User jay - MathOverflow

minimal model of $A_\infty$ structure

2013-05-16T16:08:31Z

Hi all,

I am reading about minimal model of $A_\infty$ structure. So far, I find two different ways of the construction, given by Kadeishvili and Kontsevich, Soibelman.

1) The construction of these two minimal model looks quite different, although we know they should be quasi-isomorphic. Is there any direct homomorphism between this two model?

2) Is there any other algebraic construction of minimal model of $A_\infty$ structure?

Thanks!

differential form with empty zero locus

2013-04-12T01:54:52Z

Hi there,

I am considering existence of holomorphic 2-form $\omega$ with empty zero locus on a complex manifold. Beside K3 and holomorphic symplectic manifold, is there any other example on general dimension case?

Generally, if we consider a differential forms (may not closed) with empty zero locus on a closed manifold, what will be the obstruction?

Thanks!

2-cycle of K3 surface

2012-06-29T03:02:22Z

Hi there,

I want to ask about the 2-cycle of K3 surface.

As we know, its betti number $b_2$=22, so there will be 22 2-cycle generators.

Is there any topological way to figure out such cycles direct?

For example, in the best case, if the K3 surface is elliptic and has a global section, can we use combinations of fibre and section to represent all the 22 2-cycles?

Thanks!

Answer by Jay for 2-cycle of K3 surface

2013-02-07T01:13:31Z

i find a paper by Michael B. Schulz and Elliott F. Tammaro: http://arxiv.org/abs/1206.1070

from page. 11, it gives an explicit description these cycles from the point of view of the resolution of $T^4/Z_2$, seems great!

Mirror symmetry for hyperkahler manifold

2013-01-26T00:10:59Z

Hi there,

I have some questions about the mirror symmetry of hyperkahler manifold and K3 surface.

The well-known result said: the mirror symmetry for K3 surface is just given by its hyperkahler rotation.

1) In what sense, the rotation gives the mirror map?

2) Does this means:

if we start from $(M,\omega_I,I)$, here the k3 surface $M$ has a special lagrangian fibration structure with respect to $\omega_I$ and $I$, and also a special lagrangian section. Denote its SYZ mirror by $(M^{mirror},J_{mirror})$. Then exist a (fiberwised) diffeomorphism $\phi: M \rightarrow M^{mirror}$, s.t. $\phi^{*} (J_{mirror}) = K$?

(Here $I,K$ are the standard base of the $S^2$-family of compatible complex structure of the hyperkahler metric on $M$.)

Thanks!

Fixed norm problem for analytic functions

2012-12-10T22:53:51Z

Hi there,

I have the following problems on my hand:

Given arbitrary positive sub-harmonic function $l(x,y)$ on plane, for all analytic functions: $f=a(x,y)+ib(x,y)$ on complex plane, consider the equation:

$|f|^2=l$, i.e. $a^2+b^2=l$

1) for what kind of $l(x,y)$, the equation has $C^2$ solution?

2) for what kind of $l(x,y)$, the number of solutions is finite, after quotient of $S^1$ action: $e^{i\theta}\cdot f$ ?

Thanks!

Answer by Jay for Why is the base of SLAG fibration of CY3 expected to be $S^3$?

2012-11-28T04:17:46Z

There are topological conditions for a manifold to be a base of lagrangian fibration.

For example, in the real 4 dimensional case, if you want a smooth lagrangian fibration without singular fiber, then the base have to be the unique integral affine surface, that is T^2. If you allow certain singular fibres, then the base has to be some special ones, here is a good reference: http://arxiv.org/abs/math/0312165

Similarly, in the real 6 dimensional case, there are affine structure condition on the base.

Answer by Jay for What's dual torus and mirror manifold?

2012-11-28T04:04:34Z

I think you are asking about the complex structure on tangent bundle of integral affine manifold, there we can cook up a simple one, as we did for symplectic structure on cotangent bundle.

Here is a good reference: http://www.math.cuhk.edu.hk/~kwchan/HMS_An.pdf

you can find a quick review of the construction.

Answer by Jay for Deformation of Lagrangian manifolds

2012-11-15T17:17:50Z

generally calculation is like this:

you write down the tubular neighborhood and the exp map there;
you do re-parametrization, such that your symplectic form comes in the "darboux type"

then the section of the normal bundle will be a nearby lagrangian.

there are some simple examples you can do the calculation explicitly, for example: you consider the unit circle in R^2 with the standard symplectic form, then you choose the polar coordinate to write down the exp map in the tubular neighborhood, you will find you need a simple substitution to make the symplectic form in the "darboux type"

radius of tubular neighborhood

2012-10-26T06:13:15Z

Hi there,

Is there any result about the calculation of radius of tubular neighborhood of submanifold inside a Riemannian manifold?

For example, given a simple smooth curve on R^2, what's the radius of its tubular neighborhood? (One upper bound is given by the minimal curvature, but general it is not the radius)

Maybe that is what we can expect: if the curvature of the curve is always decreasing, then the radius of the tubular neighborhood is given by the injective radius of (left) end point, it seems true, right?

For example, they curve is given by: $xy=1, x \in [1,+\infty)$

Thanks!

holomorphic sections on elliptic K3 surface

2012-03-21T15:51:21Z

Hi all,

I want to ask something about the holomorphic sections on elliptic K3:

Is there any obstruction for an ellptic K3 (as an elliptic fibration) to have holomorphic sections? Is that always some number as 240? For example, how about E(2) or Fermat's quartic?

Thanks a lot! :)

question about the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$

2012-02-03T18:15:12Z

Hi all,

I am trying to slove the recursion equation: $x_{n+1}x_{n−1}=x_n^2(1−4x_n)$ in the form of $x_n=x_n(x_1,x_2)$ or $x_n=x_n(c_1,c_2)$, and finally get the limit of the ratio: $\dfrac{x_n}{x_{n+1}}$.

I tried the way of setting: $x_n=f(n)$, and use the 1st order taylor expansion of $f(n+1)$ and $f(n−1)$, then get the first differential equation: $(f')2=4f^3$, the general solution is: $f(x)=\dfrac{4}{(2x−c_1)^2}$, but it does not fit the original recursion equation.

Then I write the equation in the form: $f''=6f^2$, the computer provides some approach by Weierstrass elliptic function (http://www.wolframalpha.com/input/?i=d^2y%2Fdx^2%3D6y^2), but it seems the Weierstrass elliptic function still has no such property as the recursion formula.

Any method I cam apply to get the final limit of ratio, maybe without solving the general soltuions? Thanks!

Answer by Jay for Singular K3 -- mathematical meaning?

2012-02-06T18:49:19Z

by the way, "singular k3 surface" has a quite different meaning it the literature: it refers to a smooth K3 surface with Picard number 20. reference: Shioda, "singular" K3 surfaces

construct the elliptic fibration of elliptic k3 surface

2012-02-06T05:35:25Z

Hi all,

As we know, every elliptic k3 surface admits an elliptic fibration over $P^1$, but generally how do we construct this fibration? For example, how to get such a fibration for Fermat quartic?

Moreover, as we know all (elliptic) k3 surfaces are differential equivalent to each other, does this mean: topologically the elliptic fibration we get for each elliptic fibraion is the same, which is just the torus fibration over $S^2$ with 24 node singularities? Or, the totally space is the same, but different complex data(structure) provides different way or "direction" of projection onto $S^2$, thus induces different type of fibrations?

Thanks!

Answer by Jay for To what extent can I think of a Lagrangian fibration in a symplectic manifold as T*N?

2012-02-04T03:27:23Z

good reference: Duistermaat, J. J. (1980), On global action-angle coordinates. Communications on Pure and Applied Mathematics, 33: 687–706

Comment by Jay

2013-04-12T14:39:41Z

maybe I should say on closed manifold.. thanks MTS

Comment by Jay

2013-03-16T05:51:39Z

thanks for the reference

Comment by Jay

2013-02-06T18:48:06Z

thank you so much! i have not expect such direct approach.. the kahler form gives the deformation of the complex structure directly through Tian-Todorov coordinates. inspiring!

Comment by Jay

2013-01-26T21:22:17Z

In the toric calabi yau case, which is not compact, Leung and his gropu have series of papers on this: arxiv.org/pdf/1006.3830.pdf

Comment by Jay

2013-01-26T21:16:37Z

I find Leung and his group follow the similar idea and get an explicit construction of mirror complex manifold for toric CY by instant corrections of open Gromov-Witten invariants: arxiv.org/pdf/1006.3830.pdf. Do you have any comments on it?

Comment by Jay

2013-01-25T23:26:55Z

thanks a lot! the relation between harmonic function and holomorphic function is essentially important here...

Comment by Jay

2012-12-13T15:47:30Z

Thank you so much! In this case, we have $log l$ is indeed harmonic. But, how to prove it is sufficient condition? I still have no clue of this.. Do you have any reference? Thanks!

Comment by Jay

2012-11-16T17:33:00Z

Hi Spiro, thank you for the reference. By the way, have you get some examples you can do the explicit calculation following there proofs?

Comment by Jay

2012-11-15T17:30:04Z

General do we have a natural map from a manifold to its its mirror? For example, in the $TB$ and $T^{*}B$ case, seems we need a legendre transform to do this, but that required extre data.. Also about K3 case, do we have such map between manifolds?

Comment by Jay

2012-10-26T16:28:38Z

so.. that is an extra restriction on $\theta \in [0, 2\pi]$? :)

Comment by Jay

2012-10-26T15:13:12Z

Thanks a lot! Maybe that is what I want: if the curvature of the curve is always decreasing, then the radius of the tubular neighborhood is given by the injective radius of (left) end point, it seems true, right?

Comment by Jay

2012-06-30T14:39:38Z

we lose some data there?

Comment by Jay

2012-02-08T05:11:08Z

that is great~ thanks! by the way, do you know anything about the existence of sections? maybe just a topological section...

Comment by Jay

2012-02-07T05:35:25Z

Thanks a lot! It is quite helpful!

Comment by Jay

2012-02-06T02:31:15Z

thanks! could you try the way: x_0=1, x_1=s and find how their ratio goes? :)