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Seiberg-Witten invariants for manifolds with $$b_+ = 1$$ and the universal wall crossing formula. (English) Zbl 0959.57029
From the introduction: The purpose of this paper is to give a systematic description of the Seiberg-Witten invariants, which were introduced in [E. Witten, Math. Res. Lett. 1, No. 6, 769-796 (1994; Zbl 0867.57029)], for manifolds with $$b_+=1$$. In this situation, compared to the general case $$b_+>1$$, several new features arise.
One of the main results of this paper is the proof of a universal wall crossing formula. This formula, which generalizes previous results of [E. Witten, loc. cit.; P. B. Kronheimer and T. S. Mrowka, ibid., 797-808 (1994; Zbl 0851.57023)] and [T. J. Li and A. Liu, ibid. 2, No. 6, 797-810 (1995; Zbl 0871.57017)] describes the difference $$SW_{X,({\mathcal O}_1, {\mathbf H}_0)} ({\mathfrak c})(+)- SW_{X,({\mathcal O}_1, {\mathbf H}_0)} ({\mathfrak c})(-)$$ as an abelian $$\text{Spin}^c (4)$$-form.
More precisely, on elements $$\lambda\in \Lambda^r (H_1 (X, \mathbb{Z})/ \text{Tors})$$ with $$0\leq r\leq\min(b_1,w_c)$$, we have: $\bigl[SW_{X, ({\mathcal O}_1, {\mathbf H}_0)} ({\mathfrak c})(+)-SW_{X, ({\mathcal O}_1,{\mathbf H}_0)} ({\mathfrak c})(-)\bigr] (\lambda)=\bigl\langle \lambda\wedge \exp(-u_c), l_{{\mathcal O}_1} \bigr\rangle,$ where $$u_c\in \Lambda^2 (H_1(X,\mathbb{Z})/ \text{Tors})$$ is given by $$u_c(a\wedge b)={1\over 2}\langle a\cup b\cup c,[X] \rangle$$, $$a,b\in H^1(X,\mathbb{Z})$$, and $$l_{{\mathcal O}_1}\in \Lambda^{b_1} H^1(X,\mathbb{Z})$$ represents the orientation $${\mathcal O}_1$$ of $$H^1(X,\mathbb{R})$$. Here $$c$$ is the Chern class of $${\mathfrak c}$$ and $$w_c:={1\over 4}(c^2-3\sigma(X)-2e(X))$$ is the index of $$c$$. This formula has same important consequences, e.g. it shows that Seiberg-Witten invariants of manifolds with positive scalar curvature metrics are essentially topological invariants. According to Witten’s vanishing theorem [E. Witten, loc. cit.], one has $$SW_{X,({\mathcal O}_1,{\mathbf H}_0)}({\mathfrak c})(\pm)=0$$ for at least one element of $$\{\pm\}$$, and the other value is determined by the wall crossing formula.
In the final part of the paper we show how to calculate $$SW_{X, ({\mathcal O}_1, {\mathbf H}_0)}({\mathfrak c}) (\pm)$$ for Kählerian surfaces. The relevant Seiberg-Witten moduli spaces have in this case a purely complex analytic description as Douady spaces of curves representing given homology classes: this description is essentially the Kobayashi-Hitchin correspondence obtained in [C. Okonek and A. Teleman, Int. J. Math. 6, No. 6, 893-910 (1995; Zbl 0846.57013)].
Witten has shown that nontrivial invariants of Kählerian surfaces with $$b_+>1$$ must necessarily have index 0. This is not the case for surfaces with $$b_+=1$$. We show that a Kählerian surface with $$b_+ =1$$ and $$b_1=0$$ has $$SW_{X,({\mathcal O}_1,{\mathbf H}_0)}({\mathfrak c}) (\{\pm\})= \{0, 1\}$$ of $$SW_{X,({\mathcal O}_1, {\mathbf H}_0)}({\mathfrak c}) (\{\pm\})=\{0,-1\}$$ as soon as the index of $${\mathfrak c}$$ is non-negative. For these surfaces the invariants are therefore completely determined by their reductions modulo 2.
There exist examples of 4-manifolds with $$b_+=1$$ which possess – for every prescribed non-negative index – infinitely many classes $${\mathfrak c}$$ of $$\text{Spin}^c(4)$$-structures with $$SW_{X,({\mathcal O}_1, {\mathbf H}_0)} ({\mathfrak c})\not \equiv 0$$.

MSC:
 57R57 Applications of global analysis to structures on manifolds 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R55 Differentiable structures in differential topology 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
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