germ and describe intrinsically the Massey products. Received: 27 November 2001 / Revised version: 23 September 2002 Published online: 24 January 2003 Mathematics Subject Classification (2000): 32 G 13, 14 A 22, 14 B 12, 22 E 65     

Cohomology of tori over p-adic curves

 We establish a duality in the cohomology of arbitrary tori over smooth but not necessarily projective curves over a p-adic field. This generalises Lichtenbaum–Tate duality between the Picard group and the Brauer group of a smooth projective curve. Received: 28 January 2002 / Published online: 28 March 2003 Mathematics Subject Classification (2000): 14G20, 14F22, 14L15, 11S25     

Deformation of hom-Lie-Rinehart algebras

Abstract

We study formal deformations of hom-Lie-Rinehart algebras. The associated deformation cohomology that controls deformations is constructed using multiderivations of hom-Lie-Rinehart algebras.     

Rigid and Complete Intersection Lagrangian Singularities

In this article we prove a rigidity theorem for lagrangian singularities by studying the local cohomology of the lagrangian de Rham complex that was introduced in [SvS03]. The result can be applied to show the rigidity of all open swallowtails of dimension 2. In the case of lagrangian complete intersection singularities the lagrangian de Rham complex turns out to be perverse. We also show that lagrangian complete intersections in dimension greater than two cannot be regular in codimension one.     

Constructions and Cohomology of Hom–Lie Color Algebras

The main purpose of this paper is to define representations and a cohomology of Hom–Lie color algebras and to study some key constructions and properties. We describe Hartwig–Larsson–Silvestrov Theorem in the case of Γ-graded algebras, study one-parameter formal deformations, discuss α ^k -generalized derivations and provide examples.     

The contact boundary of a complex polynomial

 We define the contact boundary of a complex polynomial f : ℂ ⁿ → ℂ as the intersection of some generic fiber with a large sphere. We show that, up to contact isotopy, this does not depend on the choice of the fiber (provided it is generic) and is invariant under polynomial automorphism of ℂ ⁿ . We next prove that the formal homotopy class of this contact boundary is invariant in a large family of deformations of polynomials, which are not necessarily topologically trivial. Received: 15 November 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 32S55, 53D15, 32S50     

The universality of equivariant complex bordism

We show that if A is an abelian compact Lie group, all A-equivariant complex vector bundles are orientable over a complex orientable equivariant cohomology theory. In the process, we calculate the complex orientable homology and cohomology of all complex Grassmannians. Received: 14 February 2000; in final form: 4 August 2000 / Published online: 19 October 2001     

Deformations with Section: Cotangent Cohomology, Flatness Conditions, and Modular Subgerms

We study modular subspaces corresponding to two deformation functors associated with an isolated singularity X ₀: the functor of deformations of X ₀ and the functor of deformations with section of X ₀. After recalling some standard facts on the cotangent cohomology of analytic algebras and the general theory of deformations with section, we give several criteria for modularity in terms of the relative cotangent cohomology modules of a deformation. In particular, it is shown that the modular strata for the functors and of quasi-homogeneous complete intersection singularities coincide. Then flatness conditions for the first cotangent cohomology modules of the deformation functors under consideration are compared. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 15, Theory of Functions, 2004.     

Cohomology Theories of Hopf Bimodules and Cup-Product

Given a Hopf algebra A, there exist various cohomology theories for the category of Hopf bimodules over A, introduced by M. Gerstenhaber and S. D. Schack, and by C. Ospel. We prove, when A is finite-dimensional, that they are all equal to the Ext functor on the module category of an associative algebra associated to A, described by C. Cibils and M. Rosso. We also give an expression for a cup-product in the cohomology defined by C. Ospel, and prove that it corresponds to the Yoneda product of extensions.