User xiao - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T09:09:56Z http://mathoverflow.net/feeds/user/11850 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69980/give-a-restriction-to-ensure-a-surgery-of-a-balanced-manifold-is-still-balanced/71125#71125 Answer by xiao for Give a restriction to ensure a surgery of a balanced manifold is still balanced. xiao 2011-07-24T13:12:54Z 2011-07-24T14:04:13Z <p>Maybe you mean a complex manifold $M$ of dimension $n$ is balanced iff it admits a hermitian metric $\omega$ satisfying $d\omega^{n-1}=0$.</p> <p>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:</p> <p>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.</p> <p>For the details,you should read their paper.</p> <p>In addition,if you mean "balance" in the Kahler-Einstein problem，the following two papers maybe helpful.</p> <p>a)S.K. DONALDSON:SCALAR CURVATURE AND PROJECTIVE EMBEDDINGS I </p> <p>b）CLAUDIO AREZZO AND FRANK PACARD：BLOWING UP AND DESINGULARIZING CONSTANT SCALAR CURVATURE KAHLER MANIFOLDS</p> http://mathoverflow.net/questions/65810/are-these-two-definitions-of-nef-ness-equivalent-for-moishezon-manifolds Are these two definitions of nef-ness equivalent for Moishezon manifolds? xiao 2011-05-24T03:38:34Z 2011-06-15T02:32:19Z <p>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 $X$.</p> <p>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).</p> <p><strong>Question:</strong> Is this equivalence also true for Moishezon manifolds?</p> <p>I don't know of any counterexamples. If it is not true, could someone give me a counterexample?</p> http://mathoverflow.net/questions/57535/examples-of-non-kahler-surfaces-with-explicit-non-kahler-metric Examples of non-Kahler surfaces with explicit non-Kahler metric xiao 2011-03-06T04:28:21Z 2011-03-07T09:59:29Z <p>Hi,everyone.Can someone give me some examples of non-Kahler surfaces whose complex structure and metric structure are all clear?</p> http://mathoverflow.net/questions/51747/on-pi-1f-omega-with-omega-convex On $\pi_{1}(f(\Omega))$ with $\Omega$ convex xiao 2011-01-11T10:20:56Z 2011-01-11T16:01:16Z <p>Suppose $\Omega\subset R^{n}$ is an open,convex and bounded set,$f:\Omega\to\mathbb{C}$ is a smooth map.</p> <p>My question:</p> <blockquote> <p>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 $f$?</p> </blockquote> <p>What's more,if we do not need $\pi_{1}(f(\Omega))=\lbrace 1 \rbrace$,then comes the following:</p> <blockquote> <p>2)How does $f$ affect $\pi_{1}(f(\Omega))$?</p> </blockquote> http://mathoverflow.net/questions/50635/boundary-behavior-of-kahler-cone-with-curvature-restriction Boundary behavior of Kähler cone with curvature restriction xiao 2010-12-29T08:40:31Z 2010-12-30T09:44:28Z <p>Let $(M,\omega)$ be a compact Kähler manifold. The boundary behavior of Kähler cone is very interesting; however,it's hard to understand.</p> <p>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.</p> <p>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.</p> <p>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.</p> <p>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.</p> <p>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.</p> <p>I hope some expert can give me some insight.</p> http://mathoverflow.net/questions/50493/kolodziejs-acta-paper-the-complex-monge-ampere-equationa-detailed-ploblem Kolodziej's acta paper "the complex monge-ampere equation"——a detailed ploblem xiao 2010-12-27T13:54:37Z 2010-12-29T09:54:29Z <p>Recently,I am reading kolodziej's acta paper,there are some ditails that i do not know clearly.</p> <p>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.</p> <p>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!</p> <p>I hope some expert in this field can help me.Thanks.</p> http://mathoverflow.net/questions/50553/some-non-trivial-and-explicit-shape-of-kahler-cone Some non-trivial and explicit shape of Kähler cone? xiao 2010-12-28T10:31:08Z 2010-12-28T13:12:32Z <p>It may be difficult to give some special and non-trivial examples of Kähler cones.The examples I know are the following:</p> <ol> <li>for complex tori, the Kähler cone is just the set of positive hermitian matrices;</li> <li>for a Kähler manifold $M$ with $h^{1,1}(M)=1$, then the Kähler cone is just $\mathbb{R}_{+}$.</li> </ol> <p>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:</p> <ol> <li>the cone equation or the boundary of the cone is clear;</li> <li>$c_{1}(M)>0$, or the sign of $c_{1}(M)$ is not definite (are there some intresting manifolds of this type?).</li> </ol> <p>Of course, any explicit shape will be welcome.</p> http://mathoverflow.net/questions/65810/are-these-two-definitions-of-nef-ness-equivalent-for-moishezon-manifolds/67821#67821 Comment by xiao xiao 2011-06-17T02:19:10Z 2011-06-17T02:19:10Z Thank you very much.And I think I should take some time to learn some French first... http://mathoverflow.net/questions/51747/on-pi-1f-omega-with-omega-convex/51756#51756 Comment by xiao xiao 2011-01-11T14:42:17Z 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. http://mathoverflow.net/questions/51747/on-pi-1f-omega-with-omega-convex/51756#51756 Comment by xiao xiao 2011-01-11T14:33:43Z 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,$f(\Omega) \subset \mathbb{C}$. http://mathoverflow.net/questions/51747/on-pi-1f-omega-with-omega-convex/51756#51756 Comment by xiao xiao 2011-01-11T14:14:53Z 2011-01-11T14:14:53Z What's more,if $n&gt;2$,I think $f$ can not be injective in intuitution.And I assume $f$ to be smooth,because I think differential topology may help. http://mathoverflow.net/questions/51747/on-pi-1f-omega-with-omega-convex/51756#51756 Comment by xiao xiao 2011-01-11T14:04:31Z 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. http://mathoverflow.net/questions/50080/surgery-in-complex-geometry Comment by xiao xiao 2011-01-02T14:36:13Z 2011-01-02T14:36:13Z Note that we have surgery in Ricci flow pioneered by Perelman &quot;Ricci flow with surgery on three-manifolds&quot;.So when we want to know more about Kahler-Ricci flow,it's unavoidable to consider some surgeries on complex manifolds. http://mathoverflow.net/questions/50635/boundary-behavior-of-kahler-cone-with-curvature-restriction Comment by xiao xiao 2010-12-30T15:09:12Z 2010-12-30T15:09:12Z Dear Wadim Zudilin,I would like to accept your advice. http://mathoverflow.net/questions/50493/kolodziejs-acta-paper-the-complex-monge-ampere-equationa-detailed-ploblem/50499#50499 Comment by xiao xiao 2010-12-29T10:40:29Z 2010-12-29T10:40:29Z Thanks. I should admit that I know so little about geometric measure theory.