Perverse sheaves on real loop Grassmannians

The aim of this paper is to identify a certain tensor category of perverse sheaves on the loop Grassmannian Gr of a real form G of a connected reductive complex algebraic group G with the category of finite-dimensional representations of a connected reductive complex algebraic subgroup of the dual group . The root system of is closely related to the restricted root system of G. The fact that is reductive implies that an interesting family of real algebraic maps satisfies the conclusion of the Decomposition Theorem of Beilinson-Bernstein-Deligne.     

Mixed perverse sheaves for schemes over number fields

This paper generalizes the definition of mixed perverse sheavesto schemes of finite type over a number field.Their basic properties, e.g., characterization of simple objects, are shown.     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

Dualizing complexes and perverse sheaves on noncommutative ringed schemes

A quasi-coherent ringed scheme is a pair (X, $$ \mathcal{A} $$), where X is a scheme, and $$ \mathcal{A} $$ is a noncomutative quasi-coherent $$ \mathcal{O}_X $$ -ring. We introduce dualizing complexes over quasi-coherent ringed schemes and study their properties. For a separated differential quasi-coherent ringed scheme of finite type over a field, we prove existence and uniqueness of a rigid dualizing complex. In the proof we use the theory of perverse coherent sheaves in order to glue local pieces of the rigid dualizing complex into a global complex.     

Quantum Pieri rules for isotropic Grassmannians

We study the three point genus zero Gromov-Witten invariants on the Grassmannians which parametrize non-maximal isotropic subspaces in a vector space equipped with a nondegenerate symmetric or skew-symmetric form. We establish Pieri rules for the classical cohomology and the small quantum cohomology ring of these varieties, which give a combinatorial formula for the product of any Schubert class with certain special Schubert classes. We also give presentations of these rings, with integer coefficients, in terms of special Schubert class generators and relations.     

Characteristic cycles of perverse sheaves and Milnor fibers

We show that the Milnor monodromy and the Milnor numbers naturally appear in the characteristic cycles of irreducible perverse sheaves having the singularities of a complex hypersurface. We find also a new numerical constraint on the Milnor monodromy for general hypersurface singularities.Mathematics Subject Classification (1991): 14B05, 32C38, 32S40, 35A27Dedicated to the 60th anniversary of Prof P. Schapira who taught us the theory of D-modules     

Quantum Giambelli formulas for isotropic Grassmannians

Let X be a symplectic or odd orthogonal Grassmannian which parametrizes isotropic subspaces in a vector space equipped with a nondegenerate (skew) symmetric form. We prove quantum Giambelli formulas which express an arbitrary Schubert class in the small quantum cohomology ring of X as a polynomial in certain special Schubert classes, extending the authors?? cohomological Giambelli formulas.     

Exceptional collections for Grassmannians of isotropic lines

We construct a full exceptional collection of vector bundlesin the derived categories of coherent sheaves on the Grassmannianof isotropic two-dimensional subspaces in a symplectic vectorspace of dimension 2n and in an orthogonal vector space of dimension2n + 1 for all n.     

Structure sheaves of definable additive categories

We prove that the 2-category of small abelian categories with exact functors is anti-equivalent to the 2-category of definable additive categories. We define and compare sheaves of localisations associated to the objects of these categories. We investigate the natural image of the free abelian category over a ring in the module category over that ring and use this to describe a basis for the Ziegler topology on injectives; the last can be viewed model-theoretically as an elimination of imaginaries result.     

Locally finitely presented categories of sheaves

Let A be a locally finitely presented Grothendieck category. It is shown that a class of localizations of A in the sense of Bousfield is again locally finitely presented. The criterion is applied to torsion-free classes in A, sheaves and separated presheaves on a generalized ringed space, and representations of partially ordered sets.