In short, I'm hoping for some reading recommendations.
I'm starting to do some work with Calabi-Yau manifolds, though my prerequisites are fairly minimal in differential geometry. I've taken a relaxed reading course on differential geometry, hopping around Spivak, and I'm taking a full course this semester. I've also taken a reading course on elliptic curves and a full course in algebraic geometry, building up to Riemann-Roch, in case there are some reads that build from that angle.
So, does anyone have any recommendations for good reading material for my situation?
Mostly I'm asking because I've been informed I have to read papers and present (relaxed) talks on the material, and I'd like to not fall flat on my face on the very first talk!
Thanks!
(Also, any recommendations on presenting material you very not familiar with is welcome.)
Best Answer
Excellent, if you continue to pursue arithmetic properties of Calabi-Yau's we should talk in more depth in the future since these are the types of properties I'm interested in. The first thing I'd do is familiarize myself with basics of elliptic curves (sounds like you have that one covered) and then K3 surfaces, because a CY in algebraic geometry is just generalizing the definition of these two things. When thinking about something I find it invaluable to keep going back and figuring out if or why it is true in these low dimensional cases first.
I'll try to give some references at the end, but here are some exercises that will get the definition flowing and give you some concrete examples to work with (try these on your own, but they are certain written down in places if you get stuck):
1) A smooth hypersurface in $\mathbb{P}^3$ is a K3 surface if and only if the hypersurface has degree 4.
2) A little bit more general: A smooth complete intersection in $\mathbb{P}^n$ is a K3 surface if and only if it is one of three types: a) $n=3$ and then it is problem 1, b) $n=4$ and then it is the intersection of a quadric and a cubic, c) $n=5$ and then it is the triple intersection of a quintic
3) Carefully analyze how you did problem 2 because a general principle is at work that should allow you to conjecture which complete intersections are Calabi-Yau for any dimension. Check your conjecture.
Also, the standard way I know how to prove 2 involves Bertini's Theorem, but there is a way around it that you should figure out because for arithmetic purposes you will often not be in characteristic $0$ which is needed for Bertini to work.
There are other examples that you may want to look up, but working through their construction isn't that big of a deal right now like a Kummer K3 which is moding out an abelian surface by multiplication by $-1$ and then resolving the singular points, or the example of a K3 by taking a cyclic branched cover of $\mathbb{P}^2$. Probably figure these out at some point because both construction do generalize to give higher dimensional CY's.
Another exercise that may or may not be worth doing is computing the Betti numbers of a K3 surface which will get you used to cohomological properties or maybe proving that every K3 surface has trivial fundamental group. The reason it might not be worth doing is that these actually don't generalize to higher dimensional CY's (but maybe figuring out why would be useful!).
If you want to spend some time with K3 surfaces (suggested, but not for an inordinate amount of time, lots does not generalize) then David Morrison actually lays out lots of basic properties pretty thoroughly here
The book Modular Calabi-Yau Threefolds by Christian Meyer provides an enormous number of examples and an introduction to what modularity means and basic arithmetic properties. I come back to this one often and it was nice when I was starting out. I'd say Arithmetic Algebraic Geometry has a bunch of overlap with this, though.
Calabi-Yau Varieties and Mirror Symmetry edited by Yui and Lewis is great too, but very much not as introductory. The title initially scared me off as I didn't think it had much arithmetic stuff in it, but the whole second half is arithmetic. I'd say keep this in the back of your head for once you have the basics down.
Lastly, aside from the things I just wrote, I have to say that I've learned far more from jumping in than from any references. It is really hard and slow at first because basically every sentence you think why is that true? What is that map? Where did that come from? It generates a series of exercises to solve, and if you actually keep solving them you learn a ton of things pretty quickly.
Hope this helps. Good luck!