Moduli of Vector Bundles on Curves in Positive Characteristics

Let X be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli scheme of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 (with determinant equal to a theta characteristic) whose Frobenius pull-back is not semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.     

Interpolation Problems for Vector Bundles on Algebraic Curves

In this paper we study meromorphic maps between vector bundles on a Riemann surface. We are mainly interested in stable vector bundles. For a huge number of numerical data we prove the existence of a meromorphic map between two vector bundles with a prescribed number of zeroes and a prescribed number of poles.     

Complete Sets of Relations in the Cohomology Rings of Moduli Spaces of Holomorphic Bundles and Parabolic Bundles Over a Riemann Surface

The cohomology ring of the moduli space of stable holomorphicvector bundles of rank n and degree d over a Riemann surfaceof genus g > 1 has a standard set of generators when n andd are coprime. When n = 2 the relations between these generatorsare well understood, and in particular a conjecture of Mumford,that a certain set of relations is a complete set, is knownto be true. In this article generalisations are given of Mumford'srelations to the cases when n > 2 and also when the bundlesare parabolic bundles, and these are shown to form completesets of relations. 2000 Mathematics Subject Classification 14H60.     

On Framed Instanton Bundles and Their Deformations

We consider a compact twistor space P and assume that there is a surface S ⊂ P, which has degree one on twistor fibres and contains a twistor fibre F, e.g. P a LeBrun twistor space ([20], [18]). Similar to [6] and [5] we examine the restriction of an instanton bundle V equipped with a fixed trivialization along F to a framed vector bundle over (S, F). First we develope inspired by [13] a suitable deformation theory for vector bundles over an analytic space framed by a vector bundle over a subspace of arbitrary codimension. In the second section we describe the restriction as a smooth natural transformation into a fine moduli space. By considering framed U(r)‐instanton bundles as a real structure on framed instanton bundles over P, we show that the bijection between isomorphism classes of framed U(r)‐instanton bundles and isomorphism classes of framed vector bundles over (S, F) due to [5] is actually an isomorphism of moduli spaces.     

Poisson Structures on Moduli Spaces of Framed Vector Bundles on Surfaces

In this paper we prove that the moduli spaces of framed vector bundles over a surface X, satisfying certain conditions, admit a family of Poisson structures parametrized by the global sections of a certain line bundle on X. This generalizes to the case of framed vector bundles previous results obtained in [B2] for the moduli space of vector bundles over a Poisson surface. As a corollary of this result we prove that the moduli spaces of framed SU(r) – instantons on S⁴ = ℝ⁴ ∪ {∞} admit a natural holomorphic symplectic structure.     

Restriction of Stable Rank Two Vector Bundles in Arbitrary Characteristic

Let X be a smooth variety defined over an algebraically closed field of arbitrary characteristic and 𝒪 _X (H) be a very ample line bundle on X. We show that for a semistable X-bundle E of rank two, there exists an integer m depending only on Δ(E) · H ^dim(X)?2 and H ^dim(X) such that the restriction of E to a general divisor in |mH| is again semistable. As corollaries, we obtain boundedness results, and weak versions of Bogomolov's Theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.     

Homogeneous Vector Bundles and Koszul Algebras

Let G be a reductive algebraic group defined over an algebraically closed field of characteristic zero and let P be a parabolic subgroup of G. We consider the category of homogeneous vector bundles over the flag variety G/P. This category is equivalent to a category of representations of a certain infinite quiver with relations by a generalisation of a result in [BK]. We prove that both categories are Koszul precisely when the unipotent radical P_u of P is abelian.     

New Component of the Moduli Space M(2;0,3) of Stable Vector Bundles on the Double Space P 3 of Index Two

We study the moduli scheme M(2;0,n) of rank-2 stable vector bundles with Chern classes c ₁=0, c ₂=n, on the Fano threefold X – the double space P ³ of index two. New component of this scheme is produced via the Serre construction using certain families of curves on X. In particular, we show that the Abel–Jacobi map :HJ(X) of any irreducible component H of the Hilbert scheme of X containing smooth elliptic quintics on X into the intermediate Jacobian J(X) of X factors by Stein through the quasi-finite (probably birational) map g:M of (an open part of) a component M of the scheme M(2;0,3) to a translate of the theta-divisor of J(X).     

Euler Characteristic and Fixed-Point Theorems

We propose a geometric definition of the Euler characteristic (M) for the class of compact epi-Lipschitzian sets MRⁿ and we provide existence theorems of (generalized) equilibria for set-valued mappings F when the domain M of F is neither assumed to be convex, nor smooth but has a nonzero Euler characteristic.     

Quotients of Incidence Algebras and the Euler Characteristic

In this note, we show that, if A ? kQ _A /I _A is a schurian strongly simply connected algebra given by its normed presentation, and Σ is the unique poset whose Hasse quiver coincides with Q _A , then A ? kΣ if and only if I _A has a generating set consisting of exactly χ(Q _A ) elements, where χ(Q _A ) is the Euler characteristic of Q _A . We also prove that a quotient of an incidence algebra A = kΣ/J is strongly simply connected if and only if A is simply connected and kΣ is strongly simply connected.     

Ample and Spanned Rank-2 Vector Bundles with k-Jet Ample Determinant on Algebraic Surfaces

We investigate ample and spanned rank-2 vector bundles E with k-jet ample determinant on algebraic surfaces S. In paticular, we classify such polarized pairs (S,E) with small second Chern class. The case of c_2(E)=k+1 is a generalization of the result of Ballico and Lanteri.     

二次特征值问题中特征值和特征向量的可微性

二次特征值问题中特征值和特征向量的可微性,给出了其导数的计算公式及其算法.     

Pseudodifferential Operators on Prehomogeneous Vector Spaces

ABSTRACT

Let G be a connected, linear algebraic group defined over ?, acting regularly on a finite dimensional vector space V over ? with ?-structure V _?. Assume that V possesses a Zariski-dense orbit, so that (G, ?, V) becomes a prehomogeneous vector space over ?. We consider the left regular representation π of the group of ?-rational points G _? on the Banach space C₀(V _?) of continuous functions on V _? vanishing at infinity, and study the convolution operators π(f), where f is a rapidly decreasing function on the identity component of G _?. Denote the complement of the dense orbit by S, and put S _? = S ∩ V _?. It turns out that, on V _? ? S _?, π(f) is a smooth operator. If S _? = {0}, the restriction of the Schwartz kernel of π(f) to the diagonal defines a homogeneous distribution on V _? ? {0}. Its nonunique extension to V _? can then be regarded as a trace of π(f). If G is reductive, and S and S _? are irreducible hypersurfaces, π(f) corresponds, on each connected component of V _? ? S _?, to a totally characteristic pseudodifferential operator. In this case, the restriction of the Schwartz kernel of π(f) to the diagonal defines a distribution on V _? ? S _? given by some power |p(m)| ^s of a relative invariant p(m) of (G, ?, V) and, as a consequence of the Fundamental Theorem of Prehomogeneous Vector Spaces, its extension to V _?, and the complex s-plane, satisfies functional equations similar to those for local zeta functions. A trace of π(f) can then be defined by subtracting the singular contributions of the poles of the meromorphic extension.     

M.P. Appell's Function and Vector Bundles of Rank 2 on Elliptic Curves

This paper is devoted to the function introduced by M.P. Appell in connection with decomposition of elliptic functions of the third kind into simple elements. We show that this function is related to global sections of rank-2 vector bundles on elliptic curves. We derive analogues of theta-identities for this function and prove the divisibility property for the action of the modular group, that should be considered as a replacement of the functional equation.     

四元数矢丛的辛Pontryagin示性式和它的积分公式

设ρ（Ｍ）是微分流形Ｍ上的左四元数矢丛，文中给出ρ（Ｍ）的辛Ｐｏｎｔｒｙａｇｉｎ示性类和辛Ｐｏｎｔｒｙａｇｉｎ示性式的定义，并给出说明它们之间的联系的积分公式。     

Characteristic decompositions and boundary value problems for two‐dimensional steady relativistic Euler equations

In this paper, we investigate Goursat problems, and mixed initial and boundary value problems for the two‐dimensional steady relativistic Euler equations. The global existence of classical solutions to these problems are obtained by using the characteristic decomposition method. Some applications of these results in supersonic flow in two‐dimensional ducts and the two‐dimensional relativistic jet are discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type

We introduce Besov type function spaces, based on the weak L ^p -spaces instead of the standard L ^p -spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations of perfect incompressible fluid in . For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator type estimates in our weak spaces. Abbreviate title: Euler equations in Besov spaces of weak type