Related papers: Truncated modules and linear presentations of vect…
When the action of a reductive group on a projective variety has a suitable linearisation, Mumford's geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how…
We prove three results on pure resolutions of vector bundles on projective spaces. First, we show that there are simple vector bundles of rank n on Pn with arbitrary homological dimension. We then analyze the pure resolutions given by the…
To do homological algebra with unbounded chain complexes one needs to first find a way of constructing resolutions. Spaltenstein solved this problem for chain complexes of R-modules by truncating further and further to the left, resolving…
We construct vector bundles $R^r_\mu$ on a smooth projective curve $X$ having the property that for all sheaves $E$ of slope $\mu$ and rank $r$ on $X$ we have an equivalence: $E$ is a semistable vector bundle $\iff$ $Hom(R^r_\mu,E)=0$. As a…
Moduli of vector bundles on stacky curves behave similarly to moduli of vector bundles on curves, except there are additional numerical invariants giving many different notions of stability. We apply the existence criterion for good moduli…
Motivated by applications to duality theorems for $p$-adic pro-\'etale cohomology of rigid analytic spaces, we study the category of Topological Vector Spaces in the setting of condensed mathematics. We prove that it contains, as full…
We compute the cohomology of the right generalised projective Stiefel manifolds and use it to find bounds on the rank of the complementary bundle for certain vector bundles. Further the cohomology computations are also used to find bounds…
We describe a polynomial complexity algorithm for reducing transition matrices, for vector bundles glued along a clutching-type cover of a real anisotropic conic, to canonical block diagonal forms. This is a generalization, to the real…
We give a construction of the moduli space of stable maps to the classifying stack B\mu_r of a cyclic group by a sequence of r-th root constructions on M_{0, n}. We prove a closed formula for the total Chern class of \mu_r-eigenspaces of…
A structure theorem for bounded-below modules over the subalgebra A(1) of the mod 2 Steenrod algebra generated by Sq^1, Sq^2 is proved; this is applied to prove a classification theorem for a family of indecomposable A(1)-modules. The…
We construct bases for the spaces of higher order modular forms of all orders and weights. We also provide a cohomological interpretation of these forms.
Categorical resolutions of singularities are a replacement of resolution of singularities within the realm of triangulated categories. They allow the study of the derived category of a singular variety $X$ via a triangulated category that…
We present closed form expressions for the ranks of all cohomology groups of holomorphic line bundles on several Calabi-Yau threefolds realised as complete intersections in products of projective spaces. The formulae have been obtained by…
We describe derived moduli functors for a range of problems involving schemes and quasi-coherent sheaves, and give cohomological conditions for them to be representable by derived geometric n-stacks. Examples of problems represented by…
We set up a formalism of Maurer-Cartan moduli sets for L-infinity algebras and associated twistings based on the closed model category structure on formal differential graded algebras (a.k.a. differential graded coalgebras). Among other…
In this paper we give the classification of rank 3 vector bundles without "inner" cohomology on a quadric hypersurface \Q_n (n>3) by studying the associated monads.
This paper focuses on the study of a new category of vector bundles. The objects of this category, called chiral vector bundles, are pairs given by a complex vector bundle along with one of its automorphisms. We provide a classification for…
We study the moduli space of parabolic connections of rank two on the complex projective line $\mathbb{P}^1$ minus five points with fixed spectral data. This paper aims to compute the cohomology of the structure sheaf and a certain vector…
In this note we highlight a common origin for many ubiquitous geometric structures, as well as several new ones by using only the functors of differential calculus in A.M Vinogradov's original sense, adapted to special classes of (graded)…
We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.