A Note on Special Kahler Manifolds

A note of semistable Higgs bundles over compact Kähler manifolds

In this note, by using the Yang-Mills-Higgs flow, we show that semistable Higgs bundles with vanishing the first and second Chern numbers over compact Käher manifolds must admit a filtration whose quotients are Hermitian flat Higgs bundles.

Lecture notes on the Kähler-Ricci flow

These lecture notes give an introduction to the Kahler-Ricci flow. They are based on lectures given by the authors at the conference "Analytic Aspects of Complex Algebraic Geometry", held at the Centre International de Rencontres Mathematiques in Luminy, in February 2011.

A note on compact $κ$-solutions of Kähler-Ricci flow

In this paper, we give a complete classification of $κ$-solutions of Kähaler-Ricci flow on compact complex manifolds. Namely, they must be quotients of products of irreducible compact Hermitian symmetric manifolds.

KAWA lecture notes on the Kähler-Ricci flow

These lecture notes are based on a mini-course given by the author at the sixth KAWA Winter School on March 23-26, 2015 at the Centro De Giorgi of Scuola Normale Superiore in Pisa. They provide an introduction to the study of the Kahler-Ricci flow on compact Kahler manifolds, and a detailed exposition of some recent developments.

Note on the paper of Fu and Wong on strictly pseudoconvex domains with Kähler--Einstein Bergman metrics

It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.

Geometric construction of the r-map: from affine special real to special Kähler manifolds

We give an intrinsic definition of (affine very) special real manifolds and realise any such manifold $M$ as a domain in affine space equipped with a metric which is the Hessian of a cubic polynomial. We prove that the tangent bundle $N=TM$ carries a canonical structure of (affine) special Kähler manifold. This gives an intrinsic description of the $r$-map as the map $M\mapsto N=TM$. On the physics side, this map corresponds to the dimensional reduction of rigid vector multiplets from 5 to 4 space-time dimensions. We generalise this construction to the case when $M$ is any Hessian manifold.

A note on the uniformization of gradient Kähler-Ricci solitons

Applying a well known result for attracting fixed points of biholomorphisms \cite{RR, V}, we observe that one immediately obtains the following result: if $(M^n,g)$ is a complete non-compact gradient Kähler-Ricci soliton which is either steady with positive Ricci curvature so that the scalar curvature attains its maximum at some point, or expanding with non-negative Ricci curvature, then $M$ is biholomorphic to $\ce ^n$.

The Kähler-Ricci flow on compact Kähler manifolds

These notes are based on a lecture series given at the Park City Math Institute in the summer of 2013. The notes are intended as a leisurely introduction to the Kähler-Ricci flow on compact Kähler manifolds, aimed at graduate students with some background in differential geometry.

A note on the Kähler and Mori cones of hyperkähler manifolds

In the present paper we prove that, on a hyperkähler manifold, walls of the kähler cone and extremal rays of the Mori cone are determined by all divisors satisfying certain numerical conditions.

A note on Euler number of locally conformally Kähler manifolds

Let $M^{2n}$ be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call $(M,J,θ)$ a $d$(bounded) locally conformally Kähler manifold if the lifted Lee form $\tildeθ$ on the universal covering space of $M$ is $d$(bounded). We shown that if $M^{2n}$ is homeomorphic to a $d$(bounded) LCK manifold, then its Euler number satisfies the inequality $(-1)^{n}χ(M^{2n})\geq$ (resp. $>$) $0$.

A note on para-quaternion manifolds

We show that a para-quaternion nearly Kahler manifold is necessarily a para-quaternion Kahler manifold

Notes on weighted Kähler-Ricci solitons and application to Ricci-flat Kähler cone metrics

This is largely an exposition article that expands the author's talk at the Xiamen International Conference on Geometric Analysis in June 2021. We first survey the author's joint work with Jiyuan Han on the Yau-Tian-Donaldson (YTD) conjecture for $g$-weighted Kähler-Ricci solitons ($g$-solitons). We then review recent works of Apostolov-Canderbank-Jubert-Lahdili which establish a connection between a particular $g$-soliton equation with Ricci-flat Kähler cone metrics (or equivalently Sasaki-Einstein metrics). The main interest in this connection is the transformation of a possibly irregular Sa…

关于李群紧商空间上 $p$-Kähler 结构的一则注记

A note on $p$-Kähler structures on compact quotients of Lie groups

本文研究带不变复结构的李群紧商空间上的 $p$-Kähler 结构，分析四维幂零流形的 $p=2$ 情形，并证明高维近阿贝尔可解流形在 $(n-2)$-Kähler 条件下必为 Kähler。

Some special Kahler metrics on SL(2,C) and their holomorphic quantization

The group SU(2)*SU(2) acts naturally on SL(2,C) by simultaneous right and left multiplication. We study the Kahler metrics invariant under this action using global Kahler potentials. The volume growth and various curvature quantities are then explicitly computable. Examples include metrics of positive, negative and zero Ricci curvature, and the 1-lump metric of the CP^1-model on a sphere. We then look at the holomorphic quantization of these metrics, where some physically satisfactory results on the dimension of the Hilbert space can be obtained. These give rise to an interesting geometrical c…

Special Einstein's equations on Kähler manifolds

This work is devoted to the study of Einstein equations with a special shape of the energy-momentum tensor. Our results continue Stepanov's classification of Riemannian manifolds according to special properties of the energy-momentum tensor to Kähler manifolds. We show that in this case the number of classes reduces.

Special Kähler-Ricci potentials on compact Kähler manifolds

A special Kähler-Ricci potential on a Kähler manifold is any nonconstant $C^\infty$ function $τ$ such that $J(\nablaτ)$ is a Killing vector field and, at every point with $dτ\ne 0$, all nonzero tangent vectors orthogonal to $\nablaτ$ and $J(\nablaτ)$ are eigenvectors of both $\nabla dτ$ and the Ricci tensor. For instance, this is always the case if $τ$ is a nonconstant $C^\infty$ function on a Kähler manifold $(M,g)$ of complex dimension $m>2$ and the metric $\tilde g=g/τ^2$, defined wherever $τ\ne 0$, is Einstein. (When such $τ$ exists, $(M,g)$ may be called {\it almost-everywhere conformally…

Special Lagrangians in nearly Kähler $\mathbb{CP}^3$

This article explores special Lagrangian submanifolds in $\mathbb{CP}^3$, viewed as a nearly Kähler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using $\mathrm{SU}(2)$ moment-type maps. We describe new homogeneous examples, from both perspectives, and classify totally geodesic special Lagrangian submanifolds. We show that every special Lagrangian in $\mathbb{CP}^3$, or the flag manifold $\mathbb{F}_{1,2}(\mathbb{C}^3)$ admitting a symmetry of an $\mathrm{SU}(2)$ subgroup of nearly Kähler automorphisms is automatically homogeneous.

Special complex manifolds

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form ω. This generalises Freed's definition of (affine) special Kähler manifolds. We also define projective versions of all these geometries. Our main result is an extrinsic realisation of all simply connected (affine or projective) special complex, symplectic and Kähler manifolds. We prove that the above three types of sp…

Classification of stationary compact homogeneous special pseudo Kähler manifolds of semisimple groups

The variation of Hodge structure of a Calabi-Yau 3-fold induces a canonical Kähler metric on its Kuranishi moduli space, known as the Weil-Petersson metric. Similarly, special pseudo Kähler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classification of homogeneous special pseudo Kähler manifolds of semisimple groups with compact stabilizer.

Almost Kahler 4-manifolds with J-invariant Ricci tensor and special Weyl tensor

We study the question of integrability of a compatible almost complex structure on a compact symplectic 4-manifold, under various natural assumptions on the curvature of the associated almost Kahler metric.

Quaternionic Kähler metrics associated with special Kähler manifolds

We give an explicit formula for the quaternionic Kähler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic Kähler. A similar explicit formula is given for the analogous (K/K) correspondence between Kähler manifolds endowed with a Hamiltonian Killing vector field. As an example, we apply this formula in the case of an arbitrary conical Kähler manifold.

Special Kahler Metrics on Complex Line Bundles and the Geometry of $K3$-Surfaces

We construct metrics with the holonomy group SU(2) on the tangent bundles of weighted complex projective lines and give a geometric description of the moduli space of special Kahler metrics on a K3-surface in the neighborhood of the flat orbifold $T^4/Z_3$.

Quaternionic Kähler metrics associated to special Kähler manifolds with mutually local variations of BPS structures

We construct a quaternionic-Kähler manifold from a conical special Kähler manifold with a certain type of mutually-local variation of BPS structures. We give global and local explicit formulas for the quaternionic-Kähler metric, and specify under which conditions it is positive-definite. Locally, the metric is a deformation of the 1-loop corrected Ferrara-Sabharval metric obtained via the supergravity c-map. The type of quaternionic-Kähler metrics we obtain are related to work in the physics literature by S. Alexandrov and S. Banerjee, where they discuss the hypermultiplet moduli space metric …

关于拟统计流形上的 Kähler 和 anti-Kähler 结构的注记

Notes concerning Kähler and anti-Kähler structures on quasi-statistical manifolds

本文研究了拟统计流形上的近 Hermite 和近反 Hermite 结构，给出了近复结构可积性条件，并证明了其在特定条件下成为 Kähler、anti-Kähler 及拟 Kähler-Norden 流形的充要或必要条件。

Kähler与非Kähler双曲性的比较研究

A Comparative Study between Kähler and Non-Kähler Hyperbolicity

本文建立了SKT与平衡双曲性之间的联系，并揭示了它们与Gromov意义下的Kähler双曲性之间的关系。

Isolated singularities of affine special Kähler metrics in two dimensions

We prove that there are just two types of isolated singularities of special Kähler metrics in real dimension two provided the associated holomorphic cubic form does not have essential singularities. We also construct examples of such metrics.

Scalar Curvature Bound for Kähler-Ricci Flows over Minimal Manifolds of General Type

In this short note, we use classic computations for Kähler-Ricci flow to achieve scalar curvature bound for minimal manifold of general type.

关于 Kählerian 流形上具有 Berger 型变形 Sasaki 度量的余切丛测地线的注记

Note on geodesics of cotangent bundle with Berger-type deformed Sasaki metric over Kählerian manifold

本文在 Kählerian 流形的余切丛及单位余切丛上引入了 Berger 型变形 Sasaki 度量，研究了其 Levi-Civita 联络，并探讨了相应的测地线。

On Multiplier Hermitian Structures on Compact Kahler Manifolds

Mabuchi introduced multiplier Hermitian structures on compact Kahler manifolds and defined metrics similar to Kahler-Einstein metrics under these structures. In this note we generalize the inequality of Moser-Trudinger type on Kahler-Einstein manifolds to this case.

Special Kahler Manifolds

We give an intrinsic definition of the special geometry which arises in global N=2 supersymmetry in four dimensions. The base of an algebraic integrable system exhibits this geometry, and with an integrality hypothesis any special Kahler manifold is so related to an integrable system. The cotangent bundle of a special Kahler manifold carries a hyperkahler metric. We also define special geometry in supergravity in terms of the special geometry in global supersymmetry.

A note on para-holomorphic Riemannian Einstein manifolds

The aim of this note is the study of Einstein condition for para-holomorphic Riemannian metrics in the para-complex geometry framework. Firstly, we make some general considerations about para-complex Riemannian manifolds (not necessarily para-holomorphic). Next, using an one-to-one correspondence between para-holomorphic Riemannian metrics and para-Kähler Norden metrics, we study the Einstein condition for a para-holomorphic Riemannian metric and the associated real para-Kähler Norden metric on a para-complex manifold. Finally, it is shown that every semi-simple para-complex Lie group inherits…

特殊非Kähler度量——旧与新

Special non-Kähler metrics -- old and new

本文综述多种非Kähler度量的新旧结果，聚焦其在紧复流形上的共存、形变与blow-up行为，并介绍一种构造具有特定性质度量的nilmanifold实例的机制。

关于横截 Kähler 流形上复 Hessian 方程 $L^\infty$ 估计的 PDE 方法注记

A note on the PDE approach to the $L^\infty$ estimates for complex Hessian equations on transverse Kähler manifolds

本文改编 Guo-Phong-Tong 等人的 PDE 方法，证明横截 Kähler 流形上复 Monge-Ampère 方程的 $L^\infty$ 估计，并由此给出 Q-Gorenstein T-簇上 Calabi-Yau 锥度量正则性的纯 PDE 证明。

On non-Kähler compact complex manifolds with balanced and astheno-Kähler metrics

In this note we present, for every $n \geq 4$, a non-Kähler compact complex manifold $X$ of complex dimension $n$ admitting a balanced metric and an astheno-Kähler metric which is in addition $k$-th Gauduchon for any $1\leq k\leq n-1$.

Construction of projective special Kähler manifolds

In this paper we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on $\mathrm{SU}(n,1)/\mathrm{S}(\mathrm{U}(n)\mathrm{U}(1))$. We use this characterisation to classify 4-dimensional projective special Kähler Lie groups.

A note on a Sung-Wang's paper

The purpose of this note is to study the connectedness at infinity of manifold by using the theory of $p$-harmonic functions. We show that if the first eigenvalue $λ_{1,p}$ for the $p$-Laplacian achievies its maximal value on a Kähler manifold or a quaternionic Kähler manifold then such a manifold must be connected at infinity unless it is a topological cylinder with an explicit warped product metric.

Kahler-Ricci flow on blowups along submanifolds

In this short note, we study the behavior of Kaher-Ricci flow on Kahler manifolds which contract divisors to smooth submanifolds. We show that the Kahler potentials are Holder continuous and the flow converges sequentially in Gromov-Hausdorff topology to a compact metric space which is homeomorphic to the base manifold.

A note on mean curvature, Maslov class and symplectic area of Lagrangian immersions

In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a Kähler-Einstein manifold. An immediate consequence is that in Kähler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.

Orbifold regularity of weak Kahler-Einstein metrics

In this note, we prove that any non-collapsing and compact Gromov-Hausdorff limit of Kahler-Einstein manifolds is either smooth or is orbifold outside a subvariety of complex codimension at least 3.

Dynamical construction of Kähler-Einstein metrics on bounded pseudoconvex domains

In this note we shall prove that the complete Kähler-Einstein volume form on a bounded strongly pseudoconvex domain with $C^{\infty}$-boundary is the normalized limit of a sequence of Bergman kernels.

The space of Kähler potentials on an asymptotically locally euclidean Kähler manifold

In this note, we shall prove geodesic convexity of the space of Kähler potentials on an ALE Kähler manifold. This extends earlier results in the compact case proved in the fundamental work of X-X. Chen. We further prove the boundedness from below of the Mabuchi energy, and give an alternative proof for the uniqueness of scalar-flat metrics in the category of ALE Kähler manifold.

On rotationally symmetric Kahler-Ricci solitons

In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf.

An integral inequality for constant scalar curvature metrics on Kähler manifolds

We present in this note a lower bound for the Calabi functional in a given Kähler class. This yields an integral inequality for constant scalar curvature metrics, which can be viewed as a refined version of Yau's Chern number inequality.

关于非负曲率的凯勒轨道折面的评论

Remarks on Kähler orbifolds of non-negative Ricci curvature

本文证明了凯勒轨道折面的非负曲率版本的 Kobayashi 定理，证明了非负曲率的紧凯勒轨道折面在满足一定条件时是单连通的。

Holonomy groups and special geometric structures of pseudo-Kählerian manifolds of index 2

The problem of classification of connected holonomy groups (equivalently of holonomy algebras) for pseudo-Riemannian manifolds is open. The classification of Riemannian holonomy algebras is a classical result. The classification of Lorentzian holonomy algebras was obtained recently. In the present paper weakly-irreducible not irreducible subalgebras of $\su(1,n+1)$ ($n\geq 0$) are classified. Weakly-irreducible not irreducible holonomy algebras of pseudo-Kählerian and special pseudo-Kählerian manifolds are classified. An example of metric for each possible holonomy algebra is given. This gives…

Kähler流形上的特殊Ricci-Hessian方程

Special Ricci-Hessian equations on Kähler manifolds

本文研究Kähler流形上的特殊Ricci-Hessian方程，揭示其在复维数二时的三种类型，并通过Cartan-Kähler定理证明其非标准实现方式。

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.

全纯等距映射到齐性凯勒流形的刚性性质

Rigidity properties of holomorphic isometries into homogeneous Kähler manifolds

本文证明了齐性凯勒流形上全纯等距映射的两个刚性定理，揭示特殊旗流形、平坦空间与齐性有界域在凯勒乘积下的非相对性，并推广了多项已有结果。

On the collapsing rate of the Kähler-Ricci flow with finite-time singularity

This short note studies the collapsing behavior of the Kähler-Ricci flow on a compact Kähler manifold X admitting a holomorphic submersion X -> B where B is a Kähler manifold of lower dimension than X. We give cohomological and curvature conditions under which the fibers collapse at the optimal rate ~(T-t)^{1/2}

Existence of complete Kahler Ricci-flat metrics on crepant resolutions

In this note, we obtain existence results for complete Ricci-flat Kahler metrics on crepant resolutions of singularities of Calabi-Yau varieties. Furthermore, for certain asymptotically flat Calabi-Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial variety.

Endo-Pajitnov 流形上的特殊非 Kähler 度量

Special non-Kähler metrics on Endo-Pajitnov manifolds

本文研究了 Endo 和 Pajitnov 引入的 Inoue 曲面高维类似物的度量与上同调性质，给出了可解流形结构，并在对角化情形下证明了其形式性与不变 de Rham 上同调，同时刻画了多重闭和 astheno-Kähler 度量。

Quaternion-Kahler spaces, hyperkahler cones, and the c-map

Under the action of the c-map, special Kahler manifolds are mapped into a class of quaternion-Kahler spaces. We explicitly construct the corresponding Swann bundle or hyperkahler cone, and determine the hyperkahler potential in terms of the prepotential of the special Kahler geometry.

带拉格朗日纤维化与特殊凯勒几何的全纯辛流形中的全纯拉格朗日子簇

Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with Lagrangian fibrations and special Kahler geometry

该研究针对带拉格朗日纤维化的全纯辛凯勒流形，证明满足相交与投影条件的拉格朗日子簇是其底空间像上的环面纤维化且像为特殊凯勒的，解答了与Kapustin-Witten对偶相关的Hitchin问题。

一般特殊拉格朗日模空间（非凯勒Calabi--Yau三维流形）

Generic special Lagrangian moduli spaces of a non-Kähler Calabi--Yau threefold

针对非凯勒Calabi--Yau三维流形，引入扰动特殊拉格朗日子流形概念，并证明存在余稠密Hermitian度量使模空间为孤立点。