A note on the holomorphic invariants of Tian-Zhu

The α-Invariant on Toric Fano Manifolds

The global holomorphic α-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the α-invariant of toric Fano manifolds.

On the modified Futaki invariant of complete intersections in projective spaces

It is known that a necessary condition for the existence of Kähler-Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian-Zhu. In a recent work of Berman-Nyström, it was generalized for (singular) Fano varieties and the notion of algebro-geometric stability of the pair $(M,V)$ of a Fano manifold $M$ and a holomorphic vector field $V$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.

广义希钦-小林对应与韦伊-彼得森流

Generalized Hitchin-Kobayashi correspondence and Weil-Petersson current

本文沿全纯纤维空间推广希钦-小林对应，证明存在带韦伊-彼得森扭曲的广义厄米-爱因斯坦度量，推广了向量丛的宋-田纲领。

Grassmann嵌入的Tian定理与随机截线的退化集

Tian's theorem for Grassmannian embeddings and degeneracy sets of random sections

该论文将Tian定理推广至Grassmann嵌入，证明高次幂线丛截线拉回Chern类收敛于Chern形式幂，并给出随机截线零点分布及退化集电流期望。

A new formula for the energy functionals E_k and its applications

We give a new formula for the energy functionals E_k defined by Chen-Tian, and discuss the relations between these functionals. We also apply our formula to give a new proof of the fact that the holomorphic invariants corresponding to the E_k functionals are equal to the Futaki invariant.

闭 4 维流形上的凯勒形式空间与全纯驯服辛形式空间

The spaces of Kähler and holomorphically tamed symplectic forms on closed 4-manifolds

研究闭 4 维流形上凯勒形式空间的相关性质，受猜想启发还研究全纯驯服辛形式，提出平行问题并关联其与凯勒型辛形式问题，探讨唯一性、连通性和开放性等。

Kahler-Einstein metrics on Fano surfaces

In these notes we give an exposition of a result of G. Tian, which says that a Fano surfaces admits a Kahler-Einstein metric precisely when the Lie algebra of holomorphic vector fields is reductive.

关于曲面上投影包的相对均匀 Yau-Tian-Donaldson 对应关系

Relative uniform Yau--Tian--Donaldson correspondence for projective bundles over a curve

本文研究了曲面投影包的相对均匀 Yau-Tian-Donaldson 对应关系，通过构造兼容测试配置，证明了相对均匀稳定性与极值度量存在性的等价性。

Spin(7) instantons and the Hodge Conjecture for certain abelian four-folds: a modest proposal

The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. I advertise a class of abelian four-folds due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which can in fact be constructed in terms of of smooth line bundles - whose Chern characters are given Hodge classes. An instanton connection on these vector bundles would endow them with a holomorphic structure and thus prove that these classes are algebraic. I use complex multiplicati…

A characterization of complex space forms via Laplace operators

Inspired by the work of Z. Lu and G. Tian \cite{lutian}, in this paper we address the problem of studying those \K\ manifolds satisfying the $Δ$-property, i.e. such that on a neighborhood of each of its points the $k$-th power of the \K Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer $k$ (see below for its definition). We prove two results: 1. if a \K\ manifold satisfies the $Δ$-property then its curvature tensor is parallel; 2. if an Hermitian symmetric space of classical type satisfies the $Δ$-property then it is a complex space form (namely it…

The Riemannian and symplectic geometry of the space of generalized Kähler structures

On a compact complex manifold $(M, J)$ endowed with a holomorphic Poisson tensor $π_J$ and a deRham class $α\in H^2(M, \mathbb R)$, we study the space of generalized Kähler (GK) structures defined by a symplectic form $F\in α$ and whose holomorphic Poisson tensor is $π_J$. We define a notion of generalized Kähler class of such structures, and use the moment map framework of Boulanger and Goto to extend the Calabi program to GK geometry. We obtain generalizations of the Futaki--Mabuchi extremal vector field and Calabi--Lichnerowicz--Matsushima result for the Lie algebra of the group of automorp…

Complex algebraic compactifications of the moduli space of Hermitian-Yang-Mills connections on a projective manifold

In this paper we study the relationship between three compactifications of the moduli space of Hermitian-Yang-Mills connections on a fixed Hermitian vector bundle over a projective algebraic manifold of arbitrary dimension. Via the Donaldson-Uhlenbeck-Yau theorem, this space is analytically isomorphic to the moduli space of stable holomorphic vector bundles, and as such it admits an algebraic compactification by Gieseker-Maruyama semistable torsion-free sheaves. A recent construction due to the first and third authors gives another compactification as a moduli space of slope semistable sheaves…