Seiberg-Witten invariants for manifolds with \(b_+ = 1\) and the universal wall crossing formula.

*(English)*Zbl 0959.57029From 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 \(SW_{X,({\mathcal O}_1, {\mathbf H}_0)} ({\mathfrak c})\not \equiv 0\).

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 \(SW_{X,({\mathcal O}_1, {\mathbf H}_0)} ({\mathfrak c})\not \equiv 0\).