We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolovs inequality for semistable sheaves on a smooth projective variety of any dimension $ge 2$ without using any restriction theorems.
The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on a multiple hyperplanes and prove the existence of such bundles that do not split, if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most $3/2$ on the double plane and describe the family of isomorphism classes of them.
We conjecture that any perverse sheaf on a compact aspherical Kahler manifold has non-negative Euler characteristic. This extends the Singer-Hopf conjecture in the Kahler setting. We verify the stronger conjecture when the manifold X has non-positive holomorphic bisectional curvature. We also show that the conjecture holds when X is projective and in possession of a faithful semi-simple rigid local system. The first result is proved by expressing the Euler characteristic as an intersection number involving the characteristic cycle, and then using the curvature conditions to deduce non-negativity. For the second result, we have that the local system underlies a complex variation of Hodge structure. We then deduce the desired inequality from the curvature properties of the image of the period map.
Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localisation of Edidin-Grahams square root Euler class for $SO(r,mathbb C)$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a $K$-theoretic refinement by defining $K$-theoretic square root Euler classes and their localis
We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.