Alexander Brudnyi
In the paper we prove a factorization theorem for representations of fundamental groups of compact Kähler manifolds ({\em Kähler groups}) into solvable matrix groups. We apply this result to prove that the universal covering of a compact Kähler manifold with a residually solvable fundamental group is holomorphically convex.
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We prove several results on the structure of solvable quotients of fundamental groups of compact Kahler manifolds (Kahler groups).
We survey recent developments which led to the proof of the Benson-Gordon conjecture on Kähler quotients of solvable Lie groups. In addition we prove that the Albanese morphism of a Kähler manifold which is a homotopy torus is a biholomorphic map. The latter result then implies the classification of compact aspherical Kähler manifolds with (virtually) solvable fundamental group up to biholomorphic equivalence. They are all biholomorphic to complex manifolds which are obtained as a quotient of $\bbC^n$ by a discrete group of complex isometries.
We study the non-relativity for two real analytic Kähler manifolds and complex space forms of three types. The first one is a Kähler manifold whose polarization of local Kähler potential is a Nash function in a local coordinate. The second one is the Hartogs domain equpped with two canonical metrics whose polarizations of the Kähler potentials are the diastatic functions.
We study fine properties of quasiplurisubharmonic functions on compact Kähler manifolds. We define and study several intrinsic capacities which characterize pluripolar sets and show that locally pluripolar sets are globally "quasi-pluripolar".
We derive an interpolation version of constrained matrix Li-Yau-Hamilton estimate on Kähler manifolds. As a result, we first get a constrained matrix Li-Yau-Hamilton estimate for heat equation on a Kähler manifold with fixed Kähler metric. Secondly, we get a corresponding estimate for forward conjugate heat equation on Kähler manifolds with time dependent Kähler metrics evolving by Kähler-Ricci flow.
We show the existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler-Einstein metric using the normalized Kähler-Ricci flow.
Recently, Wu-Yau and Tosatti-Yang established the connection between the negativity of holomorphic sectional curvatures and the positivity of canonical bundles for compact Kähler manifolds. In this short note, we give anothe proof of their theorems by using the Kähler-Ricci flow.
We study the global property of local holomorphic isometric mappings from a class of Kahler manifolds into a product of projective algebraic manifolds with induced Fubini-Study metrics, where isometric factors are allowed to be negative.
Compact Kähler contact manifolds
论文证明非射影紧致凯勒接触流形形式为P T_Y，其中Y为紧致凯勒流形。
We study Nevanlinna theory on complete Kähler manifolds. As a consequence of the main result, we prove a defect relation of holomorphic mappings from complete Kähler manifolds of non-positive sectional curvature into complex projective manifolds under certain growth condition.
Upper bounds of the Laplace spectrum on compact Kähler manifolds
本文在紧致Kähler流形上，针对给定Kähler类中的所有度量，证明了每个Laplace特征值存在统一上界，从而肯定回答了Yau提出的问题。
We study many properties concerning weak Kählerianity on compact complex manifolds which admits a holomorphic submersion onto a Kähler or a balanced manifold. We get generalizations of some results of Harvey and Lawson (the Kähler case), Michelson (the balanced case), Popovici (the sG case) and others.
We construct a canonical singular hermitian metric with semipositive curvature current on the canonical line bundle of a compact Kähler manifold with pseudoeffective canonical bundle.
We consider a complex Plateau problem for strongly pseudoconvex contours in non Kähler manifolds. A positive solution in the case of manifolds carrying a pluriclosed Hermitian metric forms is given. For the general case we propose a conjecture.
A characterization of uniruled compact Kähler manifolds
本文将Bost的代数性刻画改编至紧Kähler流形中芽的情形，推广了叶状结构的代数积分准则，并证明紧Kähler流形是直纹的当且仅当其典范线丛不是伪有效的。
Holomorphic one-forms without zeros on Kähler manifolds of Kodaira codimension one
研究了Kähler流形上无零全纯1形式的双射分类，探讨其几何性质与分类方法。
We show a general existence theorem to the complex Monge-Ampère type equation on compact Kähler manifolds.
Kähler Hyperbolicity Modulus for Simply-connected Kähler Hyperbolic manifolds
本文研究完备Kähler流形上的Kähler双曲模量，利用多重次调和函数梯度长度的边界行为建立下界，并应用于有界对称域及带Kähler-Einstein度量或Bergman度量的有界强拟凸域。
In this paper, we construct local and global solutions to the Kähler-Ricci flow from a non-collapsed Kähler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kähler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete Kähler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and m…
We shall use the classical Perron envelope method to show a general existence theorem to degenerate complex Monge-Ampère type equations on compact Kähler manifolds.
We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.
We provide a new cohomological obstruction to the existence of astheno-Kahler metrics, and study relevant examples.
In this work, we study Monge-Ampere equations over closed Kähler manifolds with degenerated cohomology classes. Classic results and arguments in pluripotential theory are generalized a little bit to be applied to our situation.
For every compact Kähler manifold $X$ of algebraic dimension $a(X) = \dim X - 1$, we prove that $X$ has arbitrarily small deformations to some projective manifolds.
Logarithmic Akizuki--Nakano vanishing theorems on weakly pseudoconvex Kähler manifolds
本文针对弱伪凸凯勒流形提出对数阿基祖-纳科诺消失定理，并推广了诺里马atsu在紧凯勒流形上的研究结果。
We study complex non-Kähler manifolds with Hermitian metrics being locally conformal to metrics with special cohomological properties. In particular, we provide examples where the existence of locally conformal holomorphic-tamed structures implies the existence of locally conformal Kähler metrics, too.
A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We also prove that a twistor space (of a compact 4-manifold, a quaternion-Kahler manifold or a Riemannian m anifold) cannot admit an LCK metric, unless it is Kahler.
We study the existence of extremal Kähler metrics on Kähler manifolds. After introducing a notion of relative K-stability for Kähler manifolds, we prove that Kähler manifolds admitting extremal Kähler metrics are relatively K-stable. Along the way, we prove a general $L^p$ lower bound on the Calabi functional involving test configurations and their associated numerical invariants, answering a question of Donaldson. When the Kähler manifold is projective, our definition of relative K-stability is stronger than the usual definition given by Székelyhidi. In particular our result strengthens the k…
Holomorphic Deformations of Compact Kähler Hyperbolic Manifolds
本文研究紧致凯勒双曲流形的形变问题，通过提出凯勒双曲性的修正版本，为探讨 Gromov 经典凯勒双曲性的形变开性迈出第一步。
In this paper, we prove the Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle. The key point of the proof is the estimate for the $L^2$-norm of the scalar curvature along the Kähler-Ricci flow.
A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.
In this paper we study holomorphic vector bundles with singular Hermitian metrics whose curvature are Hermitian matrix currents. We obtain an extension theorem for holomorphic jet sections of nef holomorphic vector bundle on compact Kähler manifolds. Using it we prove that Fano manifolds with strong Griffiths nef tangent bundles are rational homogeneous spaces.
We prove compactification theorems for some complete Kähler manifolds with nonnegative Ricci curvature. Among other things, we prove that a complete noncompact Kähler Ricci flat manifold with maximal volume growth and quadratic curvature decay is a crepant resolution of a normal affine algebraic variety. Furthermore, such affine variety degenerates in two steps to the unique metric tangent cone at infinity.
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