Lecture 5. The Symplectic Case, Part II
We will now discuss the Seiberg-Witten invariant on a symplectic manifold X(with b! > 1) for the remaining Spine structures, and explain that it is equal to a
certain count of pseudoholomorphic curves in X.Summary of t he Last Lecture
Since X is symplectic, there is an identificationSx ~ H^2 (X; Z).
An element e E H^2 (X; Z) corresponds to a Spine structure withS+ = EEBK-^1 E
where c 1 (E) = e, and K is the canonical bundle coming from an w-compatible
almost complex structure J. We can regard the Seiberg-Witten invariant as a map
SW: H^2 (X; Z) ~ Z.If we write '¢ = ft( a, /3) with a E C^00 ( E) and /3 E C^00 ( K-^1 E), write A = Ao+ 2a,
and choose μ = rw - iFio, then the Seiberg-Witten equations are
Baa= -8:/3,
(5.1) -~ -
F: = 2(1 - lal^2 + l/31^2 )w - r(a/3-Ci/3).
We showed that any solution of these equations must satisfy the estimates
(5.2)(5.3)
j ( ( 1 -
2
;) IV aal^2 + r(l - lal^2 )^2 ) ~ 2n[w] · e,
J ( ~ IV~/3 1
2+ ~ l/31
4+ ~ 1/31
2
) ~ ~ J IV aal2
.Taking r > 2z, we see from (5.2) that if [w] · e < 0 then there is no solution, and if
[w] · e = 0 then e = 0 and there is a unique solution. We conclude that SW(e) = 0
for [w] · e < 0, and SW(O) = ±1. (Also SW(e) = SW(c - e) where c = c 1 (K).)
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