Rigid geometry on projective varieties

On projective varieties with strictly nef tangent bundles

In this paper, we study smooth complex projective varieties $X$ such that some exterior power $\bigwedge^r T_X$ of the tangent bundle is strictly nef. We prove that such varieties are rationally connected. We also classify the following two cases. If $T_X$ is strictly nef, then $X$ isomorphic to the projective space $\mathrm{P}^n$. If $\bigwedge^2 T_X$ is strictly nef and if $X$ has dimension at least $3$, then $X$ is either isomorphic to $\mathrm{P}^n$ or a quadric $\mathrm{Q}^n$.

CM Stability of Projective varieties

We develop the connection between equivariant completions of algebraic homogeneous spaces of reductive groups and lower bounds for the Mabuchi energy of a polarized manifold over the space of Bergman metrics. We provide a new definition of Tian's CM Polarization and discuss its properties.

Open problems on structure of positively curved projective varieties

We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic sectional curvature, pseudo-effective tangent bundle, and nef anti-canonical divisor.

Projectively flat KLT varieties

In the context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over the smooth locus. Generalising work of Jahnke-Radloff, we show that torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes. An analogous result for varieties with nef normalised cotangent sheaves follows.

关于具有伪有效切层的射影簇的极小模型计划

On the minimal model program for projective varieties with pseudo-effective tangent sheaf

本文发展正规射影簇上伪有效层的理论，并通过运行极小模型计划，证明具有伪有效切层的射影klt簇可分解为Fano簇和Q-阿贝尔簇。

光滑射影簇上的全纯抛物几何

Holomorphic parabolic geometries on smooth projective varieties

本文研究了光滑射影簇上的全纯抛物几何，通过限制数值维度来分类，并揭示了相关叶状结构和纤维结构。

On the varieties of the second row of the split Freudenthal-Tits Magic Square

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e.~the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-di\-men\-sional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type $\mathsf{E}_{6}$ in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone's approach to finite quadric Veronesean varieties. This approach takes projective properties…

Projective embeddings and Lagrangian fibrations of Kummer varieties

In this paper, we study asymptotic behavior of projective embeddings of Kummer varieties given by theta functions, and their amoebas. We prove that a Lagrangian fibration of the Kummer variety can be approximated by moment maps of the projective spaces.

射影结构的边界刚性与非刚性

Boundary rigidity, and non-rigidity, of projective structures

该研究探讨带边界连通解析流形上无挠仿射联络对应射影结构的边界刚性，证明一般射影结构具边界刚性，同时发现存在非刚性射影结构，并以边界局部射影不变量的消失刻画非刚性结构。

On a generic defining function of projective motion in the rigid 6-dimensional h-spaces

In this note we find a generic defining function of projective motion in the 6-dimensional rigid h-space.

Singular projective varieties and quantization

By the quantization condition compact quantizable Kaehler manifolds can be embedded into projective space. In this way they become projective varieties. The quantum Hilbert space of the Berezin-Toeplitz quantization (and of the geometric quantization) is the projective coordinate ring of the embedded manifold. This allows for generalization to the case of singular varieties. The set-up is explained in the first part of the contribution. The second part of the contribution is of tutorial nature. Necessary notions, concepts, and results of algebraic geometry appearing in this approach to quantiz…

Boundaries of Varieties in Projective Manifolds

We give a characterization of the boundaries of holomorphic chains in complex projective space in terms of certain non-linear moment conditions. This extends previous work of the authors and complements results of Dolbeault and Henkin.

On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties

We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety Hom(G, PO(3))//PO(3). The subset U contains all real points of S . As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties.

Complex Monge-Ampère equations on quasi-projective varieties

We introduce generalized Monge-Ampère capacities and use these to study complex Monge-Ampère equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is smooth outside the divisor.