Moduli for decorated tuples of sheaves and representation spaces for quivers

Metha-Ramanathan for ε and k-semistable Decorated Sheaves

This paper is devoted to generalizing the Mehta-Ramanathan restriction theorem to the case of ε-semistable and k-semistable decorated sheaves. After recalling the definition of decorated sheaves and their usual semistability we define the ε and k-(semi)stablility. We first prove the existence of a (unique) ε-maximal destabilizing subsheaf for decorated sheaves (Section 3.1). After some others preliminar results (such as the opennes condition for families of ε-semistable decorated sheaves) we finally prove, in Section 3.7, a restriction theorem for slope ε-semistable decorated sheaves. In Secti…

倾斜心（tilted hearts）中的装饰层与态射

Decorated sheaves and morphisms in tilted hearts

该研究在代数几何领域，于适当条件下将极限稳定对、稳定标架层分别对应到标准心倾斜中的满、单态射，建立模空间与Quot空间的等同并证明其射影性，还得到动机霍尔代数中联系倾斜Quot空间的公式。

Global Boundedness for Decorated Sheaves

An important classification problem in Algebraic Geometry deals with pairs $(\E,ϕ)$, consisting of a torsion free sheaf $\E$ and a non-trivial homomorphism $ϕ\colon (\E^{\otimes a})^{\oplus b}\lra\det(\E)^{\otimes c}\otimes Ł$ on a polarized complex projective manifold $(X,Ø_X(1))$, the input data $a$, $b$, $c$, $Ł$ as well as the Hilbert polynomial of $\E$ being fixed. The solution to the classification problem consists of a family of moduli spaces ${\cal M}^δ:={\cal M}^{δ-\rm ss}_{a/b/c/L/P}$ for the $δ$-semistable objects, where $δ\in\Q[x]$ can be any positive polynomial of degree at most $…

A Generalization of Principal Bundles With a Parabolic or Level Structure

We define a parameter dependent notion of stability for principal bundles with a certain local decoration, which generalizes both parabolic and level structures, and construct their coarse moduli space. A necessary technical step is the construction of the moduli space of tuples of vector bundles with a global and a local decoration, which we call decorated tumps. We introduce a notion of asymptotic stability for decorated tumps and show, that stable decorated principal bundles can be described as asymptotically stable decorated tumps.

The Singular Supports of IC sheaves on Quasimaps' Spaces are Irreducible

Let $C$ be a smooth projective curve of genus 0. Let $B$ be the variety of complete flags in an $n$-dimensional vector space $V$. Given an $(n-1)$-tuple $α\in N[I]$ of positive integers one can consider the space $Q_α$ of algebraic maps of degree $α$ from $C$ to $B$. This space admits some remarkable compactifications $Q^D_α$ (Quasimaps), $Q^L_α$ (Quasiflags) constructed by Drinfeld and Laumon respectively. In [Kuznetsov] it was proved that the natural map $π: Q^L_α\to Q^D_α$ is a small resolution of singularities. The aim of the present note is to study the singular support of the Goresky-Mac…

局部 $P^2$ 上的 BPS 树木学

BPS Dendroscopy on Local $P^2$

本文结合大体积与 orbifold 点附近的散射图结果，证明在非紧 Calabi–Yau 三维流形 $K_{P^2}$ 的 $Π$-稳定性物理切片上，Split Attractor Flow Conjecture 成立。