Flasque Model Structures for Simplicial Presheaves

It is well known that there are two useful families of model structures on presheaves: the injective and projective. In fact, there is at least one more: the flasque. For some purposes, both the projective and the injective structure run into technical and annoying (but surmountable) difficulties for different reasons. The flasque model structure, which possesses a combination of the convenient properties of both structures, sometimes avoids these difficulties. (Received: February 2006)     

Local Projective Model Structures on Simplicial Presheaves

We introduce different model structures on the categories of simplicial presheaves and simplicial sheaves on some essentially small Gro-then-dieck site T and give some applications of these simplified model categories. In particular, we prove that the stable homotopy categories SH((Sm/k)Nis,A1) and SH((Sch/k)cdh,A1) are equivalent. This result was first proven by Voevodsky and our proof uses many of his techniques, but it does not use his theory of -closed classes.     

Singularities of pairs via jet schemes

Let be a smooth variety and a closed subscheme. We use motivic integration on the space of arcs of to characterize the fact that is log canonical or log terminal using the dimension of the jet schemes of . This gives a formula for the log canonical threshold of , which we use to prove a result of Demailly and Kollár on the semicontinuity of log canonical thresholds.

     

The stack of formal groups in stable homotopy theory

We construct the algebraic stack of formal groups and use it to provide a new perspective onto a recent result of M. Hovey and N. Strickland on comodule categories for Landweber exact algebras. This leads to a geometric understanding of their results as well as to a generalisation.     

Completion of G-spectra and stable maps between classifying spaces

We prove structural theorems for computing the completion of a G-spectrum at the augmentation ideal of the Burnside ring of a finite group G. First we show that a G-spectrum can be replaced by a spectrum obtained by allowing only isotropy groups of prime power order without changing the homotopy type of the completion. We then show that this completion can be computed as a homotopy colimit of completions of spectra obtained by further restricting isotropy to one prime at a time, and that these completions can be computed in terms of completion at a prime.As an application, we show that the spectrum of stable maps from BG to the classifying space of a compact Lie group K splits non-equivariantly as a wedge sum of p-completed suspension spectra of classifying spaces of certain subquotients of G×K. In particular this describes the dual of BG.     

Deligne's duality for de Rham realizations of 1‐motives

Let M be a 1‐motive over a base scheme S and M ′ its Cartier dual. We show the existence of a canonical duality between the de Rham realizations of M and M ′; this generalizes a result in [5]. Furthermore, we study universal extensions of 1‐motives and their relation with ?‐extensions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some pairs of adjoint functors involving track homotopy categories

K. A. Hardie and K. H. Kamps investigated the track homotopy categoryH _B over a fixed spaceB ([1]). They have introduced two pairs of adjoint functors:P _B N _B andm _* m ^*, whereP _B :H _B →H ^B , andm _*:H _A →H _B for a fixed mapm:A→B. We have introduced a split fibration of categoriesL:H _b →H _B and provedL J, J L in [2]. This paper first extendsP _B N _B to for any fixed mapb: . Moreover we also extend these results to obtain two pairs of adjoint functors involving track homotopy categoriesH _b andH ^b whereH ^b is the dual ofH _b . One of our results isN _b P _b . This differs fromP _B N _B . Supported by National Natural Science Foundation of China     

Stable Projective Homotopy Theory of Modules,Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.     

Gonality for stable curves and their maps with a smooth curve as their target

Here we study the deformation theory of some maps f: X → ℙ ^r , r = 1, 2, where X is a nodal curve and f|T is not constant for every irreducible component T of X. For r = 1 we show that the “stratification by gonality” for any subset of with fixed topological type behaves like the stratification by gonality of M _g.      

The Convergence of (~γ)s(b0hn - h1bn-1)

This paper computes the Thom map on γ2 and proves that it is represented by 2b2,0h1,2 in the ASS. The authors also compute the higher May differential of b2,0, from which it is proved that (~γ)s(b0hn - h1bn-1) for 2 ≤ s ＜ p - 1 are permanent cycles in the ASS.     

The Convergence of (r|~)_s(b_0h_n-h_1b_(n-1))

This paper computes the Thorn map onγ2 and proves that it is represented by 2b2,0h1,2 in the ASS. The authors also compute the higher May differential of 62,0, from which it is proved thatγs(b0hn-h1bn-1) for 2≤s < p - 1 are permanent cycles in the ASS.     

球面稳定同伦元素α1β1βs的非平凡性

决定球面稳定同伦群是同伦论中的核心问题之一,是非常重要的．该文证明:球面稳定同伦元素α1β1βs是一个阶为p的非平凡元素,其中p≥5是任意奇素数,1≤s相似文献   

On Homotopy Groups of Real Algebraic Varieties and their Complexifications

Let X ₀ be a topological component of a nonsingular real algebraic variety and i:X → X _C is a nonsingular projective complexification of X. In this paper, we will study the homomorphism on homotopy groups induced by the inclusion map i:X ₀ → X _C and obtain several results using rational homotopy theory and other standard tools of homotopy theory.     

球面稳定同伦群的回访新元素族

本文构造了在Adams谱序列中由hng0γ3∈E26,t所表示的球面稳定同伦群πt-6S的新元素族,回访了文[1]中构造的bn-1g0γ3-元素族∈πt-7S,其中t=2pn(p-1)+6(p2+P+1)(p-1),P≥7是素数, n≥4.     

Examples of rational homotopy types of blow-ups

We give an example of two homotopic embeddings of manifolds with isomorphic complex normal bundles but such that the blow-ups of along and along have different rational homotopy types.

     

Two New Families in the Stable Homotopy Groups of Sphere and Moore Spectrum

This paper proves the existence of an order p element in the stable homotopy group of sphere spectrum of degree pnq pmq q - 4 and a nontrivial element in the stable homotopy group of Moore spectum of degree pnq pmq q - 3 which are represented by h0(hmbn-1-hnbm-1) and i*(h0hnhm) in the E2-terms of the Adams spectral sequence respectively, where p≥7 is a prime, n≥m 2≥4, q = 2(p - 1).     

Hodge polynomials of the moduli spaces of rank 3 pairs

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple on X consists of two holomorphic vector bundles E ₁ and E ₂ over X and a holomorphic map . There is a concept of stability for triples which depends on a real parameter σ. In this paper, we determine the Hodge polynomials of the moduli spaces of σ-stable triples with rk(E ₁) = 3, rk(E ₂) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.