Invariant sheaves

Invariant Sheaves

§0. Introduction The sheaves of tangent vector fields, differential forms or differential operators are canonical. Namely they are invariant by the coordinate transformations. We call such sheaves invariant sheaves. More precisely for a positive integer n, an invariant sheaf on n-manifold is given by the data: coherent OX -module FX for each smooth variety X of dimension n and an isomorphism β(...

Equivariant sheaves and their applications to invariant theory

Counting invariant of perverse coherent sheaves and its wall-crossing

We introduce moduli spaces of stable perverse coherent systems on small crepant resolutions of Calabi-Yau 3-folds and consider their DonaldsonThomas type counting invariants. The stability depends on the choice of a component (= a chamber) in the complement of finitely many lines (= walls) in the plane. We determine all walls and compute generating functions of invariants for all choices of cha...

Coherent Sheaves and Cohesive Sheaves

We consider coherent and cohesive sheaves of O–modules over open sets Ω ⊂ Cn. We prove that coherent sheaves, and certain other sheaves derived from them, are cohesive; and conversely, certain sheaves derived from cohesive sheaves are coherent. An important tool in all this, also proved here, is that the sheaf of Banach space valued holomorphic germs is flat. To Linda Rothschild on her birthday

Sheaves on fibered threefolds and quiver sheaves

This paper classifies a class of holomorphic D-branes, closely related to framed torsion-free sheaves, on threefolds fibered in resolved ADE surfaces over a general curve C, in terms of representations with relations of a twisted Kronheimer– Nakajima-type quiver in the category Coh(C) of coherent sheaves on C. For the local Calabi–Yau case C ∼= A and special choice of framing, one recovers the ...