Robert Treger
The aim of the note is to extend the uniformization theorem to compact Kahler spaces X with mild singularities and establish a kind of rigidity of their universal coverings. We assume the fundamental group of X is large, residually finite and nonamenable.
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Theorem (uniformization). Let X be a compact Kahler manifold of dimension n with large, residually finite and nonamenable fundamental group. Then its universal covering is a bounded domain in the n-dimensional affine space.
Equality in the Miyaoka-Yau inequality and uniformization of non-positively curved klt pairs
本文研究了紧Kähler klt对在orbifold Miyaoka-Yau不等式中等号成立的条件，证明了此时其orbifold泛覆盖为单位球或仿射空间。
In this paper, with the aim of establishing a structure theorem for a compact Kähler manifold $X$ with semi-positive holomorphic sectional curvature, we study a morphism $ϕ: X \to Y$ to a compact Kähler manifold $Y$ with pseudo-effective canonical bundle. We prove that the morphism $ϕ$ is always smooth (that is, a submersion), the image $Y$ admits a finite etale cover $T \to Y$ by a complex torus $T$, and further that all the fibers are isomorphic when $X$ is projective. Moreover, by applying a modified method to maximal rationally connected fibrations, we show that $X$ is rationally connected…
Non-abelian Hodge correspondence over singular Kähler spaces
本文在奇异凯勒空间上建立了非阿贝尔霍奇对应，并通过调和丛和降维定理扩展了射影klt品种的结果，应用于具有大canonical除数的品种的准统一化定理。
Let $X$ be a non-uniruled compact Kähler space of dimension 3. We show that the group of bimeromorphic automorphisms of $X$ is Jordan. More generally, the same result holds for any compact Kähler space admitting a quasi-minimal model.
We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.
We prove that for any compact quasi-smooth strictly $k$-analytic space $X$ there exist a finite extension $l/k$ and a quasi-étale covering $X'\to X\otimes_kl$ such that $X'$ possesses a strictly semistable formal model. This extends a theorem of U. Hartl to the case of the ground field with a non-discrete valuation.
We prove a version of Temkin's local altered uniformization theorem. We show that for any rig-smooth, quasi-compact and quasi-separated admissible formal $\mathcal{O}_K$-model $\mathfrak{X}$, there is a finite extension $K'/K$ such that $\mathfrak{X}_{\mathcal{O}_{K'}}$ locally admits a rig-étale morphism $g\colon \mathfrak{X}' \to \mathfrak{X}_{\mathcal{O}_{K'}}$ and a rig-isomorphism $h\colon \mathfrak{X}" \to \mathfrak{X}'$ with $\mathfrak{X}'$ being a successive semi-stable curve fibration over $\mathcal{O}_{K'}$ and $\mathfrak{X}"$ being a poly-stable formal $\mathcal{O}_{K'}$-scheme. Mor…
We show that the orbifold fundamental group of an effective compact K{ä}hler orbifold with nef anticanonical bundle has polynomial growth, which generalizes M.P ăun's results for manifolds [P ău97, Theorem 1,Theorem 2]
Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces
本文证明了在既约、局部不可约的紧致Kähler空间上复Monge-Ampère方程有界解的连续性，确认了奇异Kähler-Einstein势的连续性，肯定回答了连续逼近性质的猜想，并解决了具对数终端奇异的紧致Kähler簇上弱Kähler-Ricci流势的连续性问题。
A note on compact Kähler varieties with pseudo-effective tangent sheaf
本文证明具有商奇点且切层伪有效的紧致Kähler流形经有限拟平展覆盖后平坦纤维化为复环面，纤维有理连通，并在klt情形及射影、低维情形下推广。
We extend the results of generic vanishing theory to polarizable real Hodge modules on compact complex tori, and from there to arbitrary compact Kähler manifolds. As applications, we obtain a bimeromorphic characterization of compact complex tori among compact Kähler manifolds, semi-positivity results, and a description of the Leray filtration for maps to tori.
Let $X$ be a compact Kähler manifold and let $(L, φ)$ be a pseudo-effective line bundle on $X$. We first define a notion of numerical dimension of pseudo-effective line bundles with singular metrics, and then discuss the properties of this type numerical dimension. We finally prove a very general Kawamata-Viehweg-Nadel type vanishing theorem on an arbitrary compact Kähler manifold.
In this article we prove a theorem of Ohsawa-Takegoshi type on compact Kähler manifolds. Our arguments follow the "standard" approach for this kind of extension results; however, there are many complications arising from the regularization process of quasi-psh functions on compact Kähler manifolds, and unfortunately we only obtain a particular case of the expected result. We remark that the additional hypothesis we are forced to make are natural, since they are in many situations; we hope to remove them in a near future.
On the fundamental group of the regular part of Fujiki's compact Kahler symplectic orbifolds
本文计算了Fujiki构造的紧Kahler辛轨道流形正则部分的基本群，并确定了其中的不可约辛轨道流形，回答了A.Perego提出的问题。
We prove a theorem of Tits type about automorphism groups for compact Kahler manifolds, which has been conjectured in the paper [KOZ].
In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kähler and non-Kähler complex structures. Such examples were constructed independently by M. Atiyah, A. Blanchard and E. Calabi in the $1950$'s. In the $1980$'s V. Tsanov gave an example of a simply connected manifold that admits both Kähler and non-Kähler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperkähler manifold has the same property.
We construct a geometrically compactified moduli algebraic space of Kahler-Einstein Fano manifolds.
We prove that the Gromov-Hausdorff compactification of the moduli space of Kahler-Einstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tian's theorem on the existence of Kahler-Einstein metrics on smooth Del Pezzo surfaces and classifies the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.
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