A Construction of the Rational Function Sheaves on Elliptic Curves

Abstract The authors introduce an effective method to construct the rational function sheaf κ on an elliptic curve E, and further study the relationship between κ and any coherent sheaf on E. Finally, it is shown that the category of all coherent sheaves of finite length on E is completely characterized by κ.     

Character sheaves on the semi-stable locus of a group compactification

We study the intermediate extension of the character sheaves on an adjoint group to the semi-stable locus of its wonderful compactification. We show that the intermediate extension can be described by a direct image construction. As a consequence, we show that the “ordinary” restriction of a character sheaf on the compactification to a “semi-stable stratum” is a shift of semisimple perverse sheaf and is closely related to Lusztig's restriction functor (from a character sheaf on a reductive group to a direct sum of character sheaves on a Levi subgroup). We also provide a (conjectural) formula for the boundary values inside the semi-stable locus of an irreducible character of a finite group of Lie type, which gives a partial answer to a question of Springer (2006) [21]. This formula holds for Steinberg character and characters coming from generic character sheaves. In the end, we verify Lusztig's conjecture Lusztig (2004) [16, 12.6] inside the semi-stable locus of the wonderful compactification.     

Coherent Sheaves on P1: their Families and their Deformations Related to a Behavioral Approach to Singular Systems

Recently, Lomadze, Ravi, Rosenthal and Schumacher gave a natural and geometrically meaningful one-to-one correspondence between abstract linar behaviors and coherent sheaves on P ¹. Motivated by their result, here we give a complete picture of the deformation theory of coherent sheaves on P ¹. We use our results to study deformations over the field R obtaining a connectedness result for the real locus of the algebraic parameter spaces.     

Cohomology of Sheaves of Frechet Algebras and Spectral Theory

We propose a holomorphic functional calculus for a noncommutative operator family generating a supernilpotent Lie subalgebra. This calculus extends Taylor's holomorphic functional calculus.     

Singular Supports for Character Sheaves on a Group Compactification

Let G be a semisimple adjoint group over C and Ḡ be the De Concini–Procesi completion of G. In this paper, we define a Lagrangian subvariety Λ of the cotangent bundle of Ḡ such that the singular support of any character sheaf on Ḡ is contained in Λ. Received: May 2006, Accepted: November 2006     

Generalized parabolic sheaves on an integral projective curve

We extend the notion of a parabolic vector bundle on a smooth curve to define generalized parabolic sheaves (GPS) on any integral projective curve X. We construct the moduli spacesM(X) of GPS of certain type onX. IfX is obtained by blowing up finitely many nodes inY then we show that there is a surjective birational morphism from M(X) to M (Y). In particular, we get partial desingularisations of the moduli of torsion-free sheaves on a nodal curveY.     

Exceptional Pairs in Hereditary Categories

We study the category 𝒞(X, Y) generated by an exceptional pair (X, Y) in a hereditary category ?. If r = dim _k Hom(X, Y) ≥ 1 we show that there are exactly 3 possible types for 𝒞(X, Y), all derived equivalent to the category of finite dimensional modules mod(H _r ) over the r-Kronecker algebra H _r . In general 𝒞(X, Y) will not be equivalent to a module category. More specifically, if ? is the category of coherent sheaves over a weighted projective line 𝕏, then 𝒞(X, Y) is equivalent to the category of coherent sheaves on the projective line ?¹ or to mod(H _r ) and, if 𝕏 is wild, then every r ≥ 1 can occur in this way.     

Weights of twisted exponential sums

Let k be a finite field of characteristic p, l a prime number different from p, a nontrivial additive character, and a character on . Then ψ defines an Artin-Schreier sheaf on the affine line , and χ defines a Kummer sheaf on the n-dimensional torus . Let be a Laurent polynomial. It defines a k-morphism . In this paper, we calculate the weights of under some non-degeneracy conditions on f. Our results can be used to estimate sums of the form

Vertex Algebra Sheaf Structure on Torus

In this paper, we first give a 1-1 corresponds between torus C/Λ and cubic curve C in P_C~2. As complex manifold, they are isomorphic, therefore we can treat C/Λ as a variety and construction a vertex algebra sheaf on it.     

Recognition in a Distributed System

A general design for a distributed system is presented which characterizes system recognition of external events as a sheaf in a Grothendieck topos on a category of diagrams. The information that the system may derive from this sheaf involves transitions through a state space constructed from a collection of topoi interconnected by geometric morphisms. Similarities of this design with natural neural systems leads to the premise that cognitive processes involve the construction and manipulation of sheaf-like structures.     

Moduli of Framed Parabolic Sheaves

This paper gives a construction of the moduli space of framed parabolic sheaves on a Riemann surface. This space serves as a universal, master, space for the well known moduli space of parabolic bundles, as well as moduli spaces of vector bundles, which can all be obtained from this space by torus quotients. The construction is given for the structure group SL(N, C), and indeed is adapted to this case. At the end of the paper, an approach is suggested for dealing with the case of arbitrary reductive groups, involving the associated loop group.     

Bounded complexes of cotorsion sheaves

We provide a necessary and su?cient condition which ensure that every flat quasi-coherent sheaf has finite cotorsion dimension. Also, we will show that every locally noetherian scheme with a dualizing complex has this requirement.     

Adjoint Linear Systems on Normal Log Surfaces

We prove a Reider type theorem for separating any cluster by an adjoint system to a pseudoeffective divisor on a normal surface. As a corollary we get a Reider type theorem for adjoint linear systems (to a nef Q-divisor) on normal log surfaces. This theorem is new even for smooth surfaces.     

Some remarks on the Brylinski-Radon and the Fourier transforms

We will study a relationship between the Brylinski-Radon and the Fourier transforms of a monodromic perverse sheaf. In fact we will show that they are equivalent modulo a constant perverse sheaf. As an application, we will discuss a relation between the Sklyanin transform and the Brylinski-Radon transform.     

Functors of the category of abelian sheaves on regular affine scheme

In this short paper, we prove that ifR is a regular local ring of unequal characteristic, then there exists an additive covariant functorG from the category of abelian sheaves on SpecR to the category of abelian groups such that id _R (G(R))>dimG(R). This result shows that the answer to the question 3.8 (ii) in [3] may be negative.     

Finiteness of Total Cofibres

The suspension of the total cofibre (Y) of a diagram Y of infinite CW complexes, indexed by the face lattice of a polytope, is in general not homotopy equivalent to a finite CW complex. However, if Y satisfies additional conditions (making it into a locally finite non-linear sheaf on a projective toric variety) we can show that (Y) is a homotopy finite space. The main tools used in the proof are a Bousfield–Kan spectral sequence for total cofibres, and a modification of the ech construction to calculate sheaf cohomology groups.     

C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module*

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C^*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C^*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of L(H), where L(H) denotes the set of all adjointable operators on H. The C^*-subalgebras of L(H) generated by elements in {P - P ∧ Q, Q - P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π, X e) of i(P, Q, H) such that e π(P - P ∧ Q) = π(P) - π(P) ∧ π(Q),eπ(Q - P ∧ Q) = π(Q) - π(P) ∧ π(Q).When (P, Q) is semi-harmonious, that is, R(P + Q) and R(2I - P - Q) are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I - Q, I - P, H) are unitarily equivalent via a unitary operator in L(H). A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I - Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C^*-modules are also provided.     

Moduli of twisted orbifold sheaves

We study stacks of slope-semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope-semistable sheaves on smooth projective surfaces.