关于4价曲面奇点的有理同调圆盘光滑化

Smoothing nodal Calabi-Yau n-folds

Let X be an n-dimensional Calabi-Yau with ordinary double points, where n is odd. Friedman showed that for n=3 the existence of a smoothing of X implies a specific type of relation between homology classes on a resolution of X. (The converse is also true, due to work of Friedman, Kawamata and Tian.) We sketch a more topological proof of this result, and then extend it to higher dimensions. For n>3 the "Yukawa product" on the middle dimensional (co)homology plays an unexpected role. We also discuss a converse, proving it for nodal Calabi-Yau hypersurfaces in projective space.

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…

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。

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。

Milnor and Tjurina numbers for smoothings of surface singularities

For an isolated hypersurface singularity $f=0$, the Milnor number $μ$ is greater than or equal to the Tjurina number $τ$ (the dimension of the base of the semi-universal deformation), with equality if $f$ is quasi-homogeneous. K. Saito proved the converse. The same result is true for complete intersections, but is much harder. For a Gorenstein surface singularity $(V,0)$, the difference $μ- τ$ can be defined whether or not $V$ is smoothable; it was proved in [23] that it is non-negative, and equal to 0 iff $(V,0)$ is quasi-homogeneous. We conjecture a similar result for non-Gorenstein surface …

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。

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。

KSBA surfaces with elliptic quotient singularities, $π_1=1$, $p_g=0$, and $K^2=1,2$

Among log canonical surface singularities, the ones which have a rational homology disk smoothing are the cyclic quotient singularities $\frac{1}{n^2}(1,na-1)$ with gcd$(a,n)=1$, and three distinguished elliptic quotient singularities. We show the existence of smoothable KSBA normal surfaces with $π_1=1$, $p_g=0$, and $K^2=1,2$ for each of these three singularities. We also give a list of new (and old) normal surface singularities in smoothable KSBA surfaces for invariants $π_1=1$, $p_g=0$, and $K^2=1,2,3,4$.

论四维Dehn扭转与Milnor纤维化

On four-dimensional Dehn twists and Milnor fibrations

该研究借助磁单极Floer同调进展计算族Seiberg--Witten不变量，证明极小椭圆奇点链不定辛填充的边界Dehn扭转无限阶非平凡，揭示了四维空间中三类全新的拓扑与辛几何现象。