A note on Bridgeland stability conditions and Catalan numbers

Note on the motivic DT/PT correspondence and the motivic Flop formula

We prove the motivic version of the DT/PT-correspondence in \cite{PT} and the motivic flop formula of the curve counting invariants in the derived category of smooth Calabi-Yau threefold DM stacks. The main method we use is Bridgeland's Hall algebra identities and the motivic integration map of Bridgeland and Joyce from the motivic Hall algebra of the abelian category of coherent sheaves on a Calabi-Yau threefold DM stack $Y$ to the motivic quantum torus, which is a Poisson algebra homomorphism.

Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces

We survey the basic theory of non-commutative K3 surfaces, with a particular emphasis to the ones arising from cubic fourfolds. We focus on the problem of constructing Bridgeland stability conditions on these categories and we then investigate the geometry of the corresponding moduli spaces of stable objects. We discuss a number of consequences related to cubic fourfolds including new proofs of the Torelli theorem and of the integral Hodge conjecture, the extension of a result of Addington and Thomas and various applications to hyperkähler manifolds. These notes originated from the lecture ser…

Stability conditions for Slodowy slices and real variations of stability

We provide examples of an explicit submanifold in Bridgeland stabilities space of a local Calabi-Yau, and propose a new variant of definition of stabilities on a triangulated category, which we call a "real variation of stability conditions". We discuss its relation to Bridgeland's definition; the main theorem provides an illustration of such a relation. We also state a conjecture by the second author and Okounkov relating this structure to quantum cohomology of symplectic resolutions and establish its validity in some special cases. More precisely, let X be the standard resolution of a transv…

Introduction to stability conditions

These are notes of a course given at the 'school on moduli spaces' at the Newton Institute in January 2011. The abstract theory of stability conditions (due to Bridgeland and Douglas) on abelian and triangulated categories is developed via tilting and t-structures. Special emphasis is put on the bounded derived category of coherent sheaves on smooth projective varieties (in particular for curves and K3 surfaces). The lectures were targeted at an audience with little prior knowledge of triangulated categories and stability conditions but with an interest in vector bundles on curves.

四曲面相干系统的弱稳定性条件

Weak stability conditions on coherent systems of genus four curves

本文研究四曲面相干系统的稳定性条件，证明其退化为节点三次三折面Kuznetsov组件上的稳定性条件。

关于积商曲面上的稳定性条件的注记

A note on stability conditions on product-quotient surfaces

本文研究积商曲面的稳定性条件，证明G自由作用时C₁×C₂的积Bridgeland稳定性条件可诱导X的倾斜稳定性条件，还构造了含A₁奇点的X的非几何稳定性条件，其归属取决于特定不等式。

Real variations of stability conditions for noncommutative symplectic resolutions

A localization theorem for the cyclotomic rational Cherednik algebra $H_c=H_c((\mathbb{Z}/l)^n\rtimes \mathfrak{S}_n)$ over a field of positive characteristic has been proved by Bezrukavnikov, Finkelberg and Ginzburg. Localizations with different parameters give different $t$-structures on the derived category of coherent sheaves on the Hilbert scheme of points on a surface. In this short note, we concentrate on the comparison between different $t$-structures coming from different localizations. When $n=2$, we show an explicit construction of tilting bundles that generates these $t$-structures…