User hkshlzw - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T13:39:29Z http://mathoverflow.net/feeds/user/4437 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67719/is-there-asymptotic-expansion-of-heat-kernel-of-complex-laplacian Is there asymptotic expansion of heat kernel of complex laplacian? HKSHLZW 2011-06-14T02:44:30Z 2011-06-14T02:44:30Z <p>On real Riemannian manifold , the heat kernel of the laplacian have an asymptotic expansion . But on complex manifold , i haven't seen a result like this , i.e. the heat kernel of the Kodaira Laplacian have an asymptotic expansion as the real case . Maybe I know so little , so I want to ask that Is there an asymptotic expansion of the heat kernel of the Kodaira laplacian ?</p> http://mathoverflow.net/questions/65410/how-to-prove-this-version-of-schwartz-lemma how to prove this version of Schwartz lemma ? HKSHLZW 2011-05-19T09:07:04Z 2011-05-19T09:33:19Z <p>I see this theorem in Hans Grauert &amp; Reinhold Remmert's book 'Theory of stein space' , page 190, in chapter 6 . The classical Schwartz lemma is stated as follows , but i don't know how to prove it. Let $E$ , $E'$ be disks centered at the origin in the $w$-plane with radii <code>$0&lt;r'&lt;r$</code>. Let $a:=r'r^{-1}$ . Suppose <code>$h\in {\mathcal{O}}(E)$</code> vanishes of order $e$ at the origin . Then $|h|_{E'}\leq a^{e}|h|_E$ , where $|\cdot|_E$ means the super-norm of functions defined on $E$ .</p> http://mathoverflow.net/questions/65323/flatness-of-coherent-analytic-sheaf flatness of coherent analytic sheaf HKSHLZW 2011-05-18T12:49:30Z 2011-05-18T13:07:21Z <p>I meet a problem like this : given a short exact sequence $0\rightarrow E_1\rightarrow E_2\rightarrow E_3\rightarrow 0$ , where $E_i,i=1,2,3$ are coherent sheaves over a compact complex manifold $X$ . Let $L$ be a holomorphic line bundle over $X$ , $\mathcal{O}_X(L)$ be the associated coherent analytic sheaf , can we get $0\rightarrow E_1\otimes\mathbb{O}_X(L) \rightarrow E_2\otimes\mathcal{O}_X(L) \rightarrow E_3\otimes\mathcal{O}_X(L) \rightarrow 0$ ? THen furthermore for any other coherent analytic sheaf $S$ , can we get $0\rightarrow E_1\otimes S \rightarrow E_2\otimes S \rightarrow E_3\otimes S \rightarrow 0$ ?</p> http://mathoverflow.net/questions/65201/what-is-the-formulation-of-the-precise-weitzenbock-formula What is the formulation of the precise Weitzenbock formula ? HKSHLZW 2011-05-17T03:12:43Z 2011-05-17T03:35:40Z <p>E is a holomorphic vector bundle over a complex manifold. I want to calculate the Laplacian of E-valued (p,q) forms.In P100 of Griffith-Harris' book(trivial line bundle case),they mention a "precise Weitzenbock formula".In the previous calculation,they only give the terms of order 2,what is the formulation of the lower order terms?</p> http://mathoverflow.net/questions/64268/who-can-give-me-a-example-of-coherent-sheaf who can give me a example of coherent sheaf HKSHLZW 2011-05-08T08:41:38Z 2011-05-08T13:28:51Z <p>What are examples of coherent sheaves $\mathfrak{F}$ on a compact complex $n$-fold with $\dim \operatorname{Supp} \mathfrak{F}=p$ , where $0\leq p \leq n$ ? And how can they be described in local coordinates under the equivalence with vector bundles?</p> http://mathoverflow.net/questions/64266/triviality-of-line-bundle-over-complex-manifold/64271#64271 Answer by HKSHLZW for Triviality of line bundle over complex manifold HKSHLZW 2011-05-08T08:54:53Z 2011-05-08T08:54:53Z <p>Given any holomorphic line bundle $L$ on a complex manifold $X$ , then we can get a element in $H^1(X,\mathcal{O}^\ast)$ . Then if it is zero in $H^1(X,\mathcal{O}^\ast)$ , it is trivial .</p> http://mathoverflow.net/questions/64250/how-to-prove-the-following-fact-in-sheaf-cohomology how to prove the following fact in sheaf cohomology ? HKSHLZW 2011-05-08T01:43:30Z 2011-05-08T02:49:02Z <p>Let $X$ be a compact complex n-fold . Then for every coherent sheaf $\mathfrak{F}$ on $X$ , and every holomorphic line bundle $L$ on $X$ , then the dimension of $H^0 (X,\mathfrak{F}\otimes\mathcal{O}_X(L))$ does not depend on $L$ when dim Supp$\mathfrak{F}=0$ .</p> http://mathoverflow.net/questions/57925/how-to-prove-the-relationship-between-pseudoconvexity-and-the-monge-ampere-matrix/61104#61104 Answer by HKSHLZW for how to prove the relationship between pseudoconvexity and the monge-ampere matrix? HKSHLZW 2011-04-09T01:33:43Z 2011-04-09T01:33:43Z <p>Since $\phi$ is always nonzero inside $\Omega$ , so this matrix has precise one negative eigenvalue and n positive eigenvalues is equivalent to $-\partial\bar{\partial}log\phi$ is non-negative , but which means that $-log\phi$ is a plurisubharmonic exhaustion function for the domain $\Omega$ , i.e. $\Omega$ is pseudoconvex.</p> http://mathoverflow.net/questions/57925/how-to-prove-the-relationship-between-pseudoconvexity-and-the-monge-ampere-matrix how to prove the relationship between pseudoconvexity and the monge-ampere matrix? HKSHLZW 2011-03-09T07:16:09Z 2011-04-09T01:33:43Z <p>In several complex variables , to determine the pseudoconvexity of a domain in $C^n$ is very important . There are various criterion to decide whether a domain is pseudoconvex . In particular ,if the domain is defined by a $C^\infty$ defining function $\phi>0$ , then 'Levi pseudoconvex ' is equivalent to the following matrix (which is called 'Monge-Ampere matrix') <code>$$\begin{pmatrix} -\phi &amp; -\partial_\bar{k}\phi \\ -\partial_j\phi &amp; -\partial_{j\bar{k}}^2\phi \end{pmatrix}$$</code> have precise one negative eigenvalue and n positive eigenvalues .</p> <p>So my question is how to prove this ? Anybody knows? Thanks very much!</p> http://mathoverflow.net/questions/56782/how-to-determine-the-pseudoconvexity-of-hartog-domains how to determine the pseudoconvexity of Hartog domains HKSHLZW 2011-02-27T02:27:46Z 2011-02-27T11:05:51Z <p>let $\Omega\subset C^n$ be a domain with $C^\infty$ defining function $-\phi$ and $-\psi$ ,then we consider the set $\bar{\Omega}=(z_1,z_2,z_3)\in C^{n+d_2+d_3}:{|z_2|^2\over \phi(z_1)}+{|z_3|^2\over \psi(z_1) }&lt;1$ , then what is the sufficient and necessary conditions for $\bar{\Omega}$ being pseudoconvex and why ? </p> <p>An a special type of domain in complex Euclidean space , the questions as above have their own meaning , so does there any one studied this type question systematically and any reference ?</p> http://mathoverflow.net/questions/49381/who-can-tell-me-how-to-understand-and-compute-chern-roots who can tell me how to understand and compute Chern roots HKSHLZW 2010-12-14T12:50:25Z 2010-12-14T13:06:47Z <p>As i learn the local index theory , Chern roots appears and i cannot understand what it is and i can't find any references about it .Can anyone tell me something about it and give me some references,also tell me how to compute it ?</p> http://mathoverflow.net/questions/48752/could-you-help-me-understand-an-equation-in-a-paper-by-atiyah-bott could you help me understand an equation in a paper by Atiyah & Bott? HKSHLZW 2010-12-09T12:22:42Z 2010-12-11T02:01:47Z <p>Has anyone read Atiyah and Bott's famous paper "The moment map and equivariant cohomology"?</p> <p>I have some trouble with the equaations appearing between equations (4.18) and (4.19) in page 13 of the original paper. The paper claims $$D\lambda a = D(a-i(X)a\theta) = da-i(X)da \theta + i(X)au,$$ whereas I think that it should be $$D\lambda a=D(a-i(X)a\theta)=da+i(X)da\theta + i(X)au,$$ where the minus sign is due to $\mathcal{L}(X)a=i(X)da+di(X)a = 0$.</p> <p>Can anyone tell me what it should be?</p> <p>Thank you!</p> http://mathoverflow.net/questions/48737/homology-and-cohomology-of-a-quotient-manifold homology and cohomology of a quotient manifold HKSHLZW 2010-12-09T10:02:37Z 2010-12-09T14:17:08Z <p>suppose $M$ is a manifold , $G$ is a lie group (may be finite) ,then let $G$ act on $M$ freely , $N=M/G$ is then a manifold ,so my question is what relations may be between the homology and cohomology of $M$ and $N$ ? In the surface case ,if $G$ is a finite group ,then we can get no differences between the cohomolgy and homology of $M$ and $N$ then the euler number will be the same what will mean that the order of the group must be one ,and so there are no actions of finite groups freely acting on a surface and it is a very beautiful claim.</p> <p>Does there any effective way to compute the cohomology of a quotient space when the action is not free?</p> <p>So let us consider more about the question raised above , references about this question are also welcomed!!</p> http://mathoverflow.net/questions/47400/the-slice-theorem-in-the-symplectic-manifold the slice theorem in the symplectic manifold HKSHLZW 2010-11-26T05:54:40Z 2010-11-26T08:36:14Z <p>as we all know that the slice theorem is very important in symplectic geometry , especially in the proof of marsten-sternberg-weinstein reduction theorem . so I wonder a similar question that does there is a similar theorem when the symplectic manifold have some sigular point ? and i need a original proof of the slice theorem 'Sur certains groupes de transformations de Lie' by Koszul , if you have the electric version of this paper,please send a copy to me wangzhiwei08@gmail.com. </p> http://mathoverflow.net/questions/20373/the-central-issues-in-complex-geometry the central issues in complex geometry HKSHLZW 2010-04-05T09:04:49Z 2010-09-23T06:28:01Z <p>I want to know about people in researching complex (maybe differential) geometry are careing about what currently ? For example ,$L^2$ estimate inspired by Lars Hormander is a very useful tool,and how does this theory be developed currently ? As myself , i like this method very much ,but i don't know which is the next important problem be solved by this method . How far will this method go ? As well , just like the holomorphic morse inequalities , when it is proved by Demailly in 1985,twenty years passed , it seems that during thest twenty years there are no important results comes out in complex geometry ? I'm a beginner in complex geometry, i think that i can't scratch the direction of complex geometry ? I don't know people are careing about what in complex geometry ?Only doing some little questions or leave this field? So this is just the purpose of this question i asked , i want to communicate with all who are interested in this field.</p> http://mathoverflow.net/questions/20922/what-is-the-motivation-of-shimura-variety what is the motivation of Shimura variety? HKSHLZW 2010-04-10T14:07:22Z 2010-04-10T19:54:49Z <p>Tonight, a friend of mine give me a concise introduction to Shimura variety . I only get some first impression of it. I think the hodge structure is a generalization of the cohomology ring of Kaehler manifold or algebraic manifold , and i think of the Shimura variety an anologue of the analytic familly of complex manifolds , just as introduced in Kodaira's book Complex manifolds.And i suspect that there maybe some anologue theorem's as what Kodaira had done by deformation of complex structures . I'm just doing some imagination unboundedless , do not laugh at me !Heh!</p> http://mathoverflow.net/questions/20872/when-can-we-cancel-vector-bundles-from-tensor-products/20921#20921 Answer by HKSHLZW for When can we cancel vector bundles from tensor products? HKSHLZW 2010-04-10T13:50:33Z 2010-04-10T13:50:33Z <p>you may get some indications from Atiyah's book :K-theory.</p> http://mathoverflow.net/questions/19485/a-small-questions-about-hopf-theorem a small questions about hopf theorem HKSHLZW 2010-03-27T04:43:23Z 2010-03-28T00:39:59Z <p>The famous hopf theorem says that a smooth map from a oriented closed dimension p manifold to S^{p} is homotopic if and only if f and g have the same brower degree. To prove the theorem Milnor suggested us three theorems in the book 'topology from the differential view point':</p> <ul> <li><p>Theorem A: any two such homotopic smooth mapping induce the framed cobordant Pontryagin manifold .</p></li> <li><p>Theorem B: If two Pontryagin manifold induced by f and g are frame cobordant, the f and g are homotopic (smooth).</p></li> <li><p>Theorem C: any frame cobordism Pontryagin manifold are induced by some smooth mapping f.</p></li> </ul> <p>First, it is well-know that if f and g is smooth homotopic, then they have the same brower degree.</p> <p>Second, we need to prove that if f and g have the same degree, then they are homotopic. From above three theorems, we only have to prove that f and g have the frame cobordant Pontryagin manifold. Since dim M=p=dim of p-sphere, so the corresponding frame are of 0 dim, i.e. discrete points in M, so if we define sgn(x)=1 or -1 for x in this frame cobordism Pontryagin manifold due to its orientation given by the frame, we can conclude that frame cobordant Pontryain manifold have the same degree(=sum sgn(x)), but i don't know how to prove that if they have the same degree, they are frame cobordant in the particular case of dim 0 ?</p> <p>Note : we have the notations and definitions as in Milnor's book.</p> http://mathoverflow.net/questions/17341/are-all-riemannian-metrics-induced-by-euclidean-metrics-nash-embedding-theorem Are all Riemannian metrics induced by Euclidean metrics? [Nash Embedding Theorem] HKSHLZW 2010-03-07T01:30:06Z 2010-03-07T03:30:32Z <p>Let $M$ be a smooth manifold. We can get a Riemannian metric on $M$ by at least two methods: first by partitions of unity and second by the Whitney embedding theorem: we can embed $M$ into a Euclidean space of sufficiently large dimension, and we thus get a Riemannian metric on $M$ by restricting the Euclidean metric on the ambient space. </p> <p>Can all Riemannian metrics on $M$ be constructed in the second way above?</p> <p>[Edited for for punctuation, grammar and clarity -- PLC]</p> http://mathoverflow.net/questions/17345/the-cech-cohomology-of-the-sheaf-of-germs-of-plurisubharmonic-functions-defined-o the Cech-cohomology of the sheaf of germs of plurisubharmonic functions defined on a domain in C^n HKSHLZW 2010-03-07T01:56:21Z 2010-03-07T02:28:27Z <p>we all know that if we consider the sheaf of germs of a holomorphic functions defined on a domain in C^n,we have too many beautiful theorems characterizing the geometry of the domain by consider the Cech-cohomology of the sheaf.Then i think that plurisubharmonic functions is in some sense a weaker function than holomorphic functions.So we may get some beautiful theorems as the case of holomorphic case, for example if we can proof that for a domain in C^n,the first Cech-cohomology of the sheaf of germs of plurisubharmonic functions vanishes ,we then can choose any good plurisubharmonic functions as we want. What i want to ask is that have you ever considered such a question ,and i don't know whether this is a good question ? I want to hear some suggestions.</p> http://mathoverflow.net/questions/67719/is-there-asymptotic-expansion-of-heat-kernel-of-complex-laplacian Comment by HKSHLZW HKSHLZW 2011-06-14T14:30:44Z 2011-06-14T14:30:44Z Thanks very much! http://mathoverflow.net/questions/65410/how-to-prove-this-version-of-schwartz-lemma Comment by HKSHLZW HKSHLZW 2011-05-19T09:18:03Z 2011-05-19T09:18:03Z I see this theorem in Hans Grauert &amp; Reinhold Remmert's book 'Theory of stein space' , page 190, in chapter 6 . http://mathoverflow.net/questions/65410/how-to-prove-this-version-of-schwartz-lemma Comment by HKSHLZW HKSHLZW 2011-05-19T09:16:47Z 2011-05-19T09:16:47Z Sorry , I really can't see what you have fixed ! I think there may be something wrong with my computer ! http://mathoverflow.net/questions/65410/how-to-prove-this-version-of-schwartz-lemma Comment by HKSHLZW HKSHLZW 2011-05-19T09:14:37Z 2011-05-19T09:14:37Z does there anything wrong in the mathoverflow , why i typed what i want to ask ,but it only demonstrate part of my words ? http://mathoverflow.net/questions/65323/flatness-of-coherent-analytic-sheaf Comment by HKSHLZW HKSHLZW 2011-05-19T06:13:39Z 2011-05-19T06:13:39Z thank you very much ! And how to determine the flatness of a giving coherent sheaf in general , does there exist some kind of obstruction ? http://mathoverflow.net/questions/64250/how-to-prove-the-following-fact-in-sheaf-cohomology/64253#64253 Comment by HKSHLZW HKSHLZW 2011-05-08T09:31:28Z 2011-05-08T09:31:28Z Thank you very much ! http://mathoverflow.net/questions/64250/how-to-prove-the-following-fact-in-sheaf-cohomology/64253#64253 Comment by HKSHLZW HKSHLZW 2011-05-08T09:06:14Z 2011-05-08T09:06:14Z how to get this isomorphism ? http://mathoverflow.net/questions/48997/griffiths-and-harris-reference Comment by HKSHLZW HKSHLZW 2010-12-11T06:14:05Z 2010-12-11T06:14:05Z There is another book 'topology' by Lefschtz also referred in the reference of the zero chapter in GH's book. I think it maybe a good choice! http://mathoverflow.net/questions/48752/could-you-help-me-understand-an-equation-in-a-paper-by-atiyah-bott Comment by HKSHLZW HKSHLZW 2010-12-11T01:05:42Z 2010-12-11T01:05:42Z Sorry for the comment,i only persue the truth but not mend to comment someone needless to say Atiyah-Bott, so if you have the answer or you 're sure of it ,why not show me the correct.By the way , i also thinks Atiyah-Bott are of the most important celebrates in the 20'!! http://mathoverflow.net/questions/20373/the-central-issues-in-complex-geometry/20436#20436 Comment by HKSHLZW HKSHLZW 2010-12-10T04:44:26Z 2010-12-10T04:44:26Z thanks very much! http://mathoverflow.net/questions/20373/the-central-issues-in-complex-geometry/39701#39701 Comment by HKSHLZW HKSHLZW 2010-12-10T04:44:12Z 2010-12-10T04:44:12Z thanks very much! http://mathoverflow.net/questions/48737/homology-and-cohomology-of-a-quotient-manifold/48741#48741 Comment by HKSHLZW HKSHLZW 2010-12-09T12:13:39Z 2010-12-09T12:13:39Z yeah,i have got it ! But it may doesn't anything to the case when the action is not free which is what i want to consider! Any ideas? http://mathoverflow.net/questions/48737/homology-and-cohomology-of-a-quotient-manifold Comment by HKSHLZW HKSHLZW 2010-12-09T10:33:30Z 2010-12-09T10:33:30Z Or, if essentially my assertion about surface is wrong in any way , so what i concern is just the question: can we analys the cohomology of $N$ from the one of the $M$? http://mathoverflow.net/questions/48737/homology-and-cohomology-of-a-quotient-manifold Comment by HKSHLZW HKSHLZW 2010-12-09T10:31:00Z 2010-12-09T10:31:00Z sorry, i didn't know what the wiki is ,then i think it as a good choice . So , if adding a condition that if $N$ is oriented ,so the situation works as i concern works? http://mathoverflow.net/questions/47400/the-slice-theorem-in-the-symplectic-manifold/47408#47408 Comment by HKSHLZW HKSHLZW 2010-11-26T09:25:26Z 2010-11-26T09:25:26Z Yeah , what you said is correct ,but what i want is in the not free case , ie the stratified symplectic space even clearly in the symplectic orbifold case ,does there exists such an analogue slice theorem ?