We give an alternative proof of Kovacs's vanishing theorem. Our proof is based on the standard arguments of the minimal model theory. We do not need the notion of Du Bois pairs. We reduce Kovacs's vanishing theorem to the well-known relative Kawamata-Viehweg-Nadel vanishing theorem.
In this, the third paper of the series, we construct a large family of representations of the quantum toroidal gl(1)-algebra whose bases are parameterized by plane partitions with various boundary conditions and restrictions. We study the corresponding formal characters. As an application We obtain a Gelfand-Zetlin-type basis for a class of irreducible lowest weight gl(infinity)-modules.
Except for a few corrections added in 2010, the article reproduces notes, sent to Nagata in the 1970s, which were a translation in the language of schemes of his articles [2], [3]. The proofs are a “constructive” version of his proofs. Making the proofs constructives allowed us to proceed without using valuations. A more detailed exposition and additional results are in Conrad [1]. Unless otherwise mentioned, all the schemes considered are supposed to be Noetherian. 0. Rappels sur les éclatements 0.1. Si a est un idéal de l’anneau A, le schéma déduit de X = Spec(A) en éclatant a est réunion d’ouverts D(x) pour x parcourant un système de générateurs de a, et D(x) est le spectre du sous-anneau A[ax−1] de A[x−1]. De plus, D(x) ∩ X − V (a) = Spec(A[x−1]). Si c est un second idéal de A, définissant un sous-schéma V (c) de X , la trace sur D(x) du transformé pur de V (c) est définie par l’idéal de A[ax−1] trace de l’idéal de A[x−1] engendré par c. LEMME 0.2 Soient X un schéma, a et b deux faisceaux d’idéaux sur X et X̃ le schéma éclaté de X le long de a+ b. Alors, l’ouvert de X̃ réunion des D(x) pour x ∈ b contient le transformé pur de V (a). En effet, X̃ est réunion des D(x) pour x ∈ a ou x ∈ b, et pour x ∈ a, D(x) est disjoint du transformé pur de V (a). Kyoto Journal of Mathematics, Vol. 50, No. 4 (2010), 661–670 DOI 10.1215/0023608X-2010-009, © 2010 by Kyoto University Received January 27, 2010. Revised June 10, 2010. Accepted June 14, 2010. 2010 Mathematics Subject Classification: 14E25.
Suppose M is a complete n-dimensional manifold, n >= 2, with a metric (g) over bar (ij)(x, t) that evolves by the Ricci flow partial derivative(t)(g) over bar (i) (j)= -2 (2) over bar ij in M x (0, T). For any 0 < p < 1 (P(0), t(0)) is an element of M x (0, T) q is an element of M, we define the L(p)-length between p(0) and q, L(p)-geodesic, the generalized reduced distance l(p) and the generalized reduced volume (N) over tilde (p)(tau), tau = t(0) - t, corresponding to the L(p)-geodesic at the point p(0) at time t(0). Under the condition (R) over bar (ij) >= -c(1)(g) over bar (ij) on M x (0, t(0)) for some constant c(1) > 0, we will prove the existence of a L(p)-geodesic which minimize the L(p) (q, (tau) over bar)-length between p(0) and q for any (tau) over bar > 0. This result for the case p = 1/2 was mentioned and used many times by G. Perelman but no proof of it was given in Perelman's papers on Ricci flow. Let g(tau) = (g) over bar (t(0) - tau) and let (V) over tilde ((tau) over bar)(p) be the rescaled generalized reduced volume. Suppose M also has nonnegative curvature operator with respect to the metric (t) for any t is an element of (0, T) and when 1/2 < p < 1, M has uniformly bounded scalar curvature on (0, T). Let 0 < c < 1 and let tau(0) = min((2(1 - p))(-1/(2p-1)), t(0)). For any 1/2 <= p < 1 we prove that there exists a constant A(0) >= 0 with A(0) = 0 for p = 1/2 such that e-(A0 tau)<(V)over tilde>(p)(tau) is a nionotone decreasing function in (0, (tau) over bar)(1) where (tau) over bar (1) = (1 - c)tau(0) if 1/2 < p < 1 and (tau) over bar (1) = to if p = 1/2. When (M, (g) over bar) is an ancient kappa-solution of the Ricci flow, we will prove a monotonicity property of the rescaled generalized volume with respect to T for any 1/2 <= p < 1. When p = 1/2, the L(p)-length, L(p)-geodesic, the l(p) function and <(V)over tilde>(p)(tau) are equal to the L-length, L-geodesic, the reduced distance l and the reduced volume (V) over tilde(tau) introduced by Perelman in his papers on Ricci flow. We will also prove a result on the reduced distance 1 and the reduced volume (V) over tilde which was used by Perelman without proof in [18].
Let $H$ be an ample line bundle on a non-singular projective surface $X$, and $M(H)$ the coarse moduli scheme of rank-two $H$-semistable sheaves with fixed Chern classes on $X$. We show that if $H$ changes and passes through walls to get closer to $K_X$, then $M(H)$ undergoes natural flips with respect to canonical divisors. When $X$ is minimal and its Kodaira dimension is positive, this sequence of flips terminates in $M(H_X)$; $H_X$ is an ample line bundle lying so closely to $K_X$ that the canonical divisor of $M(H_X)$ is nef. Remark that so-called Thaddeus-type flips somewhat differ from flips with respect to canonical divisors.
We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.
IntroductionIn [28], Kitchloo constructed a map f : BX → BK ∧ p where K is a certain Kac-Moody group of rank two, X is a rank two mod p finite loop space and f is such that it induces an isomorphism between even dimensional mod p cohomology groups.Here B denotes the classifying space functor and (-) ∧ p denotes the Bousfield-Kan F p -completion functor ([8]).This space X -or rather the triple (X ∧ p , BX ∧ p , e) where e : X ΩBXis a particular example of what is known as a p-compact group.These objects were introduced by Dwyer and Wilkerson in [15] as the homotopy theoretical framework to study finite loop spaces and compact Lie groups from a homotopy point of view.The foundational paper [15] together with its many sequels by Dwyer-Wilkerson and other authors represent now an active, well established research area which contains some of the most important recent advances in homotopy theory.While p-compact groups are nowadays reasonably well understood objects, our understanding of Kac-Moody groups and their classifying spaces from a homotopy point of view is far from satisfactory.The work of Kitchloo in [28] started a project which has also involved Broto, Saumell, Ruiz and the present author and has produced a series of results ([2], [3], [10]) which show interesting similarities between this theory and the theory of p-compact groups, as well as non trivial challenging differences.The goal of this paper is to extend the construction of Kitchloo that we have recalled above to produce rank-preserving maps BX → BK ∧ p for a wide family of p-compact groups X.These maps can be understood as the homotopy analogues to monomorphisms, in a sense that will be made precise in Section 13.We prove: Theorem 1.1.Let p be a prime and let X be a simply connected pcompact group with Weyl group W X .Assume that the order of W X is prime to 2000
We obtain the tail estimation of the quadratic variation of a local martingale with no assumption with respect to positive jumps. Moreover, applying it, we also discuss a tail property of the first-passage times of stochastic integrals.
We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index 2 sub-group of the N\'eron-Severi group and we obtain a set of generators of this group. The N\'eron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.
Let R be a commutative Noetherian ring, and let I and J be two ideals of R. Assume that R is local with the maximal ideal m. we mainly prove that (i) there exists an equalityinf{i vertical bar H-I,J(i) (M) is not Artinian} = inf{depth M-p vertical bar p is an element of W (I, J)\{m}}for any finitely generated R-module M, where W(I, J) = {p is an element of Spec (R) vertical bar I-n subset of p + J for some positive integer n}; (ii) for any finitely generated R-module M with dim M = d, H-I,J(d) (M) is Artinian. Also, we give a characterization to the supremum of all integers r for which H-I,J(r) (M) not equal 0.
This paper is devoted to study of sufficient conditions under which a transcendental meromorphic function has no unbounded Fatou components and to extension of some results for entire functions to meromorphic functions. Actually, we shall mainly discuss non-existence of unbounded wandering domains of a meromorphic function. The case for a composition of finitely many meromorphic functions with at least one of them being transcendental can be also investigated in terms of the argument of this paper.
Let G be a connected and simply connected, nilpotent Lie group. In this paper, we show that the cortex of G is a semi-algebraic set by means of a geometric characterization. It is also shown that the cortex is the image under a linear projection of a countable union of a semi-algebraic sets lying in the tensor product T(g) circle times g*.
Using a variation from the construction of the Ornstein-Uhlenbeck process on canonical path-space C ([0 , 1]; R ) in terms of the Brownian sheet, we obtain a large class of processes, adapted to the Brownian filtration, which admit the one dimensional marginals of a martingale.
We consider the Euler type integral associated to the configuration space of points on an elliptic curve, which is an analogue of the hypergeometric function associated to the configuration space of points on a projective line. We calculate the {\it twisted homology group}, with coefficients in the local system associated to a power function $g^{\alpha}$ of an elliptic function $g$, and the intersection form. Applying these calculations, we describe the {\it connection matrices} representing the linear isomorphisms induced from analytic continuations of the functions defined by the integrations of $g^{\alpha}$ over twisted cycles.
It is classical to approximate the distribution of fractional Brownian motion by a renormalized sum $ S_n $ of dependent Gaussian random variables. In this paper we consider such a walk $ Z_n $ that collects random rewards $ \xi_j $ for $ j \in \mathbb Z$, when the ceiling of the walk $ S_n $ is located at $ j$. The random reward (or scenery) $ \xi_j $ is independent of the walk and with heavy tail. We show the convergence of the sum of independent copies of $ Z_n$ suitably renormalized to a stable motion with integral representation, whose kernel is the local time of a fractional Brownian motion (fBm). This work extends a previous work where the random walk $ S_n$ had independent increments limits.
Let A be a commutative noetherian ring. In this paper, we interpret localizing subcategories of the derived category of A by using subsets of Spec A and subcategories of the category of A-modules. We unify theorems of Gabriel, Neeman and Krause.
We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coanda and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group defined over an algebraically closed field. In course of the proofs some new insight into the structure of such principal $G$-bundles is obtained.
R n u(x)e -ix•ξ dx.We put u s = (u, u) s , (u, v) = (u, v) 0 , andThe norm of a Banach space X is denoted by • X .For 0 < T < ∞, a nonnegative integer j, and a Banach space X, we denote by C j ([0, T ]; X) the Banach space of all functions of C j -class on the interval [0, T ] with the value in X.We putA pseudo-differential operator P (D), D = (D 1 , . . ., D n ) and D j = -i∂ j , with a symbol P (ξ) is defined by P (D)u(x) = (2π) -n R n P (ξ)û(ξ)e ix•ξ dξ.We put J = 1 + |D|, so that u s = J s u .For operators A and B, we denote by [A, B] = AB -BA the commutator.Throughout this paper, we denote inessential constants by the same symbol C.
In this paper we will consider whether there exists a time periodic solution of the Navier-Stokes equations for infinite channels in R(n)(n = 2, 3). H. Beirao da Veiga [4] treated such a problem. This paper is the special case of his paper and we argue the relation between the existence of stationary and time periodic solutions of the Navier-Stokes equations.
Employing the same technique as in our previous papers, we establish an intrinsic characterization of the direct product of a complex Euclidean ball and punctured planes in the category of Stein manifolds from the viewpoint of holomorphic automorphism group.
