Abstract not found

To any dg-category T (over some base ring k), we define a D −-stack MT in the sense of [HAGII], classifying certain T op-dg-modules. When T is saturated, MT classifies compact objects in the triangulated category [T] associated to T. The main result of this work states that under certain finiteness conditions on T (e.g. if it is saturated) the D −-stack MT is locally geometric (i.e. union of open and geometric sub-stacks). As a consequence we prove the algebraicity of the group of auto-equivalences of saturated dg-categories. We also obtain the existence of reasonable moduli for perfect complexes on a smooth and proper scheme, as

Abstract. We attempt to develop a general algebro-geometric study of the moduli stack of commutative, 1-parameter formal Lie groups, in full comportment with the modern foundations of algebraic geometry. We emphasize the pro-algebraic structure of this stack: it is the inverse limit, over varying n, of moduli stacks of n-buds, and these latter stacks are algebraic. Our main theorems pertain to the height stratification relative to fixed prime p on the stacks of formal Lie groups and of n-buds. Notably, we show that the stack of n-buds of height ≥ h is smooth and universally closed over Fp of dimension −h; we characterize the stratum of n-buds of (exact) height h and the stratum of formal Lie groups of (exact) height h as classifying stacks of certain groups, smooth algebraic in the bud case; and we obtain some structure results on these groups. We also obtain a second characterization of the stratum of formal Lie groups of height h as an inverse limit of classifying stacks of certain finite étale algebraic groups.

Abstract. We associate to each toric vector bundle on a toric variety X(∆) a “branched cover ” of the fan ∆ together with a piecewise-linear function on the branched cover. This construction generalizes the usual correspondence between toric Cartier divisors and piecewise-linear functions. We apply this combinatorial geometric technique to investigate the existence of resolutions of coherent sheaves by vector bundles, using singular nonquasiprojective toric threefolds as a testing ground. Our main new result is the construction of complete toric threefolds that have no nontrivial toric vector bundles of rank less than or equal to three. The combinatorial geometric sections of the paper, which develop a theory of cone complexes and their branched covers, can be read independently. Contents

Abstract. We introduce and study kernel algebras, i.e., algebras in the category of sheaves on a square of a scheme, where the latter category is equipped with a monoidal structure via a natural convolution operation. We show that many interesting categories, such as D-modules, equivariant sheaves and their twisted versions, arise as categories of modules over kernel algebras. We develop the techniques of constructing derived equivalences between these module categories. As one application we generalize the results of [39] concerning modules over algebras of twisted differential operators on abelian varieties. As another application we recover and generalize the results of Laumon [29] concerning an analog of the Fourier transform for derived categories of quasicoherent sheaves on a dual pair of generalized 1-motives.

Abstract. Let F be a non-Archimedian field. We prove that each open and compact subset of GLn(F) can be decomposed into finitely many open, compact, and self-conjugate subsets. As a corollary, we obtain a short, elementary proof of a well-known theorem of I.M. Gelfand and D.A. Kazhdan. 1.

Abstract. We prove a formula relating the fermionic forms associated to a finite quiver and the Poincaré polynomials of quiver varieties associated to the same quiver. Applied to quivers of type ADE, our result implies a version of the fermionic Lusztig conjecture. 1.

1. Connections on groups and torsors 4 2. Gerbes and their gauge stacks 14 3. Morita theory for locally trivialized gerbes 20

Seifert fibered 3–manifolds were introduced and studied in [Sei32], see also [ST80]. Roughly speaking, these are 3–manifolds M which admit a differentiable map f: M → F to a surface F such that every fiber is a circle. Higher dimensional Seifert fibered manifolds were investigated in [OW75]. The authors observed that in many cases of interest Seifert fibered manifolds correspond to holomorphic Seifert C ∗-bundles, and started to develop a general theory of holomorphic Seifert G-bundles for any complex Lie group G. Definition 1. Let X be a normal variety (or algebraic space) over a field k and G a (reduced) algebraic group over k. A Seifert G-bundle over X is a normal variety (or algebraic space) Y together with a morphism f: Y → X and a G-action on Y satisfying the following two conditions. (1) f is affine and G-equivariant (with respect to the trivial action on X). (2) For every x ∈ X, the G-action on the reduced fiber Yx: = redf −1 (x) G × Yx → Yx is G-equivariantly isomorphic to the natural (left) G-action on G/Ix for some finite subgroup Ix ⊂ G.

and topological aspects of the schematization functor