Higher Chern Classes and Steenrod Operations in Motivic Cohomology

In this short paper we investigate the relation between higher Chern classes and reduced power operations in motivic cohomology. More precisely, we translate the well-known arguments [5] into the context of motivic cohomology and define higher Chern classes c_p,q : K _p(X) → H^2q-p (X,Z(q)) → H^2q-p(X, Z/l(q)), where X is a smooth scheme over the base field k, l is a prime number and char(k) ≠ l. The same approach produces the classes for K-theory with coefficients as well. Let further Pⁱ : H^m(X, Z/l(n)) → H^m+2i(l-1) (X, Z/l(n + i(l - 1))) denote the ith reduced power operation in motivic cohomology, constructed in [2]. The main result of the paper looks as follows.     

Completeness and compactness for varieties over a local field

This paper shows that for a local field K, a subfield k ⊂ K and a variety X over k, X is complete if and only if for every finite field extension Kʹ | K, the set X(Kʹ) is compact in its strong topology. The author likes to thank Florian Pop, Jakob Stix, Stefan Wewers, Gunther Cornelissen and his own parents for their support. Received: 13 April 2006     

Disk-like Tiles Derived from Complex Bases

For each positive integer k, the radix representation of the complex numbers in the base –k+i gives rise to a lattice self-affine tile T _k in the plane, which consists of all the complex numbers that can be expressed in the form ∑ _j≥1 d _j (–k+i)^–j , where d _j ∈{0, 1, 2, ...,k ²}. We prove that T _k is homeomorphic to the closed unit disk {z∈C:∣z∣ ≤ 1} if and only if k ≠ 2. The first author is supported by Youth Project of Tianyuan Foundation (10226031) and Zhongshan University Promotion Foundation for Young Teachers (34100-1131206); the second author is supported by National Science Foundation (10041005) and Guangdong Province Science Foundation (011221)     

Cohomology vanishing¶and a problem in approximation theory

For a simplicial subdivison Δ of a region in k ⁿ (k algebraically closed) and r∈N, there is a reflexive sheaf ? on P ⁿ , such that H ⁰(?(d)) is essentially the space of piecewise polynomial functions on Δ, of degree at most d, which meet with order of smoothness r along common faces. In [9], Elencwajg and Forster give bounds for the vanishing of the higher cohomology of a bundle ℰ on P ⁿ in terms of the top two Chern classes and the generic splitting type of ℰ. We use a spectral sequence argument similar to that of [16] to characterize those Δ for which ? is actually a bundle (which is always the case for n= 2). In this situation we can obtain a formula for H ⁰(?(d)) which involves only local data; the results of [9] cited earlier allow us to give a bound on the d where the formula applies. We also show that a major open problem in approximation theory may be formulated in terms of a cohomology vanishing on P ² and we discuss a possible connection between semi-stability and the conjectured answer to this open problem. Received: 9 April 2001     

Stein extensions of real symmetric spaces and the geometry of the flag manifold

 Let G be a connected semisimple Lie group contained in its simply connected complexification G _C . Let KG∩K _C be a maximal compact subgroup of G. Denote by X _o the unique closed G-orbit in the full flag manifold ℱ and by 𝒪 the unique open K _C -orbit in ℱ. The set consisting of the elements gK _C so that gX _o ⊂𝒪 is an Stein extension of G/K⊂G _C /K _C . There is a universal domain , natural form the point of view of group actions which has been conjectured to be Stein. The main result of this paper is the inclusion . In the second part of the paper I show, under some dominance condition in the parameter, that representations in Dolbeault cohomology can be realized as holomorphic sections of vector bundles over . Received: 9 September 2002 / Revised version: 12 July 2002 / Published online: 8 April 2003 Mathematics Subject Classification (2002): 22E30 Research partially supported by NSF grant DMS-9801605 and DMS 0074991.     

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason's theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H⁰(k,(1)) corresponding to a primitive nth root of unity.     

Cohomologie de Hochschild et espaces projectifs tronqués

In this paper, we outline the construction of obstructions tok-formality wherek is a field of arbitrary characteristic. We show that the obstructions to intrinsick-formality of a graded algebraH ^* lie in the Hochschild cohomologyH H ^m,1 (H ^*/k, H ^*). We calculate explicitly the Hochschild cohomology of the algebraH(ℂP ^∞/ℂP ⁿ ;k) and deduce that the stunted complex projective space ℂP ^∞/ℂP ⁿ is intrinsicallyk-formal for any fieldk.

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This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

M?bius transform, moment-angle complexes and Halperin–Carlsson conjecture

The motivation for this paper comes from the Halperin–Carlsson conjecture for (real) moment-angle complexes. We first give an algebraic combinatorics formula for the M?bius transform of an abstract simplicial complex K on [m]={1,…,m} in terms of the Betti numbers of the Stanley–Reisner face ring k(K) of K over a field k. We then employ a way of compressing K to provide the lower bound on the sum of those Betti numbers using our formula. Next we consider a class of generalized moment-angle complexes Z_K^{(\mathbb D, \mathbb S)}\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})}, including the moment-angle complex Z_K\mathcal{Z}_{K} and the real moment-angle complex \mathbbRZ_K\mathbb{R}\mathcal {Z}_{K} as special examples. We show that H^*(Z_K^{(\mathbb D, \mathbb S)};k)H^{*}(\mathcal{Z}_{K}^{(\underline{\mathbb{ D}}, \underline{\mathbb{ S}})};\mathbf{k}) has the same graded k-module structure as Tor  ^k[v](k(K),k). Finally we show that the Halperin–Carlsson conjecture holds for Z_K\mathcal{Z}_{K} (resp. \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}) under the restriction of the natural T ^m -action on Z_K\mathcal{Z}_{K} (resp. (ℤ₂) ^m -action on \mathbb RZ_K\mathbb{ R}\mathcal{Z}_{K}).     

The K-theory of fields in characteristic p

We show that for a field k of characteristic p, H ⁱ (k,ℤ(n)) is uniquely p-divisible for i≠n (we use higher Chow groups as our definition of motivic cohomology). This implies that the natural map K _n ^M (k)?K _n (k) from Milnor K-theory to Quillen K-theory is an isomorphism up to uniquely p-divisible groups, and that K _n ^M (k) and K _n (k) are p-torsion free. As a consequence, one can calculate the K-theory mod p of smooth varieties over perfect fields of characteristic p in terms of cohomology of logarithmic de Rham Witt sheaves, for example K _n (X,ℤ/p ^r )=0 for n>dimX. Another consequence is Gersten’s conjecture with finite coefficients for smooth varieties over discrete valuation rings with residue characteristic p. As the last consequence, Bloch’s cycle complexes localized at p satisfy all Beilinson-Lichtenbaum-Milne axioms for motivic complexes, except possibly the vanishing conjecture. Oblatum 21-I-1998 & 26-VII-1999 / Published online: 18 October 1999     

Idempotency of Extensions via the Bicompletion

Let Top ₀ be the category of topological T ₀-spaces, QU ₀ the category of quasi-uniform T ₀-spaces, T : QU ₀ → Top ₀ the usual forgetful functor and K : QU ₀ → QU ₀ the bicompletion reflector with unit k : 1 → K. Any T-section F : Top ₀ → QU ₀ is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU ₀ is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005).     

On Hochschild cohomology rings of Fibonacci algebras

Let Λ be a Fibonacci algebra over a field k. The multiplication of Hochschild cohomology ring of Λ induced by the Yoneda product is described explicitly. As a consequence, the multiplicative structure of Hochschild cohomology ring of Λ is proved to be trivial.     

Entropy of Stationary Measures and Bounded Tangential de-Rham Cohomology of Semisimple Lie Group Actions

Let G be a connected noncompact semisimple Lie group with finite center, K a maximal compact subgroup, and X a compact manifold (or more generally, a Borel space) on which G acts. Assume that ν is a μ -stationary measure on X, where μ is an admissible measure on G, and that the G-action is essentially free. We consider the foliation of K\ X with Riemmanian leaves isometric to the symmetric space K\ G, and the associated tangential bounded de-Rham cohomology, which we show is an invariant of the action. We prove both vanishing and nonvanishing results for bounded tangential cohomology, whose range is dictated by the size of the maximal projective factor G/Q of (X, ν). We give examples showing that the results are often best possible. For the proofs we formulate a bounded tangential version of Stokes’ theorem, and establish a bounded tangential version of Poincaré’s Lemma. These results are made possible by the structure theory of semisimple Lie groups actions with stationary measure developed in Nevo and Zimmer [Ann of Math. 156, 565--594]. The structure theory assert, in particular, that the G-action is orbit equivalent to an action of a uniquely determined parabolic subgroup Q. The existence of Q allows us to establish Stokes’ and Poincaré’s Lemmas, and we show that it is the size of Q (determined by the entropy) which controls the bounded tangential cohomology. Supported by BSF and ISF. Supported by BSF and NSF.     

Nikulin involutions on <Emphasis Type="Italic">K</Emphasis>3 surfaces

We study the maps induced on cohomology by a Nikulin (i.e. a symplectic) involution on a K3 surface. We parametrize the 11-dimensional irreducible components of the moduli space of algebraic K3 surfaces with a Nikulin involution and we give examples of the general K3 surface in various components. We conclude with some remarks on Morrison–Nikulin involutions, these are Nikulin involutions which interchange two copies of E ₈(−1) in the Néron Severi group. The second author is supported by DFG Research Grant SA 1380/1-1.     

P4k?1-factorization of bipartite multigraphs

Let λK _m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P _v-factorization of λK _m,n is a set of edge-disjoint P _v -factors of λK _m,n which partition the set of edges of λK _m,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence of a P _v -factorization of λK _m,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P _4k−1-factorization of λK _m,n is (1) (2k − 1)m ⩽ 2kn, (2) (2k − 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k − 1), (4) λ(4k − 1)mn/[2(2k − 1)(m + n)] is an integer.     

Duality in the cohomology of crystalline local systems

Let k be a perfect field of a positive characteristic p, K-the fraction field of the ring of Witt vectors W(k) Let X be a smooth and proper scheme over W(k). We present a candidate for a cohomology theory with coefficients in crystalline local systems: p -adic étale local systems on X_K characterized by associating to them so called Fontaine-crystals on the crystalline site of the special fiber X _k. We show that this cohomology satysfies a duality theorem.     

On K 1,k -factorization of bipartite multigraphs

A K1,k-factorization of λKm,n is a set of edge-disjoint K1,k-factors of λKm,n, which partition the set of edges of λKm,n. In this paper, it is proved that a sufficient condition for the existence of K1,k-factorization of λKm,n, whenever k is any positive integer, is that （1） m ≤ kn, （2） n ≤ km, （3） km-n = kn-m ≡ 0 （mod （k^2- 1）） and （4） λ（km-n）（kn-m） ≡ 0 （mod k（k- 1）（k^2 - 1）（m ＋ n））.     

On projectively normal embeddings of generalk-gonal curves

Fix integersg, k andt witht>0,k≥3 andtk<g/2−1. LetX be a generalk-gonal curve of genusg andR∈Pic ^k (X) the uniqueg _k ¹ onX. SetL:=K _X⊗(R ^*)^⊗t.L is very ample. Leth _L:X→P(H ⁰(X, L)^*) be the associated embedding. Here we prove thath _L(X) is projectively normal. Ifk≥4 andtk<g/2−2 the curveh _L(X) is scheme-theoretically cut out by quadrics. The author was partially supported by MURST and GNSAGA of CNR (Italy).     

On a spectral sequence for equivariant K-theory

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.