On equivalences of derived and singular categories

Let X and Y be two smooth Deligne-Mumford stacks and consider a pair of functions f: X → $ \mathbb{A}^1 $ \mathbb{A}^1 , g:Y → $ \mathbb{A}^1 $ \mathbb{A}^1 . Assuming that there exists a complex of sheaves on X × $ \mathbb{A}^1 $ \mathbb{A}^1 Y which induces an equivalence of D ^b (X) and D ^b (Y), we show that there is also an equivalence of the singular derived categories of the fibers f ⁻¹(0) and g ⁻¹(0). We apply this statement in the setting of McKay correspondence, and generalize a theorem of Orlov on the derived category of a Calabi-Yau hypersurface in a weighted projective space, to products of Calabi-Yau hypersurfaces in simplicial toric varieties with nef anticanonical class.     

Projective contact manifolds

The present work is concerned with the study of complex projective manifolds X which carry a complex contact structure. In the first part of the paper we show that if K _X is not nef, then either X is Fano and b ₂(X)=1, or X is of the form ℙ(T _Y ), where Y is a projective manifold. In the second part of the paper we consider contact manifolds where K _X is nef. Oblatum 15-X-1999 & 3-II-2000?Published online: 8 May 2000     

Manifolds polarized by vector bundles

LetX be a complex projective manifold of dimension n and let ε be an ample vector bundle of rank r. Let also τ = τ (X,ε) = min {t ∈ ℝ : K_X + t det ε is nef} be the nef value of the pair (X, ε). In this paper we classify the pairs (X, ε) such that{ Mathematics Subject Classification (2000)14J60; 14J40; 14E30     

A criterion of openness of a family of nef line bundles

Letf:X→S be a projective morphism of noetherian schemes andL a line bundle onX. In this note we will give a condition for openness of the setS _L ^nef ={∈S‖L _s is nef onX _s }. This article was processed by the author using the Springer-Verlag TEX mamath macro package 1990     

Rational points of bounded height on Del Pezzo surfaces of degree six

LetK be a number field. Denote byV ₃ a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW ₃ the open complement of the exceptional lines inV ₃. LetN _{W s}(−ω, X) be the number ofK-rational points onW ₃ whose anticanonical heightH _−ω is bounded byX. Manin has conjectured that asymptoticallyN _{W 3}(−ω, X) tends tocX(logX)³, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc _K such thatN _{W 3}(−ω, X)≤c_KX(logX)^3+2r , wherer is the rank of the group of units ofO _K. The constantc _K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN _{W 3}(−ω,X)≤cX^l+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.     

The speed of convergence of a martingale

LetX _n, n≧0, be a martingale with respect to the σ-fieldsF _n and letB _n ² =Σ_1≧n E{(X ₁−X ₁₋₁)²|F ₁₋₁} It is known that ifB ₁ ² <∞ on some set Ω₀ thenX _∞=limX _n exists and is finite a.e. on Ω₀ We show that under suitable conditions there exists a constant ν<∞ for which lim supB _n ⁻¹ {log logB _n ² }^−1/2|X _∞−X _n−1 | ≦ √2(η+1). If “the fluctuations ofB _n are small” (in the sense of the Corollary) then ν=0 and the usual upper bound of a law of the iterated logrithm results. This upper bound is not necessarily achieved, though. Research supported in part by the NSF under Grant No. MCS 72-04534A04.     

Hyperspaces of Banach Spaces with the Attouch—Wets Topology

Let X be an infinite-dimensional Banach space with weight τ. By Cld _AW (X), we denote the hyperspace of nonempty closed sets in X with the Attouch—Wets topology. By Fin _AW (X), Comp _AW (X) and Bdd _AW (X), we denote the subspaces of Cld _AW (X) consisting of finite sets, compact sets and bounded closed sets, respectively. In this paper, it is proved that Fin _AW (X)≈Comp _AW (X)≈ℓ₂(τ)×ℓ₂ ^f ℓandℓBdd _AW (X)≈ℓ₂(2^τ)×ℓ₂ ^f where ≈ means ‘is homeomorphic to’, ℓ₂(τ) is the Hilbert space with weight τ (ℓ₂(ℵ₀)=ℓ₂ the separable Hilbert space) and ℓ₂ ^f ={(x _i ) _iεN εℓ₂∣x _i =0 except for finitely many iεN}.     

Onp-absolutely summing constants of banach spaces

Given 1≦p<∞ and a real Banach spaceX, we define thep-absolutely summing constantμ _p(X) as inf{Σ _i ^=1/m |x*(x _i)|^p p Σ _i ^=1/m ‖x _i‖^p p]¹ p}, where the supremum ranges over {x*∈X*; ‖x*‖≤1} and the infimum is taken over all sets {x ₁,x ₂, …,x _m} ⊂X such that Σ _i ^=1/m ‖x _i‖>0. It follows immediately from [2] thatμ _p(X)>0 if and only ifX is finite dimensional. In this paper we find the exact values ofμ _p(X) for various spaces, and obtain some asymptotic estimates ofμ _p(X) for general finite dimensional Banach spaces. This is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. A. Dvoretzky and Prof. J. Lindenstrauss.     

Uniform rates of convergence in the CLT for quadratic forms in multidimensional spaces

Summary.   Let X,X ₁,X ₂,… be a sequence of i.i.d. random vectors taking values in a d-dimensional real linear space ℝ ^d . Assume that E X=0 and that X is not concentrated in a proper subspace of ℝ ^d . Let G denote a mean zero Gaussian random vector with the same covariance operator as that of X. We investigate the distributions of non-degenerate quadratic forms ℚ[S _N ] of the normalized sums S _N =N ^−1/2(X ₁+⋯+X _N ) and show that

On the Cantor-Bendixson derivative, resolvable ranks, and perfect set theorems of A. H. Stone

A Borel derivative on the hyperspace 2 ^X of a compactumX is a Borel monotone mapD: 2 ^X →2 ^X . The derivative determines a Cantor-Bendixson type rank δ:2^X → ω₁ ∪ {∞} . We show that ifA⊂2 ^X is analytic andZ⊂A intersects stationary many layers δ₋₁({ξ}), then for almost all σ,F∩δ₋₁({ξ}) cannot be separated fromZ ∩∪ _a<ξ ^δ−1({a}) (and also fromZ ∩∪ _a>ξ ^δ−1({a}) by anyF _σ-set. Another main result involves a natural partial order on 2 ^X related to the derivative. The results are obtained in a general framework of “resolvable ranks” introduced in the paper. During our work on this paper the second author was a Visiting Professor at the Miami University, Ohio. This author would like to express his gratitude to the Department of Mathematics and Statistics for the hospitality.     

On Banach spaces whose dual balls are not weak∗ sequentially compact

Theorem 1. LetX be a Banach space. (a) IfX ^∗ has a closed subspace in which no normalized sequence converges weak^∗ to zero, thenl ₁ is isomorphic to a subspace ofX. (b) IfX ^∗ contains a bounded sequence which has no weak^∗ convergent subsequence, thenX contains a separable subspace whose dual is not separable. The second-named author was supported in part by NSF-MPS 72-04634-A03.     

Some ℵ0-bounded subsets of stone-čech compactifications

We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2^ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 2^2ℵ0 pairwise disjoint, dense ℵ₀-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ₀-bounded.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

The plus construction, Postnikov towers and universal central module extensions

Given a connected spaceX, we consider the effect of Quillen’s plus construction on the homotopy groups ofX in terms of its Postnikov decomposition. Specifically, using universal properties of the fibration sequenceAX→X→X ⁺, we explain the contribution of π _n X to π _n X ⁺, π _n+1 X ⁺ and π _n AX, π _n+1 AX explicitly in terms of the low dimensional homology of π _n X regarded as a module over π₁ X. Key ingredients developed here for this purpose are universal II-central fibrations and a theory of universal central extensions of modules, analogous to universal central extensions of perfect groups. Research partially supported by NSERC of Canada.     

On Two Questions About Topological (and Nonstandard) Extensions

We construct, under MA, a non-Hausdorff (T₁-)topological extension ^*ω of ω, such that every function from ω to ω extends uniquely to a continuous function from ^*ω to ^*ω. We also show (in ZFC) that for every nontrivial topological extension ^*X of a countable set X there exists a topology τ_f on ^*X, strictly finer than the Star topology, and such that (^*X, τ_f) is still a topological extension of X with the same function extensions ^*f. This solves two questions raised by M. Di Nasso and M. Forti.     

Many partition relations below density

We force 2 ^λ to be large, and for many pairs in the interval (λ, 2 ^λ ) a strong version of the polarized partition relations holds. We apply this to problems in general topology. For example, consistently, every 2 ^λ is the successor of a singular and for every Hausdorff regular space X, hd(X) ≤ s(X)⁺³, hL(X) ≤ s(X)⁺³ and better when s(X) is regular, via a halfgraph partition relations. For the case s(X) = ℵ ₀ we get hd(X), hL(X) ≤ N ₂.     

Complemented copies of ℓ<Subscript>p</Subscript> spaces in tensor products

We give sufficient conditions on Banach spaces X and Y so that their projective tensor product X ⊗_π Y, their injective tensor product X ⊗_ɛ Y, or the dual (X ⊗_π Y)^* contain complemented copies of ℓ_p.     

Hyperspaces of nowhere topologically complete spaces

It is proved that ifX is a connected locally continuumwise connected coanalytic nowhere topologically complete space, then the hyperspace 2 ^X of all nonempty compact subsets ofX is strongly universal in the class of all coanalytic spaces. Moreover, 2 ^X is homeomorphic to Π₂ ifX is a Baire space, and toQ∖Π₁ ifX contains a dense absoluteG _δ-setG ⊂X such that the intersectionG ∩U is connected for any open connectedU ⊂X. (Here Π₁, Π₁⊂X are the standard subsets of the Hilbert cubeQ absorbing for the classes of analytic and coanalytic spaces, respectively.) Similar results are obtained for higher projective classes. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 35–51, July, 1997. Translated by O. V. Sipacheva