Adelic descent theory

A compactification of the moduli space of principal Higgs bundles over singular curves

Let X be a smooth projective curve defined over an algebraically closed field k, and let FX denote the absolute Frobenius morphism of X when the characteristic of k is positive. A vector bundle over X is called virtually glob- ally generated if its pull back, by some finite morphism to X from some smooth projective curve, is generated by its global sections. We prove the following. If the characteristic of k is positive, a vector bundle E over X is virtually globally generated if and only if (F m X ) ∗ E ∼ Ea ⊕ E f for some m, where Ea is some ample vector bundle and E f is some finite vector bundle over X (either of Ea and E f are allowed to be zero). If the characteristic of k is zero, a vector bundle E over X is virtually globally generated if and only if E is a direct sum of an ample vector bundle and a finite vector bundle over X (either of them are allowed to be zero).

In this chapter we treat the theory of principal bundles and connections on them for families of supermanifolds. The most important examples of principal bundles are frame bundles of vector bundles. Many extra structures on vector bundles, such as metrics or almost complex structures can actually be formulated in terms of a reduction of the structure group of the frame bundle of the vector bundle. The theory of connections on principal bundles sheds light on properties of covariant derivatives that are compatible with such extra structures.

Let H be the upper half plane and T a discrete subgroup of AutH.Suppose that H mod Y is of finite measure.This work stems from the question whether there is an algebraic interpretation for the moduli of unitary representations of T similar to the case whenWe show that this is indeed the case via the moduli of vector bundles on the compactification of H mod r, provided with some additional structures which we propose to call parabolic structures.The idea of parabolic structures is inspired from A. Weil's work [6,§2, Chapter I, p. 56].Let Xbe a smooth, irreducible, projective curve defined, say, over an algebraically closed field k.By vector bundles on X we understand algebraic vector bundles.DEFINITION 1.Let V be a vector bundle on X and Q G X. Then a quasiparabolic structure of Vat Q is giving a flag on the fibre VQ of Fat Ô, i.e., giving linear subspaces F*VQ of VQ,We call I = (l l9 . . .9 l r ) the type (or flag type) of the quasi-parabolic structure.Let k x = l x -/ 2 , k 2 = l 2 -/ 3 , . . ., fc fWl = /,._! -l r , k r = l r \ then k t are called the multiplicities of the quasi-parabolic structure.DEFINITION 2. Let V be a vector bundle on X and Q G X. Then a parabolic structure of V at Q is giving (i) a quasi-parabolic structure of V at Q; say I = (l t , . . ., l r ) is its type and {k.} its multiplicities, and(ii) constants a = (a 1 , . . ., a n ) called the weights of the parabolic structure such that 0 < a x < a 2 < • • • < ot n < 1 and there are r distinct elements among a, say a' = (o^, . . ., Û£), 0 < a\ < a 2 < • • • < a r < 1, such that OL\ occurs k x times, a 2 occurs k 2 times, . . ., a r occurs k r times among a.We call a. the weight ofF l V Q .Note that l x = « = rkV.Let F, H> be vector bundles on X with quasi-parabolic structures at Q.An isomorphism ƒ: F -• W of vector bundles is said to be a quasi-parabloic isomorphism if the types of V, W at Q are the same and fçiF'Vç) = F 1 WQ (/^:

On delta for parameterized curve singularities

Several intrinsic topological ways to encode connections on vector bundles on smooth complex algebraic curves will be described. In particular the notion of Stokes decompositions will be formalised, as a convenient intermediate category between the Stokes filtrations and the Stokes local systems/wild monodromy representations. The main result establishes a new simple characterisation of the Stokes decompositions. \begin{equation*} \ \ \ \includegraphics [width=0.6\textwidth ]{fission} \end{equation*}

Let M(n, ξ) be the moduli space of stable vector bundles of rank n ≥ 3 and fixed determinant ξ over a complex smooth projective algebraic curve X of genus g ≥ 4. We use the gonality of the curve and r-Hecke morphisms to describe a smooth open set of an irreducible component of the Hilbert scheme of M(n, ξ), and to compute its dimension. We prove similar results for the scheme of morphisms \({M or_P (\mathbb{G}, M(n, \xi))}\) and the moduli space of stable bundles over \({X \times \mathbb{G}}\), where \({\mathbb{G}}\) is the Grassmannian \({\mathbb{G}(n - r, \mathbb{C}^n)}\). Moreover, we give sufficient conditions for \({M or_{2ns}(\mathbb{P}^1, M(n, \xi))}\) to be non-empty, when s ≥ 1.

In this paper, we prove that the notions of Hilbert stability and Mumford stability agree for vector bundles of arbitrary rank over smooth curves. The notion of Hilbert stability was introduced by Gieseker and Morrison in 1984, and they showed that for smooth curves and vector bundles of rank two it agrees with Mumford stability. A different proof for the rank two case was given by M. Teixidor i Bigas. Our proof uses a new approach and avoids complicated computations. Our results might serve as a first step in the construction of the Hilbert stable compactification of the universal moduli space of stable vector bundles over the moduli space of smooth curves as suggested by Teixidor.

Poincare bundles and Picard bundles on the Jacobian of a smooth curve have been studied extensively classically. Peter Newstead and his collaborators have made major contributions to the theory of Poincaré and Picard bundles over the moduli spaces of vector bundles of higher ranks. We present old and new results on the Poincare bundles and Picard bundles on the moduli spaces of vector bundles on smooth curves and their generalisations to nodal curves. Finally we consider parabolic vector bundles. We prove the parabolic stability of Poincaré bundles over the moduli spaces of parabolic vector bundles, with parabolic structure of any type, on a smooth curve.

Faithfully flat descent of almost perfect complexes in rigid geometry

Let $G$ be a reductive complex algebraic group and $P$ a complex $G$-module with algebraic quotient of dimension $\ge 1$. We construct a map from a certain moduli space of algebraic $G$-vector bundles over $P$ to a $\mathbb{C}$-module possibly of infinite dimension, which is an isomorphism under some conditions. We also show non-triviality of moduli of algebraic $G$-vector bundles over a $G$-stable affine hypersurface of some type. In particular, we show that the moduli space of algebraic $G$-vector bundles over a $G$-stable affine quadric with fixpoints and one-dimensional quotient contains $\mathbb{C}^p$.

Let X be a smooth projective curve of genus g over \(\mathbb {C}\) and M be the moduli space of stable vector bundle of rank 2 and determinant isomorphic to a fixed line bundle of degree 1 on X. Let E be the Poincare bundle on \(X \times M\) and \(\Phi _E : D^b(X) \rightarrow D^b(M)\) Fourier–Mukai functor defined by E. It was proved in our earlier paper that \(\Phi _E\) is fully faithful for every smooth projective curve of genus \(g \ge 4.\) It is proved in this present paper that the result is also true for non-hyperelliptic curves of genus 3. Combining known results in the case of hyperelliptic curves, one obtains that \(\Phi _E\) is fully faithful for all X of genus \(g \ge 2\).

We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle EH → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad (EH) is homogeneous. Fix a faithful H-module V. We also show that EH is homogeneous if the vector bundle EH ×H V associated to it for the H-module V is homogeneous.

It follows from the work of Burban and Drozd arXiv:0905.1231 that for nodal curves $C$, the derived category of modules over the Auslander order $\mathcal{A}_C$ provides a categorical (smooth and proper) resolution of the category of perfect complexes $\mathrm{Perf}(C)$. On the A-side, it follows from the work of Haiden-Katzarkov-Kontsevich arXiv:1409.8611 that for punctured surfaces $X$ with stops $\Lambda$ at their boundary, the partially wrapped Fukaya category $\mathcal{W}(X,\Lambda)$ provides a categorical (smooth and proper) resolution of the compact Fukaya category $\mathcal{F}(X)$. Inspired by this analogy, we establish an equivalence between the derived category of modules over the Auslander orders over certain nodal stacky curves and partially wrapped Fukaya categories associated to punctured surfaces of arbitrary genus equipped with stops at their boundary. As an application, we deduce equivalences between derived categories of coherent sheaves (resp. perfect complexes) on such nodal stacky curves and the wrapped (resp. compact) Fukaya categories of punctured surfaces of arbitrary genus.

The Auslander-Reiten quiver of perfect complexes for a self-injective algebra