Archive for October, 2008

Kaisa’s talk today

Posted by mathwww on October 31, 2008

So, as I expected in the last post, I got lost in Professor Li’s talk today, if not completely, so I am not blogging his talk without notes in hand–even with notes, I can still not write down his ideas here because today’s talk contained too much information for me. Instead, Kaisa’s talk today is interesting and comprehensive, but most of which is facts well-known to GW users, so I will just put down something I didn’t know before. (which might still be well-known… but anyway, this is why I set up this blog!)

The example I still remember is the moduli space $\bar{\mathcal{M}_{0,2}}(\mathbb{P}^2,1)$ , where we cook up a $(\mathbb{C}^*)^3$ -action on it, by the action on $\mathbb{P}^2$ . The action on projective space is easy: $(z_1,z_2,z_3): [w_1,w_2,w_3]\rightarrow[z_1w_1,z_2w_2,z_3w_3]$ . The point to transfer this action to the moduli space, as indicated by Kaisa’s graph on board–a triagle with numbered vertices, is to observe that there’s an invariant $\mathbb{P}^1$ (represented by an edge on the graph) connecting 2 fixed points(rep. by vertices on the graph). So each edge will represent a fixed point on $\bar{\mathcal{M}_{0,2}}(\mathbb{P}^2,1)$ where the curve is mapped to the $\mathbb{P}^1$ connecting the 2 fixed points(two of [0,0,1],[0,1,0],[1,0,0]) on $\mathbb{P}^2$ , with the 2 marked points goes to the fixed points. The rest is an application of Atiyah-Bott’s localization formula on these fixed points to localize the integration in the definition of GW-invt to a sum of fractions w.r.t. the fixed points ( $\mathbb{P}^1$ represented by the edges). In this simple case, basically the only thing that shows up is when 2 marked points goes to the same fixed point, in which case we need a nodal curve to really accomodate these 2 marks. In the general situation, as pointed out by Kaisa, there could be more nodals, and multiple covers could be the most dangerous issue. I asked a question about if the collapsing component would always contribute just 1/-1 to the fraction it belongs to, Kaisa said when there are more marks and more components collapse at the same time, it should not be the case. I guess the right question to ask is that if a single collapsing component always contribute 1/-1, and more generally, if the collapsing nodal curve just contain minimal number of marks, do they contribute 1/-1. Anyways, I think I should go to the expression of A-B localization before asking this kind of vague questions.

As I wrote the post, I was thinking of degree 2 case. the first thought is to figure out an action on $\mathbb{P}^2$ so that it maintains some degree 2 curves canonically. But then it came up to my mind that possibly the most natural way to think of this is still to use the same action, but the fixed points inherited becomes multiple curves(double edges) or nodal(union of 2 cuves). So in this way the computation of all GW invts of genus 0 in $\mathbb{P}^2$ will become computations over some graph on the triangle.

This is also connected to my reading course on GW theory with Professor Ciocan-Fontanine, which I intended to learn something about localization at the beginning. So Kaisa’s talk gave a good introduction to the idea of localization method; on the other hand, I am still stuck by the unfortunate chapter of constructing virtual fundamental cycles… Anyone who would like to teach me something about deformation and stacks will be BG!

Well… something just hit me on my head, gosh! I am meeting Ionut tomorrow! I can’t believe I wrote this post…

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(Advertisement) WordPress 上的blog

Posted by mathwww on October 30, 2008

seminararea.wordpress.com

我最近注册的blog。基本上是想弄成一个比较casual 的大家都可以post 自己idea 或者notes 以及互相讨论的地方。我开了一个公共的posting access:

friendofdog3, password:post_math

欢迎大家上去post，虽然即使基本只有我在上面post 我也做好心理准备了……

如果有人想用自己的帐号post 就更欢迎了，在wordpress 上面登记个帐号，然后告诉我让我加你的权限吧。

同时，如果有人对该blog 的名称和副标题有好的建议的请告诉我…… 我还是不怎么会起名字的。

Hope we will have fun there!

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Professor Li’s talk on Tuesday

Posted by mathwww on October 29, 2008

At the very beginning, Professor Li introduced the concept of “(birational) cobordism” of symplectic manifold.

Suppose we have two symplectic manifolds $(X,\omega), (X',\omega')$ of dimension 2n, we say they are cobordism iff exist $(W^{2n+2},\Omega)$ , and an $S^1$ action on which possibly fixed point as the only subset that’s not free. Now on $W$ , the moment map $L: W\rightarrow \mathbb{R}$ has $L^{-1}(\alpha_i)/S^1=X^i, i=1,2$ , where $\alpha_{1,2}$ are the sup and inf, resp.

I asked a question about the relation between this definition of cobordism and linear equivalence because if we have a mapping $f:W\rightarrow\mathbb{C}P^1$ , we shall inherit from this mapping naturally an $S^1$ action, and something like moment map as above, the inverse image of 2 poles would be the cobordism object. Professor Li and Weiyi mentioned something about toric varieties, symplectic cut, which I completely didn’t understand. But one thing is clear: if we have a cobordism above we don’t necessarily get a mapping from $W$ to $\mathbb{C}P^1$ because that requires a section of the “quotient space” $W/S^1$

After several mins’ discussion, the talk went on. Professor Li mentioned the work of Gullemin & Sternberg, in 1989, which states the following result: any symplectic cobordism can be decomposed into 4 kinds of elementary cobordism type, as described below:

1. $\mathbb{Z}$ -linear deformation, which is a deformation of symplectic structure $\omega\rightarrow\omega+tk$ , where $k\in H^2(X,\mathbb{Q})$ . The graph Professor Li drew was like a square(to represent $W$ , mapped to a straight line next to it, to represent the moment map), which I was not sure the upper or lower edge is longer, but they are supposed to be cobordism objects. Also the Morse-Bott index is not specified, probably because the fixed point doesn’t exist in the construction.

2,3. Symplectic blow-up or blow-down: the graph is like square with the “blown-up-side” shorter (because, as Professor Li pointed out, the volume became smaller). For blow up, the M-B index is (2p,0), blow down (0,2q), where 2p,2q are the dimension of the blown-up/down base. M-B index is the index of some fixed point living in $W$ .

4. As Professor Li pointed out, this is the most important case. I forgot the precise statement, but the picture is still a square with fixed point inside, and the M-B index is (2p,2q). From my own understanding, Professor Li was talking about blowing up this point could separate the fixed point into 2, and the index is (0,2q) and (2p,0) resp. so that the cobordism becomes a combination of blow-up/down.

I also guess the theorem of Gulllemin and Sternberg is proved by the decomposition of the index so that to fit them into the deform-blowup/down procedure.

Professor Li then mentioned the result in 4-dimensional, people could prove that corbordism is possible to be made first blow-up+deformation to some $Y$ , then blow-down+deformation, which is algebraically called strong (cobordism?equivalence?). He also mentioned that it is a consequence existence of minimal model in dim-4. Seppi asked if the first blow-down+deformation to MM, then blow-up+deformation also count as strong cobordism, and the answer seems to be no. Seppi also asked about if it is possible to put all these diagrams and arrows to algebraic setting, so that to make everything into a category(could be I misunderstood his comment).

Anyways, Professor Li then compare the situation with Morse function, and pointed out that the strong cobordism is parable to finding a Morse function which the value of critical points are also in the same order as their index, which might suggest that the MM proof of the above theorem could be just to cook up a function out of cobordism through MM then revert it.

At this time things just started to become interesting, unfortunately at least 2 of us have to teach. That makes a continuation of this talk tomorrow. Hopefully I can still blog the seminar tmr: things become interesting always suggest higher risk of being lost…

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Matrix expression for future use

Posted by mathwww on October 29, 2008

If you want to input a matrix in your post, just copy the expression below:

$ \begin{pmatrix}x’\\ y’\\ z’\\ w’\end{pmatrix}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\cos\theta&-\sin\theta\\ 0&0&\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}$

As one adds “latex” right behind the first “$”, the code will turn out:

$\begin{pmatrix}x'\\ y'\\ z'\\ w'\end{pmatrix}=\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&\cos\theta&-\sin\theta\\ 0&0&\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x\\ y\\ z\\ w\end{pmatrix}$