User 喻yuwei - MathOverflow

integral hodge classes of the Calabi-Yau 3-fold

2012-10-24T16:00:12Z

I have been read many papers,But I don"t know a integral hodge class of the calabi-Yau 3-fold is algebraic or non-algebraic?Hope give some help and nice reference. Calabi-Yau 3-fold is a Kahler 3-fold with trival canonical bundle.Is it a open question？

How do Hodge classes for Calabi-Yau 4-folds compare with the classes for tori?

2012-09-25T07:12:00Z

Let $X$ be a Calabi-Yau 4-fold, i.e., a connected 4-dimensional compact Kahler manifold with $K_{X} \cong \mathscr{O}_{X}$ and $h^{i} (X,O_{X} )= 0$ for $0 \lt i \lt 4$. Given a general 4-dimensional Weil torus $T$, one has a Hodge class contained in $H^{2,2}( X )$. I would like to ask whether the following equality holds:

$$ Hg^{2} ( X)_{prim}= Hg^{2}\left ( T \right )_{prim}$$

where $Hg^{2}\left ( X \right )_{prim}$ is a primitive Hodge class, and $Hg^2 ( \star )$ is defined as $ H^4 ( \star ,\mathbb{Q} )\cap H^{2,2} ( \star )$ for $\star = X,T$. If the equality is faulty, I would like to know why. If Hodge class is not unique, how do we get the classification of Hodge classes？

estimate over simply-connected Riemannian manifold with non-positive sectional curvature

2012-08-31T12:02:48Z

Let $M$ be a Complete simply-connected n-dimensional Riemannian manifold with nonpositive curvature, $\Omega $ is a open subset of $M$ ,If $n\geq 4$ Is anyone give a estimate of $\frac{Vol_{n}\left ( \Omega \right )}{Vol_{n-1}\left ( \Omega \right )^{\frac{n}{n-1}}}$ ?,If $\Omega $ replace by $B^{n}\left ( 1 \right )$ ,it is a unit sphere,,estimate?

Isoperimetric inequality from Cartan-Hadamard manifold extend to Alexandrov space

2012-09-01T02:25:21Z

We known that Croke proved a isoperimetric inequality for the four dimensional Cartan-Hadamard manifold,I want to ask extend to Alexandrov space that whether have same isoperimetric inequality ?Whether may be added to some condition,Is $CD\left ( n,k \right )$ (curvature dimension),Is $MCP\left ( n,k \right )$ (measure contraction property) and create to New condition?Of course,Maybe inequality is same.

Line bundle over compact simply-connected topologically regular Alexandrov space

2012-09-01T04:01:34Z

Let $L$ be a line bundle over compact simply-connected topologically regular Alexandrov space $X$ of demension 4 generated by fixed point set of smooth $S^{1}-action$ , $S^{1}-action$ is a section of $L$ ,Garcia proved that $X$ with effective isometric action of $S^{1}$ is Locally smmoth,For locally smooth topologically regular Alexandrov space $X$ , I want to ask the following that whether isomorphism?\[H^{0}(X,L)\simeq H^{1}(X,X^{S^{1}})\] where $X^{S^{1}}$ is smooth $S^{1}-action$ .Do you have good idea?

"implication" question about mean curvature flow

2012-09-01T05:20:31Z

About paper's problems: Let $H$ be a mean curvature vector field. We have the differential inequality

\[ \frac{\partial }{\partial t} \left | H \right |^{2} \leq \Delta \left | H \right |^{2} + \beta \left | H \right |^{2} \]

where $\beta$ is a positive conatant. Putting $f=\left | H \right |^{2}$ and $B(R^{'}) = B_{g_{0}}(x,R^{'})$, and $q\geq 2$, then the above differential inequality apparently implies that

\[ \frac{1}{q} \frac{\partial}{\partial t} \int_{B(R^{'})} f^{q} \eta^{2} d\mu_{t} \leq \int_{B(R^{'})} (\eta ^{2} f^{q-1} \Delta f d\mu _{t} + \beta f^{q} \eta ^{2}) d\mu_{t} + \int_{B(R^{'})} \frac{1}{q} f^{q} \eta^{2} \frac{\partial }{\partial t} d\mu _{t} \]

where $\eta =0, 1$ , and $d\mu _{t}$ satisfies $\frac{\partial }{\partial t} d\mu_{t} = -\left | H \right |^{2} d\mu_{t}$ . My question is: "Why implies?，“How implies”，I don’t understand this implies？

Answer by 喻yuwei for de Rham vs Dolbeault Cohomology

2012-04-27T17:09:41Z

5.The cohomology of the projective space ，see akhil mathew's math bolg:projective space

Answer by 喻yuwei for de Rham vs Dolbeault Cohomology

2012-04-27T16:37:29Z

1.De Rham cohomology is a cohomology of k-forms on complex manifold。 Dolbeault Cohomology is a cohomology of smooth sections of the vector bundle of complex differential forms of degree (p,q) on complex manifold。so Dolbeault Cohomology is subcohomology of the De Rham cohomology.2.when p+q=0

Answer by 喻yuwei for What is Mordell-Weil lattice?

2012-04-27T16:09:11Z

[T Shioda ：Mordell-Weil lattice][1]

[1]: http://www.rkmath.rikkyo.ac.jp/math/shioda/papers/mwl.pdf more Basic，you also see the homepage of Chao Li about Elliptic Surfaces and Mordell-Weil Lattices in harvard university。

Comment by 喻yuwei

2012-10-24T15:25:45Z

Is it a false question?

Comment by 喻yuwei

2012-10-22T12:17:06Z

Please using Latex.

Comment by 喻yuwei

2012-10-16T13:40:29Z

What is the Hodge rhomb?

Comment by 喻yuwei

2012-10-03T04:03:16Z

Should be close it.

Comment by 喻yuwei

2012-09-26T12:38:28Z

It is not a real question.Who close it？

Comment by 喻yuwei

2012-09-26T02:53:06Z

Explain ：the subspace$X^{'}$ of a Weil torus$X$ consists of the Hodge classes,and belongs to$H^{4}\left ( X,\mathbb{Q} \right )$ ,by Hodge decomposition，it contained in $H^{2,2}\left ( X \right )$ ，If it is not Hodge classes，maybe contained in$H^{1,3}\left ( X \right )$ ，$H^{3,1}\left ( X \right )$ ，$H^{0,4}\left ( X \right )$ or$H^{4,0}\left ( X \right )$ .Of course,the Hodge classes of$T$ whether exist in $X$ ,I hope have proof or counterexample .Prove its existence is a very diffficult.

Comment by 喻yuwei

2012-09-25T12:46:55Z

It is not real problem.Who close it

Comment by 喻yuwei

2012-09-25T10:18:40Z

Carnahan,thank you to fix English grammar fault，About the definition of Weil tori，see Voisin's papers and search in internet.

Comment by 喻yuwei

2012-09-24T16:09:48Z

Calabi-Yau theorem says that $Ric\left ( \omega \right )=\Omega $ ,and $X$ has Ricci-flat metric.

Comment by 喻yuwei

2012-09-24T02:20:24Z

Professor Yang，yuwei$\neq $ bruno，doesn't concern me

Comment by 喻yuwei

2012-09-23T18:14:24Z

Only show the canonical bundle $K_{X}$of $X$ is trivial.

Comment by 喻yuwei

2012-09-23T16:13:36Z

It should be $X$ be a compact n-dimensional kahler manifold, has a Kahler metric with vanishing Ricci curvature,vanishing first Chern number 。

Comment by 喻yuwei

2012-09-01T15:50:45Z

Professor Yang， Can you give some details？Thank you.