We compare realizability models over partial combinatory algebras by embedding them into sheaf toposes. We then use the machinery of Grothendieck toposes and geometric morphisms to study the relationship between realizability models over different partial combinatory algebras. This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction o...

Earlier we defined the Proj of a graded ring. In these notes we introduce a relative version of this construction, which is the Proj of a sheaf of graded algebras S over a scheme X. This construction is useful in particular because it allows us to construct the projective space bundle associated to a locally free sheaf E , and it allows us to give a definition of blowing up with respect to an a...

Monday December 9: We discussed the sheaf ΩX of regular differential forms on a variety X , which is sheaf of OX-modules, locally free IFF X is smooth, in which case its rank is equal to the dimension. We computed several examples: the sheaf on A is freely generated (on every open set) by dx1, . . . , dxn. On the parabola defined by y = x in A, the elements dx and dy generate on every open set,...

Let θ : g → g be an involution of a complex semisimple Lie algebra, k ⊂ g the fixed points of θ, and V = g/k the corresponding symmetric space. The adjoint form K of k naturally acts on V . The orbits and invariants of this representation were studied by Kostant and Rallis in [KR]. Let X = K\\V be the invariant theory quotient, and f : V → X be the quotient map. The space X is isomorphic to C. ...

which is a linear combination of closures of conormal bundles to submanifolds of X. Intuitively, the microlocal multiplicities cα( ) measure the singularity of at α. In settings related to representation theory, a group G acts on X, is G-equivariant, and the microlocal multiplicities play a significant but only partially understood role in representation theory (see [Ro], [SV], [ABV], and [KaSa...

A notion of Poincaré series was introduced in [1]. It was developed in [2] for a multi-index filtration corresponding to the sequence of blow-ups. The present paper suggests the way to generalize the notion of Poincaré series to the case of arbitrary locally free sheaf on the modification of complex plane C 2. This series is expressed through the topological invariants of the sheaf. For the she...

We generalize the theory of sheaves to chamber systems. We prove that, given a chamber system C and a family R of proper residues of C containing all residues of rank c1, every sheaf defined over R admits a completion which extends C. We also prove that, under suitable hypotheses, a sheaf defined over a truncation of C can be extended to a sheaf for C. In the last section of this paper, we appl...

In this note I would like to introduce a new approach to (or rather a new language for) representation theory of groups. Namely, I propose to consider a (complex) representation of a group G as a sheaf on some geometric object. This point of view necessarily leads to a conclusion that the standard approach to (continuous) representations of algebraic groups should be modified. Let us start with...

After a brief survey of the primary ideas involved in the theory of connections on vector and principal sheaves (studied in [7], [8], [14], [15]), we examine the behaviour of connections under various types of morphisms between sheaves of the considered category. The results thus obtained are useful in the development of a non-smooth geometry in the aforementioned abstract framework and related...

We show that Boehmians defined over open sets of R constitute a sheaf. In particular, it is shown that such Boehmians satisfy the gluing property of sheaves over topological spaces. MSC: Primary 44A40, 46F99; Secondary 44A35, 18F20