Generalized Kummer surfaces and differentiable invariants of Noether-Horikawa surfaces. I. (English) Zbl 0872.14031
de Bartolomeis, Paolo (ed.) et al., Manifolds and geometry. Proceedings of a conference, held in Pisa, Italy, September 1993. Cambridge: Cambridge University Press. Symp. Math. 36, 132-177 (1996).
A surface of general type with canonical bundle \(K\) is said of type \(N\)-\(H\) if \(K^2= 2\chi-6\) where \(\chi\) is the Euler-Poincaré characteristic. For such surfaces, E. Horikawa [Ann. Math., II. Ser. 104, 357-387 (1976; Zbl 0339.14024)] showed that its moduli space is disconnected if \(16|K^2\); the 2 components of the moduli space are called of type \(C\) (connected branch locus for the canonical map) and of type \(N\) (non connected branch locus). Following M. H. Freedman [J. Differ. Geom. 17, 357-453 (1982; Zbl 0528.57011)] \(N\)-\(H\) surfaces of type \(C\) and \(N\) are homeomorphic iff \(16|K^2\). In this paper, the author is motivated by the following question:
If \(16|K^2\), are \(N\)-\(H\) surfaces of type \(N\) and \(C\) diffeomorphic?
Now a construction of P. B. Kronheimer [Duke Math. J. 64, No. 2, 229-241 (1991; Zbl 0754.57015)] showed that, for a given compact oriented differentiable 4-manifold, a principal \(SO(3)\) topological bundle \(P\) is completely determined by the Pontrjagin number \(p_1(P)\) and by the Stiefel-Whitney class \(w_2 (P)= :w\) in \(H^2(S, \mathbb{Z}/2)\)\((w^2 \equiv p_1 \bmod 4 \mathbb{Z})\). Now if one fixes \(P\), for a generic Riemannian metric \(g\), the dimension of the moduli space \({\mathcal M} (P,g)\) of the anti self dual connections modulo gauge equivalence is equal to \(-2p_1-6\). The interesting case occurs when such a dimension equals zero, in which case a differential invariant \(q(S,P)\) is well defined as the number of points of \({\mathcal M} (P,g)\) counted with multiplicity. With this in mind, the author answers the above question by the following result:
Main theorem: Let \(X\) be a generalized Kummer surface of type \((6,2m)\), i.e. \(X\) is a singular surface which can be realized as the double cover of \(\mathbb{P}_1 \times \mathbb{P}_1\) branched over the union of 6 vertical lines and \(2m\) horizontal lines, where \(m\) is an integer \(\geq 2\), and let \(S\) be the \(N\)-\(H\) surface which is the minimal resolution of singularities of \(X\).
Then there are certain classes \(w\in H^2 (S,Z/2)\) with \(w^2\equiv -1 +2m \pmod 4\), such that, for a bundle \(P\) with Stiefel-Whitney class \(w_2(P) -w\) and Pontrjagin class \(p_1(P)= -3\chi= -6m+3\), the differential invariant \(q(S,P)\) equals \(2^{2(m-2)}\).
For the entire collection see [Zbl 0840.00037].
If \(16|K^2\), are \(N\)-\(H\) surfaces of type \(N\) and \(C\) diffeomorphic?
Now a construction of P. B. Kronheimer [Duke Math. J. 64, No. 2, 229-241 (1991; Zbl 0754.57015)] showed that, for a given compact oriented differentiable 4-manifold, a principal \(SO(3)\) topological bundle \(P\) is completely determined by the Pontrjagin number \(p_1(P)\) and by the Stiefel-Whitney class \(w_2 (P)= :w\) in \(H^2(S, \mathbb{Z}/2)\)\((w^2 \equiv p_1 \bmod 4 \mathbb{Z})\). Now if one fixes \(P\), for a generic Riemannian metric \(g\), the dimension of the moduli space \({\mathcal M} (P,g)\) of the anti self dual connections modulo gauge equivalence is equal to \(-2p_1-6\). The interesting case occurs when such a dimension equals zero, in which case a differential invariant \(q(S,P)\) is well defined as the number of points of \({\mathcal M} (P,g)\) counted with multiplicity. With this in mind, the author answers the above question by the following result:
Main theorem: Let \(X\) be a generalized Kummer surface of type \((6,2m)\), i.e. \(X\) is a singular surface which can be realized as the double cover of \(\mathbb{P}_1 \times \mathbb{P}_1\) branched over the union of 6 vertical lines and \(2m\) horizontal lines, where \(m\) is an integer \(\geq 2\), and let \(S\) be the \(N\)-\(H\) surface which is the minimal resolution of singularities of \(X\).
Then there are certain classes \(w\in H^2 (S,Z/2)\) with \(w^2\equiv -1 +2m \pmod 4\), such that, for a bundle \(P\) with Stiefel-Whitney class \(w_2(P) -w\) and Pontrjagin class \(p_1(P)= -3\chi= -6m+3\), the differential invariant \(q(S,P)\) equals \(2^{2(m-2)}\).
For the entire collection see [Zbl 0840.00037].
Reviewer: Vo Van Tan (Boston)
MSC:
| 14J28 | \(K3\) surfaces and Enriques surfaces |
| 57R50 | Differential topological aspects of diffeomorphisms |
