User xiao - MathOverflow

Answer by xiao for Give a restriction to ensure a surgery of a balanced manifold is still balanced.

2011-07-24T13:12:54Z

Maybe you mean a complex manifold $M$ of dimension $n$ is balanced iff it admits a hermitian metric $\omega$ satisfying $d\omega^{n-1}=0$.

In the paper:Metric properties of manifolds bimeromorphic to compact Kahler spaces. (JDG.v37.1993.95-121), L. Alessandrini and G. Bassanelli had proved the following result:

Let $M$ and $N$ be compact complex manifolds and $f:N\longrightarrow M$ be a modification,then 1) $M$ is balanced $\Longrightarrow N$ is balanced. 2) $N$ is balanced and satisfies a cohomological condition (it 's called B in the above paper)$\Longrightarrow M$ is balanced.

For the details,you should read their paper.

In addition,if you mean "balance" in the Kahler-Einstein problem，the following two papers maybe helpful.

a)S.K. DONALDSON:SCALAR CURVATURE AND PROJECTIVE EMBEDDINGS I

b）CLAUDIO AREZZO AND FRANK PACARD：BLOWING UP AND DESINGULARIZING CONSTANT SCALAR CURVATURE KAHLER MANIFOLDS

Are these two definitions of nef-ness equivalent for Moishezon manifolds?

2011-05-24T03:38:34Z

Recently, I have been learning about nef line bundles. I know that when $X$ is projective or Moishezon, a line bundle $L$ over $X$ is said to be nef iff $$L.C=\int_{C}c_{1}(L)\ge 0$$ for every curve $C$ in $X$.

Demailly gave a definition of nefness that works on an arbitrary compact complex manifold, i.e., a line bundle $L$ over $X$ is said to be nef if for every $\varepsilon >0$ there exists a smooth hermitian metric $h_{\varepsilon}$ on $L$ such that its curvature $\Theta_{h_{\varepsilon}}(L)\ge -\varepsilon\omega$. For projective manifolds, Demailly's definition coincides with the above one given by integration (this is an easy consequence of Seshadri's ampleness criterion).

Question: Is this equivalence also true for Moishezon manifolds?

I don't know of any counterexamples. If it is not true, could someone give me a counterexample?

Examples of non-Kahler surfaces with explicit non-Kahler metric

2011-03-06T04:28:21Z

Hi,everyone.Can someone give me some examples of non-Kahler surfaces whose complex structure and metric structure are all clear?

On $\pi_{1}(f(\Omega))$ with $\Omega$ convex

2011-01-11T10:20:56Z

Suppose $\Omega\subset R^{n}$ is an open,convex and bounded set,$f:\Omega\to\mathbb{C}$ is a smooth map.

My question:

1)when $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$? Or in order to make $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$, whether there is some non-trivial restrictions on $f$?

What's more,if we do not need $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$,then comes the following:

2)How does $f$ affect $\pi_{1}(f(\Omega))$?

Boundary behavior of Kähler cone with curvature restriction

2010-12-29T08:40:31Z

Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand.

A fundamental result is due to Demailly and Paun: they proved that every nef and big class $\alpha$ contains a Kaehler current $T$. Moreover, $T$ is smooth outside some subvariety. Note that here 'big' means its mass on the manifold is positive, and this condition is crucial in their proof.

In intuition, cuvature restriction will effect the boundary behavior.For example,if $(M,\omega)$ has positive sectional curvature (Frenkel's conjecture solved by Yau and Siu),then the cone is just $R_{+}$.So in that case,the boundary is zero. Recently, Daming Wu, Fangyang Zheng and S.T. Yau proved a result which says that every nef class (without big restriction) contains a smooth non-negtive (1,1) form under a strong condition of curvature. They proved their theorem by implicit function theorem.

In order to understand Kähler cone's boundary, we can use complex Monge-Ampère equation as a tool. Indeed,it's a family of equations with the Kähler metric changing.

I interpret this method as choosing a better approximating sequence of metrics. As I know when we deal with a single complex Monge-Ampère equation, the curvature condition does not play a very important role, e.g., Yau's 1978 paper on Calabi's conjecture.

I do believe that cuvature restriction will effect Kähler cone's boundary, then my problem is how cuvature restriction comes into that family of complex Monge-Ampère equations.

I hope some expert can give me some insight.

Kolodziej's acta paper "the complex monge-ampere equation"——a detailed ploblem

2010-12-27T13:54:37Z

Recently,I am reading kolodziej's acta paper,there are some ditails that i do not know clearly.

In the top line of page 99,"it's no restriction to assume that for each s we have $\nu(\cup_{I\in{B_s}}\partial I)=0$",i do not know why.Here $\nu=(dd^{c}v)^n$ and $v\in PSH(\Omega)\cap L^{\infty}$,$\partial I$ is the boundary of some cube.

From Kolodziej's view,a poriori $\nu(\cup_{I\in{B_s}}\partial I)$ may not be zero.However,i think $\partial I$ can be seen as a part of a pluripolar set,then according to Bedford and Taylor's reasult,we know the above monge-ampere measure concentrates no mass on $\partial I$.So we get $\nu(\cup_{I\in{B_s}}\partial I)=0$,and it should not be a assumption!

I hope some expert in this field can help me.Thanks.

Some non-trivial and explicit shape of Kähler cone?

2010-12-28T10:31:08Z

It may be difficult to give some special and non-trivial examples of Kähler cones.The examples I know are the following:

for complex tori, the Kähler cone is just the set of positive hermitian matrices;
for a Kähler manifold $M$ with $h^{1,1}(M)=1$, then the Kähler cone is just $\mathbb{R}_{+}$.

Thus in order to give a non-trivial example, firstly we must require $h^{1,1}(M)>1$. I would like to see an example such that:

the cone equation or the boundary of the cone is clear;
$c_{1}(M)>0$, or the sign of $c_{1}(M)$ is not definite (are there some intresting manifolds of this type?).

Of course, any explicit shape will be welcome.

Comment by xiao

2011-06-17T02:19:10Z

Thank you very much.And I think I should take some time to learn some French first...

Comment by xiao

2011-01-11T14:42:17Z

What I was considering is the image of simply connectted space.And I think the above is the simplest and non-trivial case.

Comment by xiao

2011-01-11T14:33:43Z

Dear Thomas Rot: I know invariance of domain holds in all dimensions.However,it is of crucial importance that $\Omega$ and $f(\Omega)$ are contained in Euclidean space of the same dimension.But in my question,$f(\Omega) \subset \mathbb{C}$.

Comment by xiao

2011-01-11T14:14:53Z

What's more,if $n>2$,I think $f$ can not be injective in intuitution.And I assume $f$ to be smooth,because I think differential topology may help.

Comment by xiao

2011-01-11T14:04:31Z

Dear Thomas Rot: In order to apply invariance of domain,$\Omega$ must be a domain in $R^{2}$.However,it is not so in my question.

Comment by xiao

2011-01-02T14:36:13Z

Note that we have surgery in Ricci flow pioneered by Perelman "Ricci flow with surgery on three-manifolds".So when we want to know more about Kahler-Ricci flow,it's unavoidable to consider some surgeries on complex manifolds.

Comment by xiao

2010-12-30T15:09:12Z

Dear Wadim Zudilin,I would like to accept your advice.

Comment by xiao

2010-12-29T10:40:29Z

Thanks. I should admit that I know so little about geometric measure theory.