几何Auslander平面度判据

Geometric Auslander criterion for openness of an algebraic morphism

We give an effective criterion for openness of a morphism of schemes of finite type over a field: Over a normal base of dimension n, failure of openness is detected by a vertical component in the n'th fibred power of the morphism. This is a topological analogue of a criterion for flatness that originates with Auslander.

概型的射影态射平坦性的无穷小判据

Infinifesimal Criterion for Flatness of Projective Morphism of Schemes

推广了著名的（希尔伯特多项式）准则平坦的射影态射的Noether计划的情况下，非约基的态射。

紧凯勒空间上伪有效层的平坦性准则

A flatness criterion for pseudo-effective sheaves on compact Kähler spaces

本文将Höring–Peternell的平坦性准则从射影簇推广至凯勒空间，证明klt紧凯勒空间上第一陈类为零的伪有效层经有限拟平覆盖后自反拉回局部自由且平坦，核心依赖两项新结果。

A fast flatness testing algorithm in characteristic zero

We prove a fast computable criterion that expresses non-flatness in terms of torsion: Let R be a regular algebra of finite type over a field K of characteristic zero and let F be a module finitely generated over an R-algebra of finite type. Given a maximal ideal m in R, let S be the coordinate ring of the blowing-up of Spec(R) at the closed point m. Then F is flat over R localized in m if and only if the tensor product of F with S over R is a torsion-free module over R localized in m. If K is the field of reals or complex numbers, we give a stronger criterion - without the regularity assumptio…

Projective flatness over klt spaces and uniformisation of varieties with nef anti-canonical divisor

We give a criterion for the projectivisation of a reflexive sheaf on a klt space to be induced by a projective representation of the fundamental group of the smooth locus. This criterion is then applied to give a characterisation of finite quotients of projective spaces and Abelian varieties by $\mathbb{Q}$-Chern class (in)equalities and a suitable stability condition. This stability condition is formulated in terms of a naturally defined extension of the tangent sheaf by the structure sheaf. We further examine cases in which this stability condition is satisfied, comparing it to K-semistabili…

迈向更高余维的上田理论

Toward a higher codimensional Ueda theory

上田理论是关于一类拓扑平凡的全纯线丛的光滑超曲面的平坦性准则的理论。我们提出了一个余维2模拟上田的理论。作为应用，我们给出了三维射影空间在$8$点爆破的反典范丛是非半充分的，但允许半正曲率的光滑Hermite度量的一个充分条件.