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!



2-cycle of K3 surface
I think the easiest place to see the $22$ is in a [Kummer surface](http://en.wikipedia.org/wiki/Kummer_surface). Let $A$ be an abelian surface, so topologically $(S^1)^4$. This clearly has $h\_2 = \binom{4}{2} = 6$, and there are obvious topological repreentatives for the $2$-cycles, given by $(S^1)^2$ in $6$ different ways.


Let $X$ be the quotient of $A$ by negation. This has $16$ singular points; the images of the $16$ $2$-torsion points of $A$. Let $Y$ be $X$ blown up at these $16$ points. Then $H\_2(Y)$ 
is (**ADDED** rationally, see below) generated by the pushforwards of the $6$ $2$-cycles from $A$, and the $16$ $\mathbb{P}^1$'s introduced by resolving the singularities. $6+16=22$.


