relations between intersection form and Chern classes

Question

Is there exist a 4-manifold which intersection form has the following property $$ (a,a) \neq 0\ \text{if}\ a\neq 0, $$ and the second (or the first) Chern class (for some almost complex stucture) vanishes?

Thanks.

Answer 1

No. The first Chern class (reduced modulo 2) is the characteristic class of the intersection index form. Hence, your manifold has even form. On the other hand, it is also definite and, by Donaldson, must be diagonalizable. (Since you speak about an almost complex structure, I presume that the manifold is smooth.) This is a contradiction.