PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces. (English) Zbl 0983.57024
This is the first of a series of two papers whose continuation is [ibid., 135-212 (2001; Zbl 0983.57025), see below].
In these two papers the authors prove Witten’s conjecture relating the Donaldson and Seiberg-Witten series for terms of degree less than or equal to \(c-2\) where \(c=-(7\chi+11\sigma)/4\) and \(\chi\) and \(\sigma\) are the Euler characteristic and signature of the four-manifold. The proof applies to four-manifolds \(X\) with \(b_{1}=0\), \(b^{+}>1\) and which are abundant, SW-simple type and effective.
A four-manifold \(X\) is abundant if there is a hyperbolic sublattice \(H \subset H^{2}(X;\mathbf{Z})\) which is orthogonal to every SW-basic class. In an appendix, the authors prove that simply connected complex algebraic surfaces with \(b^{+} >1\) are always abundant.
A four-manifold \(X\) is effective if the contribution of the SW-strata (in the different levels of the compatification of the \(\text{PU}(2)\) monopole moduli space) to the formula relating Donaldson and Seiberg-Witten series only depends on the Seiberg-Witten invariants. This will be checked to hold (as a conclusion of this and the companion papers) always for the top-level SW-strata. This is a similar condition to the conjecture in [D. Kotschick and J. W. Morgan, J. Differ. Geom. 39, No. 2, 433-456 (1994; Zbl 0828.57013)].
The paper starts with a review of the \(\text{PU}(2)\) monopoles. For this it introduces the concept of \(\text{spin}^{u}\) structure (that generalizes that of \(\text{spin}^{c}\) structure), the configuration space, gauge group and the monopole equations. Uhlenbeck compactness and transversality for PU(2) monopoles is reviewed from earlier work of the authors [PU(2) monopoles. I: Regularity, Uhlenbeck compactness, and transversality, J. Differ. Geom. 49, No. 2, 265-410 (1998; Zbl 0998.57057)].
The PU(2) monopole moduli space is compactified by adding the strata formed by the so-called ideal monopoles, i.e., limits of sequences of monopoles which have bubbled off at some points. The top-level component corresponds to monopoles in the original (main) stratum.
The moduli space contains the subspace of zero section monopoles which are identified with anti-self-dual connections. On the other hand, there is a circle action on the moduli space of PU(2) monopoles whose fixed points are the reducible PU(2) monopoles, which are identified with Seiberg-Witten solutions. The PU(2) monopole moduli space provides a cobordism between the (compactified) moduli space of anti-self-dual connections and a union of SW-strata in different levels of the compactification.
A detailed analysis of the equations at a reducible point via Kuranishi techniques allows to construct a link of the top level Seiberg-Witten moduli space inside the PU(2) monopole moduli space. Since the obstruction bundle in the Kuranishi model is in general non-zero, the PU(2) monopole moduli space must be embedded (locally around SW-strata) into a larger smooth manifold, the thickened moduli space.
Finally, studying the index bundle of the family of elliptic differential operators given by the linearizations of the PU(2) monopole equation near the SW-strata, the authors compute the Chern classes of the virtual normal bundle.
In these two papers the authors prove Witten’s conjecture relating the Donaldson and Seiberg-Witten series for terms of degree less than or equal to \(c-2\) where \(c=-(7\chi+11\sigma)/4\) and \(\chi\) and \(\sigma\) are the Euler characteristic and signature of the four-manifold. The proof applies to four-manifolds \(X\) with \(b_{1}=0\), \(b^{+}>1\) and which are abundant, SW-simple type and effective.
A four-manifold \(X\) is abundant if there is a hyperbolic sublattice \(H \subset H^{2}(X;\mathbf{Z})\) which is orthogonal to every SW-basic class. In an appendix, the authors prove that simply connected complex algebraic surfaces with \(b^{+} >1\) are always abundant.
A four-manifold \(X\) is effective if the contribution of the SW-strata (in the different levels of the compatification of the \(\text{PU}(2)\) monopole moduli space) to the formula relating Donaldson and Seiberg-Witten series only depends on the Seiberg-Witten invariants. This will be checked to hold (as a conclusion of this and the companion papers) always for the top-level SW-strata. This is a similar condition to the conjecture in [D. Kotschick and J. W. Morgan, J. Differ. Geom. 39, No. 2, 433-456 (1994; Zbl 0828.57013)].
The paper starts with a review of the \(\text{PU}(2)\) monopoles. For this it introduces the concept of \(\text{spin}^{u}\) structure (that generalizes that of \(\text{spin}^{c}\) structure), the configuration space, gauge group and the monopole equations. Uhlenbeck compactness and transversality for PU(2) monopoles is reviewed from earlier work of the authors [PU(2) monopoles. I: Regularity, Uhlenbeck compactness, and transversality, J. Differ. Geom. 49, No. 2, 265-410 (1998; Zbl 0998.57057)].
The PU(2) monopole moduli space is compactified by adding the strata formed by the so-called ideal monopoles, i.e., limits of sequences of monopoles which have bubbled off at some points. The top-level component corresponds to monopoles in the original (main) stratum.
The moduli space contains the subspace of zero section monopoles which are identified with anti-self-dual connections. On the other hand, there is a circle action on the moduli space of PU(2) monopoles whose fixed points are the reducible PU(2) monopoles, which are identified with Seiberg-Witten solutions. The PU(2) monopole moduli space provides a cobordism between the (compactified) moduli space of anti-self-dual connections and a union of SW-strata in different levels of the compactification.
A detailed analysis of the equations at a reducible point via Kuranishi techniques allows to construct a link of the top level Seiberg-Witten moduli space inside the PU(2) monopole moduli space. Since the obstruction bundle in the Kuranishi model is in general non-zero, the PU(2) monopole moduli space must be embedded (locally around SW-strata) into a larger smooth manifold, the thickened moduli space.
Finally, studying the index bundle of the family of elliptic differential operators given by the linearizations of the PU(2) monopole equation near the SW-strata, the authors compute the Chern classes of the virtual normal bundle.
Reviewer: Vicente Muñoz (Madrid)
MSC:
| 57R57 | Applications of global analysis to structures on manifolds |
| 57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |
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