Topological conformal field theories and Calabi-Yau categories

This is the first of two papers which construct a purely algebraic counterpart to the theory of Gromov-Witten invariants (at all genera). These Gromov-Witten type invariants depend on a Calabi-Yau A_∞ category, which plays the role of the target in ordinary Gromov-Witten theory. When we use an appropriate A_∞ version of the derived category of coherent sheaves on a Calabi-Yau variety, this constructs the B model at all genera. When the Fukaya category of a compact symplectic manifold X is used, it is shown, under certain assumptions, that the usual Gromov-Witten invariants are recovered. The assumptions are that open-closed Gromov-Witten theory can be constructed for X, and that the natural map from the Hochschild homology of the Fukaya category of X to the ordinary homology of X is an isomorphism.     

The derived category of quasi-coherent sheaves and axiomatic stable homotopy

We prove in this paper that for a quasi-compact and semi-separated (nonnecessarily noetherian) scheme X, the derived category of quasi-coherent sheaves over X, D(Aqc(X)), is a stable homotopy category in the sense of Hovey, Palmieri and Strickland, answering a question posed by Strickland. Moreover we show that it is unital and algebraic. We also prove that for a noetherian semi-separated formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies Dqct(X) is a stable homotopy category. It is algebraic but if the formal scheme is not a usual scheme, it is not unital, therefore its abstract nature differs essentially from that of the derived category Dqc(X) (which is equivalent to D(Aqc(X))) in the case of a usual scheme.     

Ample filters of invertible sheaves

Let X be a scheme, proper over a commutative Noetherian ring A. We introduce the concept of an ample filter of invertible sheaves on X and generalize the most important equivalent criteria for ampleness of an invertible sheaf. We also prove the Theorem of the Base for X and generalize Serre's Vanishing Theorem. We then generalize results for twisted homogeneous coordinate rings which were previously known only when X was projective over an algebraically closed field. Specifically, we show that the concepts of left and right σ-ampleness are equivalent and that the associated twisted homogeneous coordinate ring must be Noetherian.     

Perverse coherent sheaves and the geometry of special pieces in the unipotent variety

Let X be a scheme of finite type over a Noetherian base scheme S admitting a dualizing complex, and let U⊂X be an open set whose complement has codimension at least 2. We extend the Deligne-Bezrukavnikov theory of perverse coherent sheaves by showing that a coherent intermediate extension (or intersection cohomology) functor from perverse sheaves on U to perverse sheaves on X may be defined for a much broader class of perversities than has previously been known. We also introduce a derived category version of the coherent intermediate extension functor.Under suitable hypotheses, we introduce a construction (called “S₂-extension”) in terms of perverse coherent sheaves of algebras on X that takes a finite morphism to U and extends it in a canonical way to a finite morphism to X. In particular, this construction gives a canonical “S₂-ification” of appropriate X. The construction also has applications to the “Macaulayfication” problem, and it is particularly well-behaved when X is Gorenstein.Our main goal, however, is to address a conjecture of Lusztig on the geometry of special pieces (certain subvarieties of the unipotent variety of a reductive algebraic group). The conjecture asserts in part that each special piece is the quotient of some variety (previously unknown for the exceptional groups and in positive characteristic) by the action of a certain finite group. We use S₂-extension to give a uniform construction of the desired variety.     

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U _σ = Spec A ^σ . A quasi-coherent sheaf on X gives rise, by taking sections over the U _σ , to a diagram of modules over the coordinate rings A ^σ , indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Σ^op-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U _σ agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.     

On countable bounded tightness for spaces Cp(X)

It is well known that the space C_p([0,1]) has countable tightness but it is not Fréchet-Urysohn. Let X be a Cech-complete topological space. We prove that the space C_p(X) of continuous real-valued functions on X endowed with the pointwise topology is Fréchet-Urysohn if and only if C_p(X) has countable bounded tightness, i.e., for every subset A of C_p(X) and every x in the closure of A in C_p(X) there exists a countable and bounding subset of A whose closure contains x. We study also the problem when the weak topology of a locally convex space has countable bounded tightness. Additional results in this direction are provided.     

Classifying Hilbert bundles

A Hilbert bundle (p, B, X) is a type of fibre space p:B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X. We generalize the classical homotopy classification theory of vector bundles to a “homotopy” classification of certain Hilbert bundles. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. The main result here is that if A is a compact set lying in the “edge” of the metric space X (e.g. if X is a topological manifold and A is a compact subset of the boundary of X), then the problem of classifying (m, n)-bundles over (X, A) reduces to a problem in the classical theory of vector bundles. In particular, we show there is a one-to-one correspondence between the members of the orbit set, [A, G_m($\text{C}$ⁿ)]/[X, U(n)] ¦ A, and the isomorphism classes of (m, n)-bundles over (X, A) which are trivial over X, A.     

The notion of o-tightness and C-embedded subspaces of products

We consider the question: when is a dense subset of a space XC-embedded in X? We introduce the notion of o-tightness and prove that if each finite subproduct of a product X = Π_α?AX_α has a countable o-tightness and Y is a subset of X such that π_B(Y) = Π_α?BX_α for every countable B ? A, then Y is C-embedded in X. This result generalizes some of Noble and Ulmer's results on C-embedding.     

Realizing modules over the homology of a DGA

Let A be a DGA over a field and X a module over H_∗(A). Fix an A_∞-structure on H_∗(A) making it quasi-isomorphic to A. We construct an equivalence of categories between A_n+1-module structures on X and length n Postnikov systems in the derived category of A-modules based on the bar resolution of X. This implies that quasi-isomorphism classes of A_n-structures on X are in bijective correspondence with weak equivalence classes of rigidifications of the first n terms of the bar resolution of X to a complex of A-modules. The above equivalences of categories are compatible for different values of n. This implies that two obstruction theories for realizing X as the homology of an A-module coincide.     

Moduli of sheaves and the Chow group of K3 surfaces

Let X be a projective complex K  3 surface. Beauville and Voisin singled out a 0-cycle _cX$c_X$ on X of degree 1 and Huybrechts proved that the second Chern class of a rigid simple vector-bundle on X   is a multiple of _cX$c_X$ if certain hypotheses hold. We believe that the following generalization of Huybrechts? result holds. Let M be a moduli space of stable pure sheaves on X with fixed cohomological Chern character: the set whose elements are second Chern classes of sheaves parametrized by the closure of M (in the corresponding moduli spaces of semistable sheaves) depends only on the dimension of M. We will prove that the above statement holds under some additional assumptions on the Chern character.     

Lazarsfeld-Mukai reflexive sheaves and their stability

Consider an ample and globally generated line bundle L on a smooth projective variety X of dimension N≥2 over ?. Let D be a smooth divisor in the complete linear system of L. We construct reflexive sheaves on X by an elementary transformation of a trivial bundle on X along certain globally generated torsion-free sheaves on D. The dual reflexive sheaves are called the Lazarsfeld-Mukai reflexive sheaves. We prove the μ_L-(semi)stability of such reflexive sheaves under certain conditions.     

Functional characterizations of p-spaces

We show that a completely regular space Y is a p-space (a ?ech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: A → Y there are a p-embedded subset (resp. a G _δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.     

Duality and normal parts of operator modules

For an operator bimodule X over von Neumann algebras A⊆B(H) and B⊆B(K), the space of all completely bounded A,B-bimodule maps from X into B(K,H), is the bimodule dual of X. Basic duality theory is developed with a particular attention to the Haagerup tensor product over von Neumann algebras. To X a normal operator bimodule X_n is associated so that completely bounded A,B-bimodule maps from X into normal operator bimodules factorize uniquely through X_n. A construction of X_n in terms of biduals of X, A and B is presented. Various operator bimodule structures are considered on a Banach bimodule admitting a normal such structure.     

When is c(x) a Clean Ring?

An element of a ring R is called clean if it is the sum of a unit and an idempotent and a subset A of R is called clean if every element of A is clean. A topological characterization of clean elements of C(X) is given and it is shown that C(X) is clean if and only if X is strongly zero-dimensional, if and only if there exists a clean prime ideal in C(X). We will also characterize topological spaces X for which the ideal C_K(X) is clean. Whenever X is locally compact, it is shown that C_K(X) is clean if and only if X is zero-dimensional.     

Parametrized spaces model locally constant homotopy sheaves

We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that spaces parametrized over X are equivalent to spaces with an action of ΩX. This gives a homotopy-theoretic version of the correspondence between covering spaces and π₁-sets. We then use these two equivalences to study base change functors for parametrized spaces.     

Bases for modules

Let R be a ring and A an R-module. We examine different notions of bases or generating sets for A. Of particular interest is the notion of an irredundant basis for A, that is, a subset X of A that generates A but for which no proper subset of X generates A. We investigate the existence and cardinality of irredundant bases.     

On derived equivalences of categories of sheaves over finite posets

A finite poset X carries a natural structure of a topological space. Fix a field k, and denote by D^b(X) the bounded derived category of sheaves of finite dimensional k-vector spaces over X. Two posets X and Y are said to be derived equivalent if D^b(X) and D^b(Y) are equivalent as triangulated categories.We give explicit combinatorial properties of X which are invariant under derived equivalence; among them are the number of points, the Z-congruency class of the incidence matrix, and the Betti numbers. We also show that taking opposites and products preserves derived equivalence.For any closed subset Y⊆X, we construct a strongly exceptional collection in D^b(X) and use it to show an equivalence D^b(X)?D^b(A) for a finite dimensional algebra A (depending on Y). We give conditions on X and Y under which A becomes an incidence algebra of a poset.We deduce that a lexicographic sum of a collection of posets along a bipartite graph S is derived equivalent to the lexicographic sum of the same collection along the opposite .This construction produces many new derived equivalences of posets and generalizes other well-known ones.As a corollary we show that the derived equivalence class of an ordinal sum of two posets does not depend on the order of summands. We give an example that this is not true for three summands.     

Integral closure of invariant ideals, toroidal resolution, and equivariant vector bundles

We establish a connection between equivariant integrally closed ideal sheaves on a G-fibration Y over a G-spherical variety X with an affine fiber V and equivariant vector bundles on the universal toroidal resolution of X. As an application, we reduce the study of invariant integrally closed ideals of V×X to that of some smaller variety in the case of X=M_n,m. Moreover, we present an affirmative answer to a problem raised by Michel Brion [Comment. Math. Helv. 66 (1991) 237-262] for two special infinite series.