On Manifolds with trivial logarithmic tangent bundle: The non-Kaehler case

On Varieties with trivial logarithmic tangent bundle

We characterize Kaehler manifolds with trivial logarithmic tangent bundle (with respect to a divisor D) as a class of certain compatifications of complex semi-tori.

A variational approach to complex Monge-Ampere equations

We show that degenerate complex Monge-Ampere equations in a big cohomology class of a compact Kaehler manifold can be solved using a variational method independent of Yau's theorem. Our formulation yields in particular a natural pluricomplex analogue of the classical logarithmic energy of a measure. We also investigate Kaehler-Einstein equations on Fano manifolds. Using continuous geodesics in the closure of the space of Kaehler metrics and Berndtsson's positivity of direct images we extend Ding-Tian's variational characterization and Bando-Mabuchi's uniqueness result to singular Kaehler-Einst…

Homogeneous principal bundles over manifolds with trivial logarithmic tangent bundle

Winkelmann considered compact complex manifolds $X$ equipped with a reduced effective normal crossing divisor $D\, \subset\, X$ such that the logarithmic tangent bundle $TX(-\log D)$ is holomorphically trivial. He characterized them as pairs $(X,\, D)$ admitting a holomorphic action of a complex Lie group $\mathbb G$ satisfying certain conditions \cite{Wi1}, \cite{Wi2}; this $\mathbb G$ is the connected component, containing the identity element, of the group of holomorphic automorphisms of $X$ that preserve $D$. We characterize the homogeneous holomorphic principal $H$--bundles over $X$, wher…

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We…

具有平凡对数切丛的复流形上的对数 Cartan 几何

Logarithmic Cartan geometry on complex manifolds with trivial logarithmic tangent bundle

本文研究了具有平凡对数切丛的紧复流形上的全纯 Cartan 几何，证明了其同构于自同构群作用拉回的充要条件是主丛存在保持几何结构的对数连接。

有向流形上对数导数的引理

Lemma on logarithmic derivative over directed manifolds

本文将Ahlfors对数导数引理推广至有向射影流形的全纯切曲线，提出了AALD与GAALD及其线性系统变换，并以此推导出全纯曲线的第二主定理型结果。