Complex and Kaehler structures on compact homogeneous manifolds - their existence, classification and moduli problem

Classification of Kaehler homogeneous manifolds of non-compact dimension two

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup such that $X := G/H$ is Kaehler and the codimension of the top non-vanishing homology group of $X$ with coefficients in $\mathbb Z_2$ is less than or equal to two. We show that $X$ is biholomorphic to a complex homogeneous manifold constructed using well-known basic building blocks, i.e., $\mathbb C, \mathbb C^*$, Cousin groups, and flag manifolds.

Complex and Kaehler structures on compact solvmanifolds

We discuss our recent results on the existence and classification problem of complex and Kaehler structures on compact solvmanifolds. In particular, we determine in this paper all the complex surfaces which are diffeomorphic to compact solvmanifolds (and compact homogeneous manifolds in general).

Kaehler-Einstein submanifolds of the infinite dimensional projective space

This paper consists of two main results. In the first one we describe all Kaehler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kaehler-Einstein metric with negative scalar curvature which admits a Kaehler immersion into the infinite dimensional complex projective space.

The log-term of the disc bundle over a homogeneous Hodge manifold

We show the vanishing of the log-term in the Fefferman expansion of the Bergman kernel of the disk bundle over a compact simply-connected homogeneous Kaehler--Einstein manifold of classical type.

A comprehensive approach to the moduli space of quasi-homogeneous singularities

We study the relationship between singular holomorphic foliations in $(\mathbb{C}^{2},0)$ and their separatrices. Under mild conditions we describe a complete set of analytic invariants characterizing foliations with quasi-homogeneous separatrices. Further, we give the full moduli space of quasi-homogeneous plane curves. This paper has an expository character in order to make it accessible also to non-specialists.

虚拟Morse-Bott指数、配对模空间及其在光滑四维流形拓扑中的应用

Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds

本文研究了圈作用下哈密顿函数的虚拟Morse-Bott指数，证明其与经典Morse-Bott指数一致，并利用Hirzebruch-Riemann-Roch定理计算了所有临界层的虚拟指数，验证了四维光滑流形的Bogomolov-Miyaoka-Yau不等式猜想。

Moduli Spaces of PU(2)-Instantons on Minimal Class VII Surfaces with b_2=1

We describe explicitly the moduli spaces $M^{pst}_g(S,E)$ of polystable holomorphic structures $E$ with $\det E\cong K$ on a rank 2 vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kaehler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, $M^{pst}_g(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely man…

幺模Sasaki与Vaisman李群

Unimodular Sasaki and Vaisman Lie groups

本文延续齐性局部共形Kaehler与Sasaki流形研究，针对幺模李群情形显式确定所有形变，得到幺模Sasaki和Vaisman李群的完整分类，并明确各类单连通幺模Vaisman李群的双全纯类型。

关于拟齐次函数的模空间

On the moduli space of quasi-homogeneous functions

本文研究了$(\mathbb{C}^2,0)$处约化拟齐次函数解析等价芽的模空间与双Lipschitz等价类的关系，证明了具有常数Henry-Parusiński不变量的非退化函数族是解析平凡的，并得出相同不变量下双Lipschitz类数量有限且存在最大配额。

The Kaehler-Einstein metric for some Hartogs domains over bounded symmetric domains

We study the complete Kähler-Einstein metric of a Hartogs domain $\widetilde Ω$, which is obtained by inflation of an irreducible bounded symmetric domain $Ω$, using a power $N^μ$ of the generic norm of $Ω$. The generating function of the Kähler-Einstein metric satisfies a complex Monge-Ampère equation with boundary condition. The domain $\widetilde Ω$ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by $X\in\lbrack0,1[$. This allows to reduce the Monge-Ampère equation to an ordinary differential equation with limit condition. This eq…

Moduli spaces of a family of topologically non quasi-homogeneous functions

We consider a topological class of a germ of complex analytic function in two variables which does not belong to its jacobian ideal. Such a function is not quasi homogeneous. Each element f in this class induces a germ of foliation (df = 0). Proceeding similarly to the homogeneous case and the quasi homogeneous case treated by Genzmer and Paul, we describe the local moduli space of the foliations in this class and give analytic normal forms. We prove also the uniqueness of these normal forms.

The universal Kobayashi-Hitchin correspondence on Hermitian manifolds

We prove a very general Kobayashi-Hitchin correspondence on arbitrary compact Hermitian manifolds. This correspondence refers to moduli spaces of "universal holomorphic oriented pairs". Most of the classical moduli problems in complex geometry (e. g. holomorphic bundles with reductive structure groups, holomorphic pairs, holomorphic Higgs pairs, Witten triples, arbitrary quiver moduli problems) are special cases of this universal classification problem. Our Kobayashi-Hitchin correspondence relates the complex geometric concept "polystable oriented holomorphic pair" to the existence of a reduct…

Extracting Invariants of Isolated Hypersurface Singularities from their Moduli Algebras

We use classical invariant theory to construct invariants of complex graded Gorenstein algebras of finite vector space dimension. As a consequence, we obtain a way of extracting certain numerical invariants of quasi-homogeneous isolated hypersurface singularities from their moduli algebras, which extends an earlier result due to the first author. Furthermore, we conjecture that the invariants so constructed solve the biholomorphic equivalence problem in the homogeneous case. The conjecture is easily verified for binary quartics and ternary cubics. We show that it also holds for binary quintics…