Some Quot schemes in tilted hearts and moduli spaces of stable pairs

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。

倾斜心脏中的商空间与Hall代数模

Quot spaces in tilted hearts and Hall algebra modules

本文通过对应叠构造了曲线上固定斜率的半稳定凝聚层的Hall代数在倾斜心脏中某些商空间的导出范畴上的双边范畴作用。

Some metric properties of spaces of stability conditions

We show that, under mild conditions, the space of numerical Bridgeland stability conditions Stab(T) on a triangulated category T is complete. In particular the metric on a full component of Stab(T) for which the central charges factor through a finite rank quotient of the Grothendieck group K(T) is complete. As an example, we compute the metric on the space of numerical stability conditions on a smooth complex projective curve of genus greater than one, and show that in this case the quotient Stab(T)/C by the natural action of the complex numbers is isometric to the upper half plane equipped w…

Cusps of the Kähler moduli space and stability conditions on K3 surfaces

In [Ma1] S. Ma established a bijection between Fourier--Mukai partners of a K3 surface and cusps of the Kähler moduli space. The Kähler moduli space can be described as a quotient of Bridgeland's stability manifold. We study the relation between stability conditions $σ$ near to a cusp and the associated Fourier--Mukai partner Y in the following ways. (1) We compare the heart of $σ$ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of $σ$-stable objects. An appendix is devoted to the group of auto-equivalences of the derived category which respect the component $Stab^\da…

Canonical tilting relative generators

Given a relatively projective birational morphism $f\colon X\to Y$ of smooth algebraic spaces with dimension of fibers bounded by 1, we construct tilting relative (over $Y$) generators $T_{X,f}$ and $S_{X,f}$ in $\mathcal{D}^b(X)$. We develop a piece of general theory of strict admissible lattice filtrations in triangulated categories and show that $\mathcal{D}^b(X)$ has such a filtration $\mathcal{L}$ where the lattice is the set of all birational decompositions $f \colon X \xrightarrow{g} Z \xrightarrow{h} Y$ with smooth $Z$. The $t$-structures related to $T_{X,f}$ and $S_{X,f}$ are proved t…

Continuity properties of measurable group cohomology

A version of group cohomology for locally compact groups and Polish modules has previously been developed using a bar resolution restricted to measurable cochains. That theory was shown to enjoy analogs of most of the standard algebraic properties of group cohomology, but various analytic features of those cohomology groups were only partially understood. This paper re-examines some of those issues. At its heart is a simple dimension-shifting argument which enables one to `regularize' measurable cocycles, leading to some simplifications in the description of the cohomology groups. A range of c…