126 M. HUTCHINGS AND C. H. TAUBES, SEIBERG-WITTEN EQUATIONS
The invariants bi, b^1 are additive under this operation. Note that CP^2 is complex
projective space with its non-complex orientation.)
3.6. Examples and applications
For an example where the Seiberg-Witten invariant is nontrivial and has an inter-
esting application, consider the K3 surface
K3 = (zf + zi + zj + z! = 0) C CP^3.
This 4-manifold has bi = 3, b:_ = 19. On K3, there is a unique Spine structure s
with c 1 (s) = 0. For this s, SWK 3 (s) = ±1. For all other Spine structures, SW= 0.
(This will follow from the computation in the next lecture.)Theorem 3.8(c) implies that the blow up K3#CP^2 has nontrivial Seiberg-
Witten invariants. (In fact if e is the Poincare dual of the exceptional divisor,
and ifs is a Spine structure with c 1 (L) = ±e, then SW(s) = ±1. This can also
be deduced from the computation in the next lecture.) Now Freedman's theorem,
which classifies topological 4-manifolds, says that K3#CP^2 is homeomorphic to
Y = # 3 CP^2 # 20 CP^2 (because the two 4-manifolds have equivalent intersection
forms over Z). But SWy = 0 by Theorem 3.8(d). So we recover the following
theorem of Donaldson:
Theorem 3.9. K3#CP^2 and # 3 CP^2 # 20 CP^2 are not diffeomorphic, even though
they are homeomorphic.
(This theorem was first proved by Simon Donaldson using his celebrated 4-
manifold invariants [4]. See also [5].)
For another application, we will show in the next lecture that if X has a sym-
plectic structure and bi > 1 then there exists a Spine structures with SW(s) = ±1.
This combines with the vanishing theorem 3.8( d) to give an obstruction to the ex-
istence of symplectic forms. For example, we get:
Theorem 3.10. [14] EBmCP^2 does not have a symplectic form when m > l.
(This is true for homotopy theoretic reasons when m is even, so the interesting
case is when m is odd.)