Weighted homogeneous singularities and rational homology disk smoothings

简单奇点链环的切触 McKay 对应

A contact McKay correspondence for links of simple singularities

本文通过用 Morse 函数扰动退化切触形式并取作用过滤同调群的正向极限，计算了简单奇点链环的柱形切触同调，其同调秩由 $|\text{Conj}(G)|$ 给出，从而展示了 Floer 理论的 McKay 对应。

Wahl亏格与指数1覆盖导出表示的正性

Positivity of Wahl defects and specialization of reflexive powers of the canonical sheaf

该研究证明非Gorenstein ℚ-高伦斯坦奇点的Wahl亏格非负，Δ+4p_g非负，还给出有理曲面奇点光滑化的黎曼-罗赫公式，完成了容许有理同调圆盘光滑化的正规曲面奇点的解析分类。

On symplectic caps

An important class of contact 3--manifolds are those that arise as links of rational surface singularities with reduced fundamental cycle. We explicitly describe symplectic caps (concave fillings) of such contact 3--manifolds. As an application, we present a new obstruction for such singularities to admit rational homology disk smoothings.

仅具有有理奇点的复射影平面的退化

Degenerations of the complex projective plane with only rational singularities

在瓦尔猜想成立的前提下，对仅具有理奇点的复射影平面正规退化进行分类，包括马内蒂和哈金 - 普罗霍罗夫分类的由马尔可夫方程控制的曲面，以及含四个非对数典范奇点的六个新退化。

意外的有理吹管操作

An Unexpected Rational Blowdown

本文提出了一种新的辛有理吹管操作，通过构建Stein有理同调盘填充，展示了非加权齐次有理奇点的联系链接的辛填充方法。

加权奇点的单值微分同胚与Seiberg--Witten理论

The monodromy diffeomorphism of weighted singularities and Seiberg--Witten theory

本文证明加权齐次型二维孤立超曲面奇点（非有理二重点）的Milnor纤维单值微分同胚在光滑映射类群中具有无限阶，并借助等变Seiberg--Witten--Floer同调中的切触不变量类比，将该结果归结为负定向Seifert纤维化有理同调3-球面辛填充边界Dehn扭转的无限阶性质。

Bounds on Wahl singularities from symplectic topology

Let X be a minimal surface of general type with positive geometric genus ($b_+ > 1$) and let $K^2$ be the square of its canonical class. Building on work of Khodorovskiy and Rana, we prove that if X develops a Wahl singularity of length $\ell$ in a Q-Gorenstein degeneration, then $\ell \leq 4K^2 + 7$. This improves on the current best-known upper bound due to Lee ($\ell \leq 400(K^2)^4$). Our bound follows from a stronger theorem constraining symplectic embeddings of certain rational homology balls in surfaces of general type. In particular, we show that if the rational homology ball $B_{p,1}$…

Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls

The Milnor fibre of a $\mathbb{Q}$-Gorenstein smoothing of a Wahl singularity is a rational homology ball $B_{p,q}$. For a canonically polarised surface of general type $X$, it is known that there are bounds on the number $p$ for which $B_{p,q}$ admits a symplectic embedding into $X$. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded $B_{p,q}$ into surfaces of general type equipped with non-canonical symplectic forms. Ultimately, these symplectic embeddings come from Mori's theory of flips, but we give an interpretation in terms of almost toric structu…

椭圆面的有理同调盘退化

Rational homology disk degenerations of elliptic surfaces

本文分类了非奇异椭圆面的有理同调盘退化，扩展了Kawamata对只含Wahl奇点情况的分类，并构造了Dolgachev面的极小半logcanonical模型。

Obstructions to special Lagrangian desingularizations and the Lagrangian prescribed boundary problem

We exhibit infinitely many, explicit special Lagrangian isolated singularities that admit no asymptotically conical special Lagrangian smoothings. The existence/ nonexistence of such smoothings is an important component of the current efforts to understand which singular special Lagrangians arise as limits of smooth special Lagrangians. We also use soft methods from symplectic geometry (the relative version of the h--principle) and tools from algebraic topology to prove (both positive and negative) results about Lagrangian desingularizations of Lagrangian submanifolds with isolated singulariti…

Lagrangian homology spheres in (A_m) Milnor fibres via C^*-equivariant A_infinity modules

We establish restrictions on Lagrangian embeddings of rational homology spheres into certain open symplectic manifolds, namely the (A_m) Milnor fibres of odd complex dimension. This relies on general considerations about equivariant objects in module categories (which may be applicable in other situations as well), as well as results of Ishii-Uehara and Ishii-Ueda-Uehara concerning the derived categories of coherent sheaves on the resolutions of (A_m) surface singularities.

Bordism与奇点消解

Bordism and resolution of singularities

本文通过改编奇点消解算法，证明复bordism到复导出orbifold及等变bordism的同调理论映射可分裂，进而为任意闭辛流形定义复cobordism取值的Gromov-Witten不变量，并约束Hamilton纤维丛空间拓扑。