Quick Update

I will be doing the Mirror Symmetry thing. I had someone “like” the post which I’ll count as a vote for it and someone explicitly vote for it. That’s two whole people! For all I know that is half of my regular readers, so by majority rule I have to do it.

I haven’t gotten around to a post because I’ve been incredibly busy. I’ve been doing my usual research and I’m taking two classes this quarter. In addition to all that I’ve been teaching a “mini-course” in our AG club. So for the next several weeks when I have down time I’ll probably think about what I’m going to say there and/or I’ll be typing up notes for that. I should point out that mirror symmetry has two parts, the derived category side and the Fukaya category side. The derived side is something that is really close to what I do, so it is without a doubt worthwhile to blog about that half, and I plan to soon. I may be a bit sketchier on the side that isn’t as important to me.

I’ll just leave you with a quick taste of what some people mean by “Mirror Symmetry” which is a bit different than Kontsevitch Mirror Symmetry which is what I’ll be explaining at some point. Suppose you have a Calabi-Yau threefold. By this I’ll mean a smooth, projective variety of dimension three over \mathbb{C} with \omega_X\simeq \mathcal{O}_X and H^1(X,\mathcal{O}_X)=H^2(X,\mathcal{O}_X)=0. By general Hodge theory there is a symmetry in the Hodge numbers h^{pq}(X)=\dim_{\mathbb{C}}H^q(X, \Omega^p), namely that h^{pq}(X)=h^{qp}(X). Also, you can use the fact that it is Calabi-Yau to check that all Hodge numbers are completely determined (independently of X) as either 0 or 1 except h^{11} and h^{12}. Fun exercise in Serre duality!

Suppose X is a Calabi-Yau threefold. There is some specified h^{11} and h^{12} (the only two unknown Hodge numbers). A mirror pair for X is another Calabi-Yau threefold with h^{11} and h^{12} swapped. In my brief encounter with Kontsevich Mirror Symmetry (which says something about an equivalence of categories) this will follow as a special case.

Since I’m in the mood I may as well say some things that immediately pop into mind when seeing this as someone that has recently been thinking in the arithmetic world. If we are over an algebraically closed field of characteristic 0, then there is a result that says h^{11}>0. In particular, there cannot be a rigid CY 3-fold if mirror symmetry is true, since h^{12} gives the space of deformations of X. But in positive characteristic there are tons of rigid CY 3-folds! Interesting.

I’ll leave you with that little taste of what is to come.


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