Related papers: Algebroid prestacks and deformations of ringed spa…

We define localized modulation maps and modulation spaces of symbols suited to the study of Rieffel's deformation quantization pseudodifferential calculus. They are used to generate Hilbert space representations for the quantized…

Derived decompositions of abelian categories are introduced in internal terms of abelian subcategories to construct semi-orthogonal decompositions (or Bousfield localizations, or hereditary torsion pairs) in various derived categories of…

Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed…

Let $M$ be a complete nonsingular fine moduli space of modules over an algebra $S$. A set of conditions is given for the Chow ring of $M$ to be generated by the Chern classes of certain universal bundles occurring in a projective resolution…

Under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. The Tannakian reconstruction theorem provides another example where a geometric object can be reconstructed from an associated category, in…

We construct a large collection of "quantum projective spaces", in the form of Koszul, Calabi-Yau algebras with the Hilbert series of a polynomial ring. We do so by starting with the toric ones (the q-symmetric algebras), and then deforming…

This is the first in a series of papers math.AG/0503029, math.AG/0410267, math.AG/0410268 on "configurations" in an abelian category A. Given a finite partially ordered set (I,<), an (I,<)-configuration (\sigma,\iota,\pi) is a finite…

The representations of a quiver Q over a field k have been studied for a long time. It seems to be worthwhile to consider also representations of Q over arbitrary finite-dimensional k-algebras A. Here we draw the attention to the case when…

Bloch-Okounkov studied certain functions on partitions $f$ called shifted symmetric polynomials. They showed that certain $q$-series arising from these functions (the so-called \emph{$q$-brackets} $\left<f\right>_q$) are quasimodular forms.…

A relationship between curved differential algebras and corings is established and explored. In particular it is shown that the category of semi-free curved differential graded algebras is equivalent to the category of corings with…

We consider an algebraic variety X together with the choice of a subvariety Z. We show that any coherent sheaf on X can be constructed out of a coherent sheaf on the formal neighborhood of Z, a coherent sheaf on the complement of Z, and an…

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let $V$ be a vertex algebra, $H$ a finite group of automorphisms of $V$,…

We introduce graded $\mathbb{E}_{\infty}$-rings and graded modules over them, and study their properties. We construct projective schemes associated to connective $\mathbb{N}$-graded $\mathbb{E}_{\infty}$-rings in spectral algebraic…

Based on the logarithmic algebraic geometry and the theory of Deligne systems, we define an abelian category of $\ell$-adic sheaves with weight filtrations on a logarithmic scheme over a finite field, which is similar to the category of…

Let $X$ be an affine, smooth, and Noetherian scheme over $\mathbb{C}$ acted on by an affine algebraic group $G$. Applying the technique developed in Arkhipov and {\O}rsted (2018a, 2018b), we define a dg-model for the derived category of…

Let H be a finite dimensional Hopf algebra over a field k and A an H-module algebra over k. Khovanov and Qi defined acyclic objects and quasi-isomorphisms by using null-homotopy and contractible objects. They also defined the cofibrant…

A Lefschetz module is a module over a graded algebra $A$ that satisfies analogues of Poincar\'{e} duality, the Hard Lefschetz property, and the Hodge--Riemann relations with respect to an open convex cone $\mathscr{K}$ in the degree one…

By a recent result of Viehweg, projective manifolds with ample canonical class have a coarse moduli space, which is a union of quasiprojective varieties. In this paper, we prove that there are manifolds with ample canonical class that lie…

We consider algebras and Frobenius algebras, internal to a monoidal category, that are graded over a finite abelian group. For the case that A is a twisted group algebra in a linear abelian monoidal category we obtain a graded…

Here we define a series of associative algebras attached to a vertex operator algebra $V$, called mode transition algebras, showing they reflect both algebraic properties of $V$ and geometric constructions on moduli of curves. One can…