Lecture 4. The Symplectic Case, Part I
Let X be a closed smooth 4-manifold with b~ > 1. Suppose that X has a
symplectic form w (i .e. w is a 2-form, W.v = 0, and w /\ w never vanishes). In this
lecture we will compute the Seiberg-Witten invariant of X for the simplest Spin IC
structures.
4.1. Statement of the theorem
We need the following basic facts about our symplectic 4-manifold:
- X has a canonical orientation, given by w /\ w.
2. There is a canonical Spin IC structure so E S x. Thus we can identify S x '.::::'.
H^2 (X; Z), by sending s 0 to 0 and extending equivariantly. (We will explain
this shortly.)- TJ;ie moduli spaces M have canonical orientations. (This is subtle and we
will not explain it.)
These three facts allow us to regard the Seiberg-Witten invariant as a map
SW: H^2 (X; Z) __, Z.Let K = T^2 •^0 X (defined below) and c = c 1 ( K). Then:
Theorem 4.1. [14, 15] Let X be as described above. Then ,
(a) SW(O) = 1.
(b) SW(c) = ±1.
(c) SW(e) = ± SW(c - e).
(d) If SW(e) =/. 0, then0 :::; [w] · e :::; [w] · c,and if either equality holds then e = 0 or e = c.Part (c) comes from the "charge conjugation invariance" stated in Theorem 3.8(e).
Parts (a) and (c) clearly imply (b). The rest of this lecture will be devoted to
proving (a) and (d).
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