2-cycle of K3 surface 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!
 A: I think the easiest place to see the $22$ is in a 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. $6+16=22$.
A: You could try Aspinwall's paper:
"K3 Surfaces and String Duality" http://arxiv.org/pdf/hep-th/9611137v5.pdf
Sections 2.3 and 2.5 are relevant to your question.
A: i find a paper by Michael B. Schulz and Elliott F. Tammaro: http://arxiv.org/abs/1206.1070
from page. 11, it gives an explicit description these cycles from the point of view of the resolution of $T^4/Z_2$, seems great!
