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142
CastelnuovoMumford Regularity in Biprojective Spaces
, 2002
"... We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on mregularity for graded modules over polynomial algebras. ..."
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Cited by 17 (5 self)
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We define the concept of regularity for bigraded modules over a bigraded polynomial ring. In this setting we prove analogs of some of the classical results on mregularity for graded modules over polynomial algebras.
Deformation theory and Lie algebra homology
, 1997
"... 1.1. Let X be a smooth proper scheme X over a field k of characteristic 0, G an algebraic group over k and p: P − → X a Gtorsor over X. Consider the following deformation problems. Problem 1. Flat deformations of X. Problem 2. Flat deformations of the pair (X,P). ..."
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Cited by 16 (2 self)
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1.1. Let X be a smooth proper scheme X over a field k of characteristic 0, G an algebraic group over k and p: P − → X a Gtorsor over X. Consider the following deformation problems. Problem 1. Flat deformations of X. Problem 2. Flat deformations of the pair (X,P).
βnbcbases for cohomology of local systems on hyperplane complements
 TRANS. AMER. MATH. SOC
, 1997
"... We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes A. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for ..."
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Cited by 13 (1 self)
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We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes A. The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only nonvanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement A. The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain “nobrokencircuits” bases of A. The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.
Holonomy on Poisson manifolds and the modular class
"... Abstract. We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bun ..."
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Cited by 13 (1 self)
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Abstract. We introduce linear holonomy on Poisson manifolds. The linear holonomy of a Poisson structure generalizes the linearized holonomy on a regular symplectic foliation. However, for singular Poisson structures the linear holonomy is defined for the lifts of tangential path to the cotangent bundle (cotangent paths). The linear holonomy is closely related to the modular class studied by A. Weinstein. Namely, the logarithm of the determinant of the linear holonomy is equal to the integral of the modular vector field along such a lift. This assertion relies on the notion of the integral of a vector field along a cotangent path on a Poisson manifold, which is also introduced in the paper. In the second part of the paper we prove that for locally unimodular Poisson manifolds the modular class is an invariant of Morita equivalence. 1.
New Model Categories From Old
 J. Pure Appl. Algebra
, 1995
"... . We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categor ..."
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Cited by 13 (5 self)
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. We review Quillen's concept of a model category as the proper setting for defining derived functors in nonabelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas  most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in nonabelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such nonabelian derived functors is the E 2 term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...
Virtual Betti numbers of real algebraic varieties
 Comptes Rendus Acad. Sci. Paris, Ser. I
"... Abstract. The weak factorization theorem for birational maps is used to prove that for all i ≥ 0 the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number βi defined for all real algebraic varieties, such that if Y is a closed subvari ..."
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Cited by 13 (4 self)
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Abstract. The weak factorization theorem for birational maps is used to prove that for all i ≥ 0 the ith mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number βi defined for all real algebraic varieties, such that if Y is a closed subvariety of X, then βi(X) = βi(X \ Y) + βi(Y). We define an invariant of the Grothendieck ring K0(VR) of real algebraic varieties, the virtual Poincaré polynomial β(X,t), which is a ring homomorphism K0(VR) → Z[t]. For X nonsingular and compact, β(X,t) is the classical Poincaré polynomial for cohomology with Z2 coefficients. The coefficients of the virtual Poincaré polynomial are the virtual Betti numbers. By the weak factorization theorem for birational morphisms (Abramovich et al. [1]), the existence of β(X,t) follows from a simple formula for the Betti numbers of the blowup of a compact nonsingular variety along a closed nonsingular center. The existence of the virtual Betti numbers for certain real analytic spaces, including real algebraic varieties, has also been announced by Totaro [23]. Kontsevich’s motivic measure on the arc space of a complex algebraic variety takes values in the completion of the localized Grothendieck ring (cf. [20], [11], [7]). Completion with
The BoardmanVogt resolution of operads in monoidal model categories, in preparation
"... Abstract. We extend the Wconstruction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for wellpointed Σcofibrant operads. The standard simplicial resolution of Godement as well as the cobarbar chain ..."
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Cited by 11 (9 self)
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Abstract. We extend the Wconstruction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for wellpointed Σcofibrant operads. The standard simplicial resolution of Godement as well as the cobarbar chain resolution are shown to be particular instances of this generalised Wconstruction.
Theory of valuations on manifolds, I. Linear spaces
 Israel J. Math
"... This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (nont ..."
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Cited by 11 (7 self)
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This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. We introduce a space of smooth (nontranslation invariant) valuations on a linear space V. We present three descriptions of this space. We describe the canonical multiplicative structure on this space generalizing the results from [3] obtained for polynomial valuations. 0 Introduction. This is the first part of a series of articles where we are going to develop theory of valuations on manifolds generalizing the classical theory of continuous valuations on convex subsets of an affine space. In this article we still work only with linear spaces. In the subsequent parts of this series we are going to generalize constructions of this article to arbitrary smooth