Compact complex surfaces and constant scalar curvature Kähler metrics

In this article, I prove the following statement: Every compact complex surface with even first Betti number is deformation equivalent to one which admits an extremal Kähler metric. In fact, this extremal Kähler metric can even be taken to have constant scalar curvature in all but two cases: the deformation equivalence classes of the blow-up of \({\mathbb {P}_2}\) at one or two points. The explicit construction of compact complex surfaces with constant scalar curvature Kähler metrics in different deformation equivalence classes is given. The main tool repeatedly applied here is the gluing theorem of C. Arezzo and F. Pacard which states that the blow-up/resolution of a compact manifold/orbifold of discrete type, which admits cscK metrics, still admits cscK metrics.     

Constant Scalar Curvature Kähler Surfaces and Parabolic Polystability

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kähler metric of constant scalar curvature on the blow-up according to Rollin and Singer (J. Eur. Math. Soc., 2004). We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus, we can produce numerous examples of Kähler surfaces of constant scalar curvature with circle or toric symmetry.     

On Kähler manifolds with harmonic Bochner curvature tensor

We prove that every irreducible Kähler manifold with harmonic Bochner curvature tensor and constant scalar curvature is Kähler–Einstein and that every irreducible compact Kähler manifold with harmonic Bochner curvature tensor and negative semi-definite Ricci tensor is Kähler–Einstein.     

Schrödinger flow of maps into symplectic manifolds

The definition of Schrödinger flow is proposed. It is indicated that the flow of ferromagnetic chain is actually Schrödinger flow of maps intoS ², and that there exists a unique local smooth solution for the initial value problem of one-dimensional Schrödinger flow of maps into Kahler manifolds. In case the targets are Kähler manifolds with constant curvature, it is proved that one-dimensional Schrödinger flow admits a unique global smooth solution.     

Cohomogeneity-Three HyperKähler Metrics on Nilpotent Orbits

Let \({\cal O}\) be a nilpotent orbit in ?^? where G is a compact, simple group and ? = Lie(G). It is known that \({\cal O}\) carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov–Kostant–Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when \({\cal O}\) is of cohomogeneity three under the action of G. It is found that such a structure lies on a one-parameter family of hyperKähler metrics with G-invariant Kähler potentials if and only if ? is Sp3, su6, So7, So12 or e7 and otherwise is the unique G-invariant hyperKähler metric with G-invariant Kähler potential.     

Spinc Structures and Scalar Curvature Estimates

In this note, we look at estimates for the scalar curvature of a compact, connected Riemannian manifold Mwhich are related to spin ^c Dirac operators.We show that one may not enlarge a Kähler metric with positiveRicci curvature without making smaller somewhere on M.More generally, if f: N M is an area-nonincreasing map of a certain topological type,then the scalar curvature k of Ncannot be everywhere larger than f.If k f, then N is isometric to M × F, where F possesses a parallel untwisted spinor.We also give explicit upper bounds for min for arbitrary Riemannian metrics on certainsubmanifolds of complex projective space.In certain cases, these estimates are sharp:we give examples where equality is obtained.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

4-dimensional anti-Kähler manifolds and Weyl curvature

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.     

K?hler Manifolds with Almost Non-negative Ricci Curvature

Compact Kähler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kähler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} \), where X _j is a Calabi-Yau manifold, or a hyperKähler manifold, or X _j satisfies H ⁰(X _j , Ω ^p ) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kähler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ∈ > 0, there exists a Kähler structure (J _∈, g _∈) on M such that the volume \({\text{Vol}}_{{g_{ \in } }} {\left( M \right)} < V\), the sectional curvature |K(g _∈)| < Λ², and the Ricci-tensor Ric(g _∈)> ?∈g _∈, where V and Λ are two constants independent of ∈. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} \), where X _i is a Calabi-Yau manifold, or a hyperKähler manifold, or X _i satisfies H ⁰(X _i , Ω ^p ) = {0}, p > 0.     

Almost Kähler metrics of negative scalar curvature on symplectic manifolds

We show that every symplectic manifold of dimension ≥ 4 admits a complete compatible almost Kähler metric of negative scalar curvature. And we discuss the C ⁰-closure of the set of almost Kähler metrics of negative scalar curvature. Some local versions are also proved.     

A closed symplectic four-manifold has almost Kähler metrics of negative scalar curvature

We show that any closed symplectic four-dimensional manifold (M, ω) admits an almost Kähler metric of negative scalar curvature compatible with ω.     

Moving Symplectic Curves in Kähler-Einstein Surfaces

We derive a parabolic equation for the K?hler angle of a real surface evolving under the mean curvature flow in a K?hler-Einstein surface and show that a symplectic curve remains symplectic with the flow. Received March 29, 2000, Revised April 29, 2000, Accepted May 16, 2000     

On length and product of harmonic forms in Kähler geometry

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as Kähler and hyperbolic geometries are concerned. In the second part of the paper, we give algebraic and topological obstructions to the existence of a geometrically 2-formal Kähler metric, at the level of the second cohomology group. A strong interaction with almost Kähler geometry is to be noted. In complex dimension 3, we list all the possible values of the second Betti number of a geometrically 2-formal Kähler metric.     

A note on Chern-Yamabe problem

We propose a flow to study the Chern-Yamabe problem and discuss the long time existence of the flow. In the balanced case we show that the Chern-Yamabe problem is the Euler-Lagrange equation of some functional. The monotonicity of the functional along the flow is derived. We also show that the functional is not bounded from below.     

Kähler Surfaces of Positive Scalar Curvature

We construct Kähler metrics of positive scalar curvature on almost all blown-up ruled surfaces of arbitrary genus. The metrics have an explicit form on ruled surfaces blown up at most twice successively from a minimal model. Our surfaces are generic in the sense that they make up a dense set in the deformations of a given ruled surface.     

Kähler manifolds with quasi-constant holomorphic curvature

The aim of this article is to classify compact Kähler manifolds with quasi-constant holomorphic sectional curvature.     

S6中具有常数K?hler角和常数曲率的极小曲面

本文给出了近Ｋ?ｈｌｅｒ球面Ｓ⁶中具有常数Ｋ?ｈｌｅｒ角和常数曲率的极小曲面的例子,同时证明了两个唯一性定理．     

4-dimensional anti-Kähler manifolds

We show that every 4-dimensional anti-Kähler manifold is Einstein and locally symmetric. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.     

On isometries of Kähler manifolds

Summary We show that an isometry between not locally hyperk?hler, locally irreducible K?hler manifolds is either holomorphic or antiholomorphic.