On the moduli stack of commutative, 1-parameter formal groups

We commence a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal groups. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of n-buds, and these latter stacks are algebraic. Our main results pertain to various aspects of the height stratification relative to fixed prime p on the stacks of buds and formal groups.     

Higher Intersection Theory on Algebraic Stacks: II

This is the second part of our work on the intersection theory of algebraic stacks. The main results here are the following. We provide an intersection pairing for all smooth Artin stacks (locally of finite type over a field) which we show reduces to the known intersection pairing on the Chow groups of smooth Deligne–Mumford stacks of finite type over a field as well as on the Chow groups of quotient stacks associated to actions of linear algebraic groups on smooth quasi-projective schemes modulo torsion. The former involves also showing the existence of Adams operations on the rational étale K-theory of all smooth Deligne–Mumford stacks of finite type over a field. In addition, we show that our definition of the higher Chow groups is intrinsic to the stack for all smooth stacks and also stacks of finite type over the given field. Next we establish the existence of Chern classes and Chern character for Artin stacks with values in our Chow groups and extend these to higher Chern classes and a higher Chern character for perfect complexes on an algebraic stack, taking values in cohomology theories of algebraic stacks that are defined with respect to complexes of sheaves on a big smooth site. As a by-product of our techniques we also provide an extension of higher intersection theory to all schemes locally of finite type over a field. As the higher cycle complex, by itself, is a bit difficult to handle, the stronger results like contravariance for arbitrary maps between smooth stacks and the intersection pairing for smooth stacks are established by comparison with motivic cohomology.     

Compactified Picard stacks over the moduli stack of stable curves with marked points

In this paper we give a construction of algebraic (Artin) stacks endowed with a modular map onto the moduli stack of stable curves of genus g with n marked points. The stacks we construct are smooth, irreducible and have dimension 4g−3+n, yielding a geometrically meaningful compactification of the universal Picard stack parametrizing n-pointed smooth curves together with a line bundle.     

Twisted Bundles and Admissible Covers

Abstract

We study the structure of the stacks of twisted stable maps to the classifying stack of a finite group G—which we call the stack of twisted G-covers, or twisted G-bundles. For a suitable group Gwe show that the substack corresponding to admissible G-covers is a smooth projective fine moduli space.     

Upper bounds for the essential dimension of the moduli stack of SL n -bundles over a curve

We find upper bounds for the essential dimension of various moduli stacks of SL _n -bundles over a curve. When n is a prime power, our calculation computes the essential dimension of the stack of stable bundles exactly and the essential dimension is not equal to the dimension in this case.     

On the de Rham cohomology of differential and algebraic stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids.Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to étale equivalence.We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E₁-term consists entirely of vector bundle valued cohomology groups.Our theory works for differentiable, holomorphic and algebraic stacks.     

Riemann–Roch for Algebraic Stacks: I

In this paper we establish Riemann–Roch and Lefschtez–Riemann–Roch theorems for arbitrary proper maps of finite cohomological dimension between algebraic stacks in the sense of Artin. The Riemann–Roch theorem is established as a natural transformation between the G-theory of algebraic stacks and topological G-theory for stacks: we define the latter as the localization of G-theory by topological K-homology. The Lefschtez–Riemann–Roch is an extension of this including the action of a torus for Deligne–Mumford stacks. This generalizes the corresponding Riemann–Roch theorem (Lefschetz–Riemann–Roch theorem) for proper maps between schemes (that are also equivariant for the action of a torus, respectively) making use of some fundamental results due to Vistoli and Toen. A key result established here is that topological G-theory (as well as rational G-theory) has cohomological descent on the isovariant étale site of an algebraic stack. This extends cohomological descent for topological G-theory on schemes as proved by Thomason.     

Moduli Stacks of Permutation Classes of Pointed Stable Curves

The notion of m/Γ-pointed stable curves is introduced. It should be viewed as a generalization of the notion of m-pointed stable curves of a given genus, where the labels of the marked points are only determined up to the action of a group of permutations Γ. The classical moduli spaces and moduli stacks are generalized to this wider setting. Finally, an explicit construction of the new moduli stack of m/Γ-pointed stable curves as a quotient stack is given. Received: February 2008     

Exact Asymptotics of the Density of the Transition Probability for Discontinuous Markov Processes

In this paper we consider some Kolmogorov–Feller equations with a small parameter h. We present a method for constructing the exact (exponential) asymptotics of the fundamental solution of these equations for finite time intervals uniformly with respect to h. This means that we construct an asymptotics of the density of the transition probability for discontinuous Markov processes. We justify the asymptotic solutions constructed. We also present an algorithm for constructing all terms of the asymptotics of the logarithmic limit (logarithmic asymptotics) of the fundamental solution as t → +0 uniformly with respect to h. We write formulas of the asymptotics of the logarithmic limit for some special cases as t → +0. The method presented in this paper also allows us to construct exact asymptotics of solutions of initial–boundary value problems that are of probability meaning.     

A homotopy theory for stacks

We give a homotopy theoretic characterization of stacks on a site C as the homotopy sheaves of groupoids on C. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, we show that these model structures are Quillen equivalent to the S ²-nullification of Jardine’s model structure on sheaves of simplicial sets on C.     

Uniformization of {\mathcal {G}}-bundles

We show some of the conjectures of Pappas and Rapoport concerning the moduli stack Bun_G{{\rm Bun}_\mathcal {G}} of G{\mathcal {G}}-torsors on a curve C, where G{\mathcal {G}} is a semisimple Bruhat-Tits group scheme on C. In particular we prove the analog of the uniformization theorem of Drinfeld-Simpson in this setting. Furthermore we apply this to compute the connected components of these moduli stacks and to calculate the Picard group of Bun_G{{\rm Bun}_\mathcal {G}} in case G{\mathcal {G}} is simply connected.     

Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne–Mumford as well as Artin. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne–Mumford stacks. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. As an application of our theory we compute the higher Chow groups of Deligne–Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.     

Exact <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">2</Emphasis></Subscript>-small ball behavior of integrated Gaussian processes and spectral asymptotics of boundary value problems

We find the exact small deviation asymptotics for the L₂-norm of various m-times integrated Gaussian processes closely connected with the Wiener process and the Ornstein – Uhlenbeck process. Using a general approach from the spectral theory of linear differential operators we obtain the two-term spectral asymptotics of eigenvalues in corresponding boundary value problems. This enables us to improve the recent results from [15] on the small ball asymptotics for a class of m-times integrated Wiener processes. Moreover, the exact small ball asymptotics for the m-times integrated Brownian bridge, the m-times integrated Ornstein – Uhlenbeck process and similar processes appear as relatively simple examples illustrating the developed general theory.Partially supported by grants of RFBR 01-01-00245 and 02-01-01099.     

The focusing problem for the Eikonal equation

We study the focusing problem for the eikonal equation¶¶ ?_tu=| ?u| ², \partial _{t}u=\left| \nabla u\right| ^{2}, ¶¶i.e., the initial value problem in which the support of the initial datum is outside some compact set in R^d \mathbf{R}^{d} . The hole in the support will be filled in finite time and we are interested in the asymptotics of the hole as it closes. We show that in the radially symmetric case there are self-similar asymptotics, while in the absence of radial symmetry essentially any convex final shape is possible. However in R² \mathbf{R}^2 , for generic initial data the asymptotic shape will be either a vanishing triangle or the region between two parabolas moving in opposite directions (a closing eye). We compare these results with the known results for the porous medium pressure equation which approaches the eikonal equation in the limit as m? 1 m\rightarrow 1 .     

Heavy Tails in Multi-Server Queue

In this paper, the asymptotic behaviour of the distribution tail of the stationary waiting time W in the GI/GI/2 FCFS queue is studied. Under subexponential-type assumptions on the service time distribution, bounds and sharp asymptotics are given for the probability P{W > x}. We also get asymptotics for the distribution tail of a stationary two-dimensional workload vector and of a stationary queue length. These asymptotics depend heavily on the traffic load. AMS subject classification: 60K25     

A tree search procedure for the container pre-marshalling problem

In the container pre-marshalling problem (CPMP) n items are given that belong to G different item groups (g = 1, … , G) and that are piled up in up to S stacks with a maximum stack height H. A move can shift one item from one stack to another one. A sequence of moves of minimum length has to be determined that transforms the initial item distribution so that in each of the stacks the items are sorted by their group index g in descending order. The CPMP occurs frequently in container terminals of seaports. It has to be solved when export containers, piled up in stacks, are sorted in a pre-marshalling process so that they can be loaded afterwards onto a ship faster and more efficiently. This article presents a heuristic tree search procedure for the CPMP. The procedure is compared to solution approaches for the CPMP that were published so far and turns out to be very competitive. Moreover, computational results for new and difficult CPMP instances are presented.     

Jost functions and Jost solutions for Jacobi matrices, I. A necessary and sufficient condition for Szegő asymptotics

We provide necessary and sufficient conditions for a Jacobi matrix to produce orthogonal polynomials with Szegő asymptotics off the real axis. A key idea is to prove the equivalence of Szegő asymptotics and of Jost asymptotics for the Weyl solution. We also prove L² convergence of Szegő asymptotics on the spectrum.     

Canonical Artin stacks over log smooth schemes

We develop a theory of toric Artin stacks extending the theories of toric Deligne–Mumford stacks developed by Borisov–Chen–Smith, Fantechi–Mann–Nironi, and Iwanari. We also generalize the Chevalley–Shephard–Todd theorem to the case of diagonalizable group schemes. These are both applications of our main theorem which shows that a toroidal embedding $X$ is canonically the good moduli space (in the sense of Alper) of a smooth log smooth Artin stack whose stacky structure is supported on the singular locus of $X$ .     

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

We consider asymptotics for orthogonal polynomials with respect to varying exponential weights w_n(x)dx = e^−nV(x) dx on the line as n → ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel‐Rotach‐type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [19, 20]. The Riemann‐Hilbert problem is analyzed in turn using the steepest‐descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμ_V for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.