Cohomology of certain moduli spaces of vector bundles

LetX be a smooth irreducible projective curve of genusg over the field of complex numbers. LetM ₀ be the moduli space of semi-stable vector bundles onX of rank two and trivial determinant. A canonical desingularizationN _o ofM _o has been constructed by Seshadri [17]. In this paper we compute the third and fourth cohomology groups ofN _o. In particular we give a different proof of the theorem due to Nitsure [12], that the third cohomology group ofN _o is torsion-free.     

Irreducible buchsbaum curves

For every biliaison class C _M of Buchsbaum curves of π ³ we prove that the leftmost shift in which there are smooth and connected curves is the same as for irreducible curves. As a consequence, every irreducible Buchsbaum curve has a flat deformation with cohomology and Hartshorne-Rao module constant which is smooth and connected.     

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 a weak sum theorem in dimension theory

A metric spaceX = U _i-0 ^x X is constructed such thatX _o={x _o} consists of a single pointx _o , X _i , i=0, 1, 2, … are disjoint and closed,X _i , i=1, 2, … are open, indX _i =0 fori=0, 1, … and indX=1. The above space (proved to be, in some sense, most simple) shows also that the dimension ind of a metric space can be raised by adjoining of a single point, a fact proved recently by E.K. Van Douwen and by T. Przymusiński. Some maximality property of the family {X; IndX=0} is proved and conditions implyingP-ind=P-Ind are given. This is part of a research thesis at the Technion, Israel Institute of Technology, towards an M.Sc. degree, directed by Professor M. Reichaw.     

Properties of fourth-order strong mixing rates and its application to random set theory

We shall define the concept of fourth-order strong mixing rates and study their properties. Results are useful for establishing a condition of the form (*) Σ_a,b,c |cum(X_o, X_a, X_b, X_c)| < ∞ or ∫|cum(X_o, X_a, X_b, X_c)| da db dc < ∞ for dependent random variables {X_a}. As an application we shall consider an evaluation of a fourth-order strong mixing rate for a random closed set Z (in the sense of Matheron) and derive the condition (*) for {X_a}, X_a being an outcome of a local measurement upon Z. The result is also applicable to point processes which admit clustering representations.     

Cohomological approach to asymptotic dimension

We introduce the notion of asymptotic cohomology based on the bounded cohomology and define cohomological asymptotic dimension asdim _Z X of metric spaces. We show that it agrees with the asymptotic dimension asdim X when the later is finite. Then we use this fact to construct an example of a metric space X of bounded geometry with finite asymptotic dimension for which asdim(X × R) = asdim X. In particular, it follows for this example that the coarse asymptotic dimension defined by means of Roe’s coarse cohomology is strictly less than its asymptotic dimension.      

Torsors under finite and flat group schemes of rank <Emphasis Type="Italic">p</Emphasis> with Galois action

In this note we study the geometry of torsors under flat and finite commutative group schemes of rank p above curves in characteristic p, and above relative curves over a complete discrete valuation ring of inequal characteristic. In both cases we study the Galois action of the Galois group of the base field on these torsors. We also study the degeneration of _p -torsors, from characteritic 0 to characteristic p, and show that this degeneration is compatible with the Galois action. We then discuss the lifting of torsors under flat and commutative group schemes of rank p from positive to zero characteristics. Finally, for a proper and smooth curve X over a complete discrete valuation field, of inequal characteristic, which has good reduction, we show the existence of a canonical Galois equivariant filtration, on the first étale cohomology group of the geometric fibre of X, with values in _p .     

Weights of the l-adic cohomology of a proper toric variety

We describe the weights of the l-adic cohomology groups of a toric variety over a finite field k in terms of the Ishida complexes of Z-modules. As a consequence, we conclude that, for an r-dimensional proper toric variety X, the m-th cohomology group H^m (X?_k[kbar],Q_l) is of pure weight if m = 0,1,2,3,2r - 3,2r - 2,2r - 1,2r.Furthermore, we show that, for any m such that 3 < m < 2r - 3,there exists an r-dimensional proper toric variety whose m-th cohomology group H^m (X?_k[kbar],Q _l ) is not pure.     

Stable reduction of curves and tame ramification

We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain a new proof of Saito’s criterion, avoiding the use of ℓ-adic cohomology and vanishing cycles.     

Tikhonov regularization of nonlinear III-posed problems in hilbert scales

In this paper we consider nonlinear ill-posed problems F(x) = y ₀, where x and y ₀ are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...     

The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X,we use ↓USC F(X) and ↓C F(X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0,1] with the subspace topologies of the hyperspace Cld F(X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology.In this paper,we shall show that there exists a homeomorphism h:↓USC F(X) → Q = [1,1] ω such that h(↓CF(X))=c0 = {(xn)∈Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.     

Function Spaces and Continuous Algebraic Pairings for Varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural structure of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying these function complexes and for forming continuous pairings of such. Building on this technique, we establish several results, including (1) the existence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle theory; (3) the agreement of the motivic cohomology cup product with morphic cohomology cup product; and (4) the Whitney sum formula for the Chern classes in morphic cohomology of vector bundles.     

The normality of closures of orbits in a Lie algebra

LexX be the closure of aG-orbit in the Lie algebra of a connected reductive groupG. It seems that the varietyX is always normal. After a reduction to nilpotent orbits, this is proved for some special cases. Results on determinantal schemes are used forGl _n . IfX is small enough we use a resolution and Bott's theorem on the cohomology of homogeneous vector bundles. Our results are conclusive for groups of typeA ₁,A ₂,A ₃ andB ₂.     

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.     

Samuel multiplicity and Fredholm theory

In this note we combine methods from commutative algebra and complex analytic geometry to calculate the generic values of the cohomology dimensions of a commuting multioperator on its Fredholm domain. More precisely, we prove that, for a given Fredholm tuple T = (T ₁, ..., T _n ) of commuting bounded operators on a complex Banach space X, the limits exist and calculate the generic dimension of the cohomology groups H ^p (z − T, X) of the Koszul complex of T near z = 0. To deduce this result we show that the above limits coincide with the Samuel multiplicities of the stalks of the cohomology sheaves of the associated complex of analytic sheaves at z = 0.     

Complex cycles on algebraic models of smooth manifolds

Every compact smooth manifold M is diffeomorphic to the set X(\mathbbR){X(\mathbb{R})} of real points of a nonsingular projective real algebraic variety X, which is called an algebraic model of M. Each algebraic cycle of codimension k on the complex variety X_\mathbbC=X×_\mathbbR\mathbbC{X_{\mathbb{C}}=X\times_{\mathbb{R}}\mathbb{C}} determines a cohomology class in H^2k(X(\mathbbR);\mathbbD){H^{2k}(X(\mathbb{R});\mathbb{D})} , where \mathbbD{\mathbb{D}} denotes \mathbbZ{\mathbb{Z}} or \mathbbQ{\mathbb{Q}} . We investigate the behavior of such cohomology classes as X runs through the class of algebraic models of M.     

Admissibility of the natural estimator of the mean of a Gaussian process

Let {X(t): t [a, b]} be a Gaussian process with mean μ L₂[a, b] and continuous covariance K(s, t). When estimating μ under the loss ∫_a^b ( (t)−μ(t))² dt the natural estimator X is admissible if K is unknown. If K is known, X is minimax with risk ∫_a^b K(t, t) dt and admissible if and only if the three by three matrix whose entries are K(t_i, t_j) has a determinant which vanishes identically in t_i [a, b], i = 1, 2, 3.     

On the cohomology of orbit space of free ℤ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-actions on lens spaces

Let G = ℤ _p , p an odd prime, act freely on a finite-dimensional CW-complex X with mod p cohomology isomorphic to that of a lens space L ^2m−1(p; q ₁, …, q _m ). In this paper, we determine the mod p cohomology ring of the orbit space X/G, when p ² ∤ m.