In this chapter we write about derived categories associated to algebraic stacks. This means in particular derived categories of quasi-coherent sheaves, i.e., we prove analogues of the results on schemes (see Derived Categories of Schemes, Section 1) and algebraic spaces (see Derived Categories of Spaces, Section 1). The results in this chapter are different from those in [LMB00] mainly because...

Let X be a compact Riemann surface of genus $$g \ge 3$$ . We consider the moduli space holomorphic connections over and logarithmic singular finite subset with fixed residues. determine Chow group these spaces. compute global sections sheaves differential operators on ample line bundles their symmetric powers spaces show that they are constant under certain conditions. Torelli-type theorem for ...

Recent work on the log-minimal model program for the moduli space of curves, as well as past results of Caporaso, Pandharipande, and Simpson motivate an investigation of compactifications of the universal moduli space of slope semi-stable vector bundles over moduli spaces of curves arising in the Hassett–Keel program. Our main result is the construction of a universal moduli space of slope semi...

I introduce the notion of probability sheaf, which is a mathematical structure capturing the relationship between probabilistic concepts (such as random variable) and sample spaces. Various probability-theoretic notions can be (re)formulated in terms of category-theoretic structure on the category of probability sheaves. As a main example, I consider the Giry monad, which, in its original formu...

Moduli spaces of sheaves and Hilbert schemes of points have experienced a recent resurgence in interest in the past several years, due largely to new techniques arising from Bridgeland stability conditions and derived category methods. In particular, classical questions about the birational geometry of these spaces can be answered by using new tools such as the positivity lemma of Bayer and Mac...

We apply Preuss' concept of $mbbe$-connectedness to the categories of lattice-valued uniform convergence spaces and of lattice-valued uniform spaces. A space is uniformly $mbbe$-connected if the only uniformly continuous mappings from the space to a space in the class $mbbe$ are the constant mappings. We develop the basic theory for $mbbe$-connected sets, including the product theorem. Furtherm...

We investigate the effect on Cauchy complete objects of the change of base 2-functor V-Cat → W-Cat induced by a two-sided enrichment V → W. We restrict our study to the case of locally partially ordered bases. The reversibility notion introduced in [Wal82] is extended to two-sided enrichments and Cauchy completion. We show that a reversible left adjoint two-sided enrichment F : V → W between lo...

In this paper, we introduce new enumerative invariants of curves on Calabi-Yau 3-folds via certain stable objects in the derived category of coherent sheaves. We introduce the notion of limit stability on the category of perverse coherent sheaves, a subcategory in the derived category, and construct the moduli spaces of limit stable objects. We then define the counting invariants of limit stabl...

Interpreting the syzygy theorem for tame modules over posets in setting of derived categories subanalytically constructible sheaves proves two conjectures due to Kashiwara and Schapira concerning existence stratifications real vector spaces that play well with having microsupport a given cone or, equivalently, corresponding conic topology.

The paper introduces the notion of state for involutive bisemilattices, a variety which plays role algebraic counterpart paraconsistent weak Kleene logic and whose elements are represented as Plonka sum Boolean algebras. We investigate relations between states over an bisemilattice probability measures (Boolean) algebras in representation and, direct limit these Moreover, we study completition ...