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.     

Moduli stacks and invariants of semistable objects on K3 surfaces

For a K3 surface X and its bounded derived category of coherent sheaves D(X), we have the notion of stability conditions on D(X) in the sense of T. Bridgeland. In this paper, we show that the moduli stack of semistable objects in D(X) with a fixed numerical class and a phase is represented by an Artin stack of finite type over C. Then following D. Joyce's work, we introduce the invariants counting semistable objects in D(X), and show that the invariants are independent of a choice of a stability condition.     

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.     

A generalization of Fulton–MacPherson configuration spaces

We construct a wonderful compactification of the variety parameterizing n distinct labeled points in X away from D, where X is a nonsingular variety and D is a nonsingular proper subvariety. When D is empty, it coincides with the Fulton–MacPherson configuration space.     

Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices

Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C* and D* may overlap. When T: V W is linear and K V and D W are cones, these results will be applied to C = T(K) and D, giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D. The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T(X) = AX + X* A yields new and known results about the existence of block diagonal X's satisfying the Lyapunov condition: T(X) is an interior point of D. For the same V, W and D, T(X) = X – B* XB will be studied for certain cones K of entry-wise nonnegative X's.     

A finiteness result for groups which quasi-act on hyperbolic spaces

Let (X, d) be a Gromov-hyperbolic metric space endowed with a measure having finite entropy H and such that the measure of every ball of radius R > 0 is finite and bounded from below by a positive function of R. In this paper we look at the set Q(X; L, C, D) of the isomorphism classes of torsion-free groups Γ which admit a discrete, D-co-bounded (L, C)-quasi-action on X (D > 0, L ≥ 1, C ≥ 0) and we describe some algebraic conditions which, imposed on the groups Γ, define finite subsets of Q(X; L, C, D), provided C < ε for some ε > 0. As an example, these conditions are satisfied when Γ is assumed to admit a faithful, discrete, m-dimensional representation over some local field (in this case ε = ε(m, H, L)). In particular (set C = 0, L = 1), our results apply when the groups are assumed to act by isometries.     

Modification systems and integration in their Chow groups

We introduce a notion of integration on the category of proper birational maps to a given variety X, with value in an associated Chow group. Applications include new birational invariants; comparison results for Chern classes and numbers of nonsingular birational varieties; ‘stringy’ Chern classes of singular varieties; and a zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern–Schwartz–MacPherson classes of possibly singular varieties, placing them into a context in which a ‘change-of-variable’ formula holds.     

First-order optimality conditions in set-valued optimization

A a set-valued optimization problem min _C F(x), x ∈X ₀, is considered, where X ₀ ⊂ X, X and Y are normed spaces, F: X ₀ ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x ⁰,y ⁰), y ⁰ ∈F(x ⁰), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x ⁰, y ⁰) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.     

On derivations of algebras with involution

Summary Let X be a complex Hilbert space, let L(X) be the algebra of all bounded linear operators on X, and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D: A(X) → L(X) satisfying the relation D(AA*A) = D(A) A*A + AD(A*)A + AA*D(A), for all A ∈ A(X). In this case D is of the form D(A) = AB-BA, for all A∈ A(X) and some B ∈ L(X), which means that D is a derivation. We apply this result to semisimple H*-algebras.     

Mirabolic Langlands duality and the quantum Calogero–Moser system

We give a generic spectral decomposition of the derived category of twisted D\mathcal{D} -modules on a moduli stack of mirabolic vector bundles on a curve X in characteristic p: that is, we construct an equivalence with the derived category of quasicoherent sheaves on a moduli stack of mirabolic local systems on X. This equivalence may be understood as a tamely ramified form of the geometric Langlands equivalence. When X has genus 1, this equivalence generically solves (in the sense of noncommutative geometry) the quantum Calogero–Moser system.     

Polynomials over division rings

LetD be a division ring with a centerC, andD[X ₁, …,X _N] the ring of polynomials inN commutative indeterminates overD. The maximum numberN for which this ring of polynomials is primitive is equal to the maximal transcendence degree overC of the commutative subfields of the matrix ringsM _n(D),n=1, 2, …. The ring of fractions of the Weyl algebras are examples where this numberN is finite. A tool in the proof is a non-commutative version of one of the forms of the “Nullstellensatz”, namely, simpleD[X ₁, …,X _m]-modules are finite-dimensionalD-spaces. This paper was written while the authors were Fellows of the Institute for Advanced Studies, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem, Israel.     

On the meromorphic convexity of normality domains in a Stein manifold

Let X be a Stein manifold. Then we prove that for any family ℱ⊂?(X) the normality domain Dℱ) is a meromorphically ?(X)-convex open set of X. Received: 4 November 1999     

Banach spaces with a weak cotype 2 property

We study the Banach spacesX with the following property: there is a numberδ in ]0,1[ such that for some constantC, any finite dimensional subspaceE ⊂X contains a subspaceF ⊂E with dimF≧δ dimE which isC-isomorphic to a Euclidean space. We show that if this holds for someδ in ]0,1[ then it also holds for allδ in ]0,1[ and we estimate the functionC=C(δ). We show that this property holds iff the “volume ratio” of the finite dimensional subspaces ofX are uniformly bounded. We also show that (althoughX can have this property without being of cotype 2)L ₂(X) possesses this property iffX if of cotype 2. In the last part of the paper, we study theK-convex spaces which have a dual with the above property and we relate it to a certain extension property.     

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.     

Embedding Certain Infinitely Connected Subsets of Bordered Riemann Surfaces Properly into ℂ2

We prove that certain infinitely connected domains D in a bordered Riemann surface, which admits a holomorphic embedding into C ², admit a proper holomorphic embedding into C ². We also prove that certain infinitely connected open subsets D⊂ℂ admit a proper holomorphic embedding into ℂ².      

SO（n）-Invariant Special Lagrangian Submanifolds of C^n＋l with Fixed Loci

Abstract Let SO(n) act in the standard way on ℂⁿ and extend this action in the usual way to ℂⁿ⁺¹ = ℂ ⊕ ℂⁿ. It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂⁿ⁺¹ that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂⁿ⁺¹ in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. * Project supported by Duke University via a research grant, the NSF via DMS-0103884, the Mathematical Sciences Research Institute, and Columbia University. (Dedicated to the memory of Shiing-Shen Chern, whose beautiful works and gentle encouragement have had the most profound influence on my own research)     

L 1 (μ, X) as a constrained subspace of its bidual

In this note we consider the property of being constrained in the bidual, for the space of Bochner integrable functions. For a Banach spaceX having the Radon-Nikodym property and constrained in its bidual and forY ⊂ X, under a natural assumption onY, we show thatL ¹ (μ, X/Y) is constrained in its bidual andL ¹ (μ, Y) is a proximinal subspace ofL ¹(μ, X). As an application of these results, we show that, ifL ¹(μ, X) admits generalized centers for finite sets and ifY ⊂ X is reflexive, thenL ¹ μ, X/Y) also admits generalized centers for finite sets.     

Lower bounds for weak epsilon-nets and stair-convexity

A set N ⊂ ℝ ^d is called a weak ɛ-net (with respect to convex sets) for a finite X ⊂ ℝ ^d if N intersects every convex set C with |X ∩ C| ≥ ɛ|X|. For every fixed d ≥ 2 and every r ≥ 1 we construct sets X ⊂ ℝ ^d for which every weak 1/r -net has at least Ω(r log ^d−1 r) points; this is the first superlinear lower bound for weak ɛ-nets in a fixed dimension.     

Viability for a class of semilinear differential equations of retarded type

Let X be a Banach space, A ： D（A） X → X the generator of a compact C0- semigroup S（t） ： X → X, t ≥ 0, D a locally closed subset in X, and f ： （a, b） × X →X a function of Caratheodory type. The main result of this paper is that a necessary and sufficient condition in order to make D a viable domain of the semilinear differential equation of retarded type u＇（t） = Au（t） ＋ f（t, u（t - q））, t ∈ [to, to ＋ T], with initial condition uto = φ ∈C（[-q, 0]; X）, is the tangency condition lim infh10 h^-1d（S（h）v（O）＋hf（t, v（-q））; D） = 0 for almost every t ∈ （a, b） and every v ∈ C（[-q, 0]; X） with v（0）, v（-q）∈ D.