Kähler-Ricci Flow and the Minimal Model Program for Projective Varieties

Kähler-Ricci flow on rational homogeneous varieties

In this work, we study the Kähler-Ricci flow on rational homogeneous varieties exploring the interplay between projective algebraic geometry and representation theory which underlies the classical Borel-Weil theorem. By using elements of representation theory of semisimple Lie groups and Lie algebras, we give an explicit description for all solutions of the homogeneous Kähler-Ricci flow on rational homogeneous varieties. This description enables us to compute explicitly the maximal existence time for any homogeneous solution and obtain explicit upper and lower bounds for several geometric quan…

Moore-Tachikawa 簇：超越对偶性

Moore-Tachikawa Varieties: Beyond Duality

本文推广 Moore-Tachikawa 簇至超凯勒商目标范畴，通过推广配边算子处理缺乏重参数化不变性，进而关联复合类 S 理论的 Drinfeld 中心定义。

Infinite time singularities of the Kähler-Ricci flow

We study the long-time behavior of the Kahler-Ricci flow on compact Kahler manifolds. We give an almost complete classification of the singularity type of the flow at infinity, depending only on the underlying complex structure. If the manifold is of intermediate Kodaira dimension and has semiample canonical bundle, so that it is fibered by Calabi-Yau varieties, we show that parabolic rescalings around any point on a smooth fiber converge smoothly to a unique limit, which is the product of a Ricci-flat metric on the fiber and of a flat metric on Euclidean space. An analogous result holds for c…

The Kähler-Ricci flow on surfaces of positive Kodaira dimension

The existence of Kähler-Einstein metrics on a compact Kähler manifold has been the subject of intensive study over the last few decades, following Yau's solution to Calabi's conjecture. The Ricci flow, introduced by Richard Hamilton has become one of the most powerful tools in geometric analysis. We study the Kähler-Ricci flow on minimal surfaces of Kodaira dimension one and show that the flow collapses and converges to a unique canonical metric on its canonical model. Such a canonical is a generalized Kähler-Einstein metric. Combining the results of Cao, Tsuji, Tian and Zhang, we give a metri…

具有非平凡模的Fano三维流形上的Kähler-Ricci孤立子

Kähler-Ricci solitons on Fano threefolds with non-trivial moduli

本研究发现了具有非平凡模的复杂度2$\bbT$-Fano三维流形上的Kähler-Ricci孤立子，建立加权Abban-Zhuang估计，证明加权K-稳定性与GIT稳定性等价，推广Koiso定理到对数Fano情形，得到首批严格加权K-半稳定Fano簇等新例子。

K-Stability of Fano spherical varieties

We prove a criterion for K-stability of a $\mathbb{Q}$-Fano spherical variety with respect to equivariant special test configurations, in terms of its moment polytope and some combinatorial data associated to the open orbit. Combined with the equivariant version of the Yau-Tian-Donaldson conjecture for Fano manifolds proved by Datar and Székelyhidi, it yields a criterion for the existence of a Kähler-Einstein metric on a spherical Fano manifold. The results hold also for modified K-stability and existence of Kähler-Ricci solitons.

A Construction of Complete Ricci-flat Kähler Manifolds

We consider an extension of the results of S. Bando, R. Kobyashi, G. Tian, and S. T. Yau on the existence of Ricci-flat Kähler metrics on quasi-projective varieties Y=X\D with α[D]=c_1(X), α>1. The requirement that D admit a Kähler-Einstein metric is generalized to the condition that the link S in the normal bundle of D admits a Sasaki-Einstein structure in the Sasaki-cone of the usual Sasaki structure provided the embedding D\subset X satisfies an additional holomorphic condition. If D is a toric variety, then S always admits a Sasaki-Einstein metric. As an application we prove that every sma…

Gromov-Hausdorff limits of Kähler manifolds with Ricci curvature bounded below

We show that non-collapsed Gromov-Hausdorff limits of polarized Kahler manifolds, with Ricci curvature bounded below, are normal projective varieties, and the metric singularities of the limit space are precisely given by a countable union of analytic subvarieties. This extends a fundamental result of Donaldson-Sun, in which 2-sided Ricci curvature bounds were assumed. As a basic ingredient we show that, under lower Ricci curvature bounds, almost Euclidean balls in Kahler manifolds admit good holomorphic coordinates. Further applications are integral bounds for the scalar curvature on balls, a…

Kähler Geometry of Quiver Varieties and Machine Learning

We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli which are compatible with the GIT quotient construction by the general linear group, and show that their Ricci curvatures give a Kähler metric on the moduli. Moreover, we use toric moment maps to construct activation functions, and prove the universal approximation theorem for the multi-variable activation function constructed from the complex projective space.

复三维奇异凯勒空间上的RCD结构

RCD structures on singular Kahler spaces of complex dimension three

本文证明复三维对数终端奇点射影簇上具有有界Nash熵且Ricci曲率下有界的奇异凯勒度量可诱导出与之同胚的紧RCD空间，并由此建立klt奇点的代数、几何与解析结构之间的联系。