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1.
The integer cohomology algebra of the complement of a complex subspace arrangement with geometric intersection lattice is completely determined by the combinatorial data of the arrangement. We give a combinatorial presentation of the cohomology algebra in the spirit of the Orlik-Solomon result on the cohomology algebras of complex hyperplane arrangements. Our methods are elementary: we work with simplicial models for the complements that are induced by combinatorial stratifications of complex space. We describe simplicial cochains that generate the cohomology. Among them we distinguish a linear basis, study cup product multiplication, and derive an algebra presentation in terms of generators and relations. 相似文献
2.
We show that the cohomology algebra of the complement of a coordinate subspace arrangement in the m-dimensional complex space is isomorphic to the cohomology algebra of the StanleyReisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) After that we calculate the latter cohomology algebra by means of the standard Koszul resolution of a polynomial ring. To prove these facts, we construct a homotopy equivalence (equivariant with respect to the torus action) between the complement of a coordinate subspace arrangement and the moment-angle complex defined by a simplicial complex. The moment-angle complex is a certain subset of the unit polydisk in the m-dimensional complex space invariant with respect to the action of the m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere; otherwise, the complex has a more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the EilenbergMoore spectral sequence. We also relate our results with well-known facts in the theory of toric varieties and symplectic geometry. Bibliography: 23 titles. 相似文献
5.
We characterize finite-dimensional Lie algebras over the real numbers for which the classical Yang-Baxter equation has a non-trivial
skew-symmetric solution (resp. a non-trivial solution with invariant symmetric part). Equivalently, we obtain a characterization
of those finite-dimensional real Lie algebras which admit a non-trivial (quasi-) triangular Lie bialgebra structure. 相似文献
6.
We prove a localization formula for group-valued equivariant de Rham cohomology of a compact G-manifold. This formula is a non-trivial generalization of the localization formula of Berline-Vergne and Atiyah-Bott for
the usual equivariant de Rham cohomology. We derive from this result a Duistermaat-Heckman formula for group valued moment
maps. As an application, we prove part of Witten’s conjectures about intersection pairings on moduli spaces of flat connections
on 2-manifolds.
Oblatum 24-VI-1999 & 29-X-1999?Published online: 21 February 2000 相似文献
8.
We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces,
to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in
the simple complex Lie algebras of classical type A, B or C, as well as in the group . We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them
as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding
Poincaré series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on
its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive
degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology
of its associated complex discriminant variety.
Received August 31, 1998; in final form August 1, 1999 / Published online October 30, 2000 相似文献
9.
Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps address the problem of representing
high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data.
This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent
the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle,
for example, is one-dimensional, but its faithful representation requires two real coordinates. In this work, we present a
strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology
to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain
the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise
NLDR analysis of a broader range of realistic data sets. 相似文献
12.
Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group as defined below.
The triviality of bundles arising from orthogonal representations of Γ is characterized completely as follows. An orthogonal
representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups. Furthermore,
the subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, the subgroup of elements which arises from real orthogonal representations of Γ is an elementary abelian 2-group,
which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation.
In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi
identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid
group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure
braid groups which give rise to trivial bundles.
Received: 6 February 2002 / Revised version: 19 September 2002 /
Published online: 8 April 2003
RID="⋆"
ID="⋆" Partially supported by the NSF
RID="⋆⋆"
ID="⋆⋆" Partially supported by grant LEQSF(1999-02)-RD-A-01 from the Louisiana Board of Regents, and by grant MDA904-00-1-0038
from the National Security Agency
RID="⋆"
ID="⋆" Partially supported by the NSF
Mathematics Subject Classification (2000): 20F36, 32S22, 55N15, 55R50 相似文献
13.
In the theory of algebraic group actions on affine varieties, the concept of a Kempf-Ness set is used to replace the categorical
quotient by the quotient with respect to a maximal compact subgroup. Using recent achievements of “toric topology,” we show
that an appropriate notion of a Kempf-Ness set exists for a class of algebraic torus actions on quasiaffine varieties (coordinate
subspace arrangement complements) arising in the Batyrev-Cox “geometric invariant theory” approach to toric varieties. We
proceed by studying the cohomology of these “toric” Kempf-Ness sets. In the case of projective nonsingular toric varieties
the Kempf-Ness sets can be described as complete intersections of real quadrics in a complex space.
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 263, pp. 159–172. 相似文献
14.
We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with
a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that
this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived
category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice
stratified space to sheaves on a poset.
T. Braden’s research was supported in part by NSF grant DMS-0201823.
N. Proudfoot’s research was supported in part by an NSF Postdoctoral Research Fellowship and NSF grant DMS-0738335. 相似文献
15.
We express the cohomology of the complement of a real subspace arrangement of diagonal linear subspaces in terms of the Betti numbers of a minimal free resolution. This leads to formulas for the cohomology in some cases, and also to a cohomology vanishing theorem valid for all arrangements. 相似文献
16.
Let X and Y be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a mapping, , can be approximated by regular mappings in the space of mappings, , equipped with the topology. In this paper, we obtain some results concerning this problem when the target space is the 2-dimensional standard
sphere and X has a complexification that is a rational (complex) surface. To get the results we study the subgroup of the second cohomology group of X with integer coefficients that consists of the cohomology classes that are pullbacks, via the inclusion mapping , of the cohomology classes in represented by complex algebraic hypersurfaces.
Received December 1, 1998; in final form August 2, 1999 相似文献
17.
We study a combinatorially defined double complex structure on the ordered chains of any simplicial complex. Its columns are related to the cell complex Kn whose face poset is isomorphic to the subword ordering on words without repetition from an alphabet of size n. This complex is shellable and as an application we give a representation theoretic interpretation for derangement numbers and a related symmetric function considered by Désarménien and Wachs [11]. We analyze the two spectral sequences arising from the double complex in the case of the bar resolution for a group. This spectral sequence converges to the cohomology of the group and provides a method for computing group cohomology in terms of the cohomology of subgroups. Its behavior is influenced by the complex of oriented chains of the simplicial complex of finite subsets of the group, and we examine the Ext class of this complex. 相似文献
19.
We derive an explicit version of the Bernstein-Gel'fand-Gel'fand (BGG) correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free resolutions over its ``Koszul dual' exterior algebra. Among the facts about the BGG correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the ``linear part' of a resolution over the exterior algebra. We explore the structure of free resolutions over an exterior algebra. For example, we show that such resolutions are eventually dominated by their ``linear parts" in the sense that erasing all terms of degree 1$"> in the complex yields a new complex which is eventually exact. As applications we give a construction of the Beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms. The explicitness of our version allows us to prove two conjectures about the morphisms in the monad, and we get an efficient method for machine computation of the cohomology of sheaves. We also construct all the monads for a sheaf that can be built from sums of line bundles, and show that they are often characterized by numerical data. 相似文献
20.
We introduce hom-Lie-Rinehart algebras as an algebraic analogue of hom-Lie algebroids, and systematically describe a cohomology complex by considering coefficient modules. We define the notion of extensions for hom-Lie-Rinehart algebras. In the sequel, we deduce a characterization of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extension and the equivalence classes of those abelian extensions in the category of hom-Lie-Rinehart algebras, respectively. We also construct a canonical example of hom-Lie-Rinehart algebra associated to a given Poisson algebra and an automorphism. 相似文献
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