# Pre-requisites and references for $K3$ surfaces

I would like to know the "roadmap" to study $K3$ surfaces.

Perhaps, my background might be helpful: I am an undergraduate student, who knows the basics of Differential Geometry, Topology, Complex Analysis, Abstract Algebra. I do not know Sheaf Theory.

My question is: What should I know first before trying to understand the basics of $K3$ surfaces? What are the minimum pre-requisites? Any references that might help me build up a foundation in me?

Please suggest in comments, if you like this question to be improved for more clarity.

• Why are you wanting to study K3 surfaces in particular? Given your background it sounds like you probably don't understand their definition, so how did you decide that that was what you wanted to learn about? Commented Oct 18, 2014 at 22:53

K3-surfaces are non-trivial examples of compact complex surfaces, i.e. manifolds of complex dimension 2. Hence you need some background in the theory of several complex variables or in algebraic geometry. The first topic is the subject of

"Barth, W.; Hulek, K.; Peters, Ch.; van de Ven, A.: Compact complex surfaces."

The second theme is the subject of the classic

"Hartshorne, R.: Algebraic geometry."

For both themes you need sheaf theory and cohomology theory as a prerequisite. You find them in Hartshorne's book. In addition, some background from Riemannian surfaces, i.e. manifolds of complex dimension 1, and curves, i.e. complex spaces of complex dimension 1, is helpful.

If you set up to master K3-surfaces, you will find on your way many examples from surface theory like tori, ruled surfaces, elliptic fibrations or Hopf surfaces. These examples are easier to handle and are interesting by themselves.

In the end, you find the theory of K3-surfaces in Chapter VIII of the above mentioned book on compact complex surfaces.

• The BHP is beautiful but the viewpoint is analytic -- this might even be a good thing, given the OP's background -- so something like Griffiths & Harris or Huybrechts could serve as preparation.
– Hoot
Commented Jan 6, 2015 at 4:04
• Huybrechts' complex geometry is such a great book. I am loving it!
– user166467
Commented Apr 11, 2015 at 17:15

Here is a great reference I found (in case someone might be interested):

http://www-math.sp2mi.univ-poitiers.fr/~sarti/corso_Perego.pdf