共查询到20条相似文献,搜索用时 656 毫秒
1.
Roy Joshua 《K-Theory》2002,27(3):197-244
This is the second part of our work on the intersection theory of algebraic stacks. The main results here are the following. We provide an intersection pairing for all smooth Artin stacks (locally of finite type over a field) which we show reduces to the known intersection pairing on the Chow groups of smooth Deligne–Mumford stacks of finite type over a field as well as on the Chow groups of quotient stacks associated to actions of linear algebraic groups on smooth quasi-projective schemes modulo torsion. The former involves also showing the existence of Adams operations on the rational étale K-theory of all smooth Deligne–Mumford stacks of finite type over a field. In addition, we show that our definition of the higher Chow groups is intrinsic to the stack for all smooth stacks and also stacks of finite type over the given field. Next we establish the existence of Chern classes and Chern character for Artin stacks with values in our Chow groups and extend these to higher Chern classes and a higher Chern character for perfect complexes on an algebraic stack, taking values in cohomology theories of algebraic stacks that are defined with respect to complexes of sheaves on a big smooth site. As a by-product of our techniques we also provide an extension of higher intersection theory to all schemes locally of finite type over a field. As the higher cycle complex, by itself, is a bit difficult to handle, the stronger results like contravariance for arbitrary maps between smooth stacks and the intersection pairing for smooth stacks are established by comparison with motivic cohomology. 相似文献
2.
In this paper we study the integral Chow ring of toric Deligne–Mumford stacks. We prove that the integral Chow ring of a semi-projective
toric Deligne–Mumford stack is isomorphic to the Stanley–Reisner ring of the associated stacky fan. The integral orbifold
Chow ring is also computed. Our results are illustrated with several examples. 相似文献
3.
Margarida Melo 《Mathematische Zeitschrift》2009,263(4):939-957
We study algebraic (Artin) stacks over [`(M)]g{\overline{\mathcal M}_{g}} giving a functorial way of compactifying the relative degree d Picard variety for families of stable curves. We also describe for every d the locus of genus g stable curves over which we get Deligne–Mumford stacks strongly representable over[`(M)]g{\overline{\mathcal M}_{g}} . 相似文献
4.
Roy Joshua⋆ 《Compositio Mathematica》2003,136(2):117-169
In this paper we establish Riemann–Roch and Lefschtez–Riemann–Roch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The Riemann–Roch theorem is established as a natural transformation between the G-theory of algebraic stacks and topological G-theory for stacks: we define the latter as the localization of G-theory by topological K-homology. The Lefschtez–Riemann–Roch is an extension of this including the action of a torus for Deligne–Mumford stacks. This generalizes the corresponding Riemann–Roch theorem (Lefschetz–Riemann–Roch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G-theory (as well as rational G-theory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological G-theory on schemes as proved by Thomason. 相似文献
5.
B. Toen 《K-Theory》1999,18(1):33-76
We develop a cohomology theory for Deligne–Mumford stacks, adapted to Hirzebruch–Riemann–Roch formulas. For this, we define the cohomology with coefficients in the representations and a Chern character, and we prove a Grothendieck–Riemann–Roch formula for the associated Riemann–Roch transformation. 相似文献
6.
Oliver Petras 《Journal of Number Theory》2009,129(10):2346-2368
We prove relations between fractional linear cycles in Bloch's integral cubical higher Chow complex in codimension two of number fields, which correspond to functional equations of the dilogarithm. These relations suffice, as we shall demonstrate with a few examples, to write down enough relations in Bloch's integral higher Chow group CH2(F,3) for certain number fields F to detect torsion cycles. Using the regulator map to Deligne cohomology, one can check the non-triviality of the torsion cycles thus obtained. Using this combination of methods, we obtain explicit higher Chow cycles generating the integral motivic cohomology groups of some number fields. 相似文献
7.
We apply the “homotopy coniveau” machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized
to arbitrary X by Edidin–Graham) and an Atiyah–Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch–Lichtenbaum and Friedlander–Suslin.
The first-named author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and
support of the NSF via grants DMS-0140445 and DMS-0457195. 相似文献
8.
It is shown that the product structures of motivic cohomology groups and of higher Chow groups are compatible under the comparison isomorphism of Voevodsky (2002) [11]. This extends the result of Weibel (1999) [14], where he used the comparison isomorphism which assumed that the base field admits resolution of singularities.The mod n motivic cohomology groups and product structures in motivic homotopy theory are defined, and it is shown that the product structures are compatible under the comparison isomorphisms. 相似文献
9.
We prove the cone theorem for varieties with LCIQ singularities using deformation theory of stable maps into Deligne–Mumford
stacks. We also obtain a sharper bound on −(K
X
+ D)-degree of (K
X
+ D)-negative extremal rays for projective -factorial log terminal threefold pairs (X, D). 相似文献
10.
The Riemann–Roch theorem without denominators for the Chern class maps on higher algebraic K-groups with values in motivic cohomology groups in the context of motivic homotopy theory is proved. 相似文献
11.
In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion
of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined
over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified
extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients
are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser. 相似文献
12.
V. Trnková 《Algebra Universalis》1998,39(3-4):121-144
A finitary monosorted algebraic theory is called locally representable (or representable) in a category with finite products if every its initial segment is the domain of a full faithful finite-products-preserving functor into
(or if itself is the domain of such a functor). The question of when local representability implies representability is discussed,
and theories for which local representability always implies representability are fully characterized.
Received December 20, 1996; accepted in final form March 19, 1998. 相似文献
13.
In the present paper, we define a notion of numerical equivalence on Chow groups or Grothendieck groups of Noetherian local rings, which is an analogue of that on smooth projective varieties. Under a mild condition, it is proved that the Chow group modulo numerical equivalence is a finite dimensional -vector space, as in the case of smooth projective varieties. Numerical equivalence on local rings is deeply related to that on smooth projective varieties. For example, if Grothendiecks standard conjectures are true, then a vanishing of Chow group (of local rings) modulo numerical equivalence can be proven. Using the theory of numerical equivalence, the notion of numerically Roberts rings is defined. It is proved that a Cohen–Macaulay local ring of positive characteristic is a numerically Roberts ring if and only if the Hilbert–Kunz multiplicity of a maximal primary ideal of finite projective dimension is always equal to its colength. Numerically Roberts rings satisfy the vanishing property of intersection multiplicities. We shall prove another special case of the vanishing of intersection multiplicities using a vanishing of localized Chern characters. 相似文献
14.
In his classic book on symmetric functions, Macdonald describes a remarkable result by Green relating the character theory of the finite general linear group to transition matrices between bases of symmetric functions. This connection allows one to analyze the character theory of the general linear group via symmetric group combinatorics. Using works of Ennola, Kawanaka, Lusztig and Srinivasan, this paper describes the analogous setting for the finite unitary group. In particular, we explain the connection between Deligne–Lusztig theory and Ennola's efforts to generalize Green's work, and from this we deduce various character theoretic results. Applications include calculating certain sums of character degrees, and giving a model of Deligne–Lusztig type for the finite unitary group, which parallels results of Klyachko and Inglis and Saxl for the finite general linear group. 相似文献
15.
Lin Weng 《Mathematische Annalen》2001,320(2):239-283
In Part I, Deligne-Riemann-Roch isometry is generalized for punctured Riemann surfaces equipped with quasi-hyperbolic metrics.
This is achieved by proving the Mean Value Lemmas, which explicitly explain how metrized Deligne pairings for -admissible metrized line bundles depend on . In Part II, we first introduce several line bundles over Knudsen-Deligne-Mumford compactification of the moduli space (or
rather the algebraic stack) of stable N-pointed algebraic curves of genus g, which are rather natural and include Weil-Petersson, Takhtajan-Zograf and logarithmic Mumford line bundles. Then we use
Deligne-Riemann-Roch isomorphism and its metrized version (proved in Part I) to establish some fundamental relations among
these line bundles. Finally, we compute first Chern forms of the metrized Weil-Petersson, Takhtajan-Zograf and logarithmic
Mumford line bundles by using results of Wolpert and Takhtajan-Zograf, and show that the so-called Takhtajan-Zograf metric
on the moduli space is algebraic.
Received February 14, 2000 / Accepted August 18, 2000 / Published online February 5, 2001 相似文献
16.
When S is a surface with pg(S)>0, Mumford proved that its Chow group A*S is not finite dimensional in some sense. In this paper, we propose another definition of finite dimensionality for the Chow groups. Using this new definition, at least the Chow group of some surface S with pg(S)>0 (for example, the product of two curves) becomes finite dimensional. The finite dimensionality of the Chow groups follows from the finite dimensionality of the Chow motives. It turns out that the finite dimensionality of the Chow motives is a very strong property. For example, we can prove Blochs conjecture (representability of the Chow groups of surfaces with pg(S)=0) under the assumption that the Chow motive of S is finite dimensional.Mathematics Subject Classification (2000): 14C 相似文献
17.
The paper shows how to associate a motivic zeta function with a
large class of infinite dimensional Lie algebras. These include
loop algebras, affine Kac-Moody algebras, the Virasoro algebra
and Lie algebras of Cartan type. The concept of a motivic zeta
functions provides a good language to talk about the uniformity
in p of local
p-adic zeta functions of finite dimensional
Lie algebras. The theory of motivic integration is employed to
prove the rationality of motivic zeta functions associated to
certain classes of infinite dimensional Lie algebras. 相似文献
18.
19.
Francesco Baldassarri 《Milan Journal of Mathematics》2005,73(1):237-258
We review the notion of regular singular point of a linear differential equation with meromorphic coefficients, from the viewpoint
of algebraic geometry. We give several equivalent definitions of regularity along a divisor for a meromorphic connection on
a complex algebraic manifold and discuss the global birational theory of fuchsian differential modules over a field of algebraic
functions. We describe the generalized algebraic version of Deligne’s canonical extension, constructed in [1, I.4].
Our main interest lies in the algebraic form of Deligne’s regularity criterion [2, II.4.4 (iii)], asserting that, on a normal
compactification, only one codimensional components of the locus at infinity need to be considered. If one considers the purely
algebraic nature of the statement, it is surprising that the only existing proof of this criterion is the transcendental argument
given by Deligne in his corrigendum to loc. cit. dated April 1971. The algebraic proof given in our book [1, I.5.4] is also incorrect, as J. Bernstein kindly indicated to
us.We introduce some notions of logarithmic geometry to let the reader appreciate Bernstein’s (counter)examples to some statements
in our book [1]. Standard methods of generic projection in projective spaces reduce the question to a two-dimensional puzzle.
We report on ongoing correspondence with Y. André and N. Tsuzuki, leading to partial results and provide examples indicating
the subtlety of the problem.
Lecture held in the Seminario Matematico e Fisico on January 31, 2005 Received: June 2005 相似文献
20.