##
**Complex curves and surgery.**
*(English)*
Zbl 0696.57007

A fundamental problem in 4-dimensional topology is to find the minimal genus of a closed surface which represents a given 2-dimensional homology class in a smooth 4-manifold. The genus of a smooth algebraic curve C of degree d in the complex projective plane \({\mathbb{C}}P^ 2\) is \((d-1)(d-2).\) Let h be the standard generator for \(H_ 2({\mathbb{C}}P^ 2)\). A fundamental conjecture, ascribed to R. Thom, is:

Conjecture 1. Fo any smoothly embedded, oriented surface \(\Sigma\) in \({\mathbb{C}}P^ 2\) with homology class \([\Sigma]=d.h\), \(d>0\), we have \(genus(\Sigma)\geq (d-1)(d-2).\)

An extension of this conjecture is:

Conjecture 2. For any smooth complex algebraic curve C in a smooth complex projective surface S and smoothly embedded, oriented surface \(\Sigma\) homologous to C, we have genus\((\Sigma)\geq\) genus\((C).\)

Many of the early techniques in the study of 4-manifolds were a result of attempts to find the minimal genus of surfaces representing a given homology class. M. A. Kervaire and J. W. Milnor [Proc. Natl. Acad. Sci. USA 47, 1651-1657 (1961; Zbl 0107.403)] showed that in \({\mathbb{C}}P^ 2\), 3h is not represented by a sphere, thus verifying Conjecture 1 for \(d\leq 3\). W. C. Hsiang and R. H. Szczarba [Proc. Symp. Pure Math. 22, 97-103 (1971; Zbl 0234.57009)] and V. A. Rokhlin [Funkts. Anal. Prilozh. 5, No.1, 48-60 (1971; Zbl 0268.57019)] used the G-signature theorem to produce lower bounds for the minimal genus. More recently, techniques from Yang-Mills theory have been applied to these problems by K. Kuga [Topology 23, 133-137 (1984; Zbl 0551.57019)], R. Fintushel and R. J. Stern [Ann. Math., II. Ser. 122, 335-364 (1985; Zbl 0602.57013)], and Furuta [On pseudofree self-dual connections on orbifolds, Preprint] to show that certain homology classes cannot be represented by spheres. Utilizing Freedman’s results, L. Rudolph [Comment. Math. Helv. 59, 592-599 (1984; Zbl 0575.57003)] has shown that Conjecture 1 is false if one replaces “smoothly embedded” with “topologically locally flat” surfaces.

One way to reduce the genus of a surface \(\Sigma\) representing a given homology class in a 4-manifold S is to surger a loop \(\delta\) in \(\Sigma\) with [\(\delta\) ] non-zero in \(H_ 1(\Sigma)\); i.e. suppose there is an embedded 2-disk in S with boundary \(\delta\) whose interior is disjoint from \(\Sigma\) and such that the non-zero section of the normal bundle of \(\delta\) in \(\Sigma\) extends over this disk. Then there is a pair of disjoint parallel disks that can be used to surger \(\Sigma\) to obtain a surface of lower genus representing the same homology class as \(\Sigma\). The main result of the paper under review is to show that one cannot obtain a counterexample to Conjecture 2 via this obvious approach. More precisely:

Theorem. Let S be a simply connected complex projective surface and C a complex curve in S with associated holomorphic bundle being ample. Then it is impossible to perform smooth surgery within S on any loop \(\delta\) in C (not homologous to zero).

This theorem is shown to follow from the author’s amazing theorem [Topology 29, No.3, 257-315 (1990)]: Theorem. Let Z be a simply connected complex projective surface. If Z can be decomposed as a smooth connected sum \(Z=X\#(S^ 2\times S^ 2)\), then the 4-manifold X has a negative definite intersection form.

Conjecture 1. Fo any smoothly embedded, oriented surface \(\Sigma\) in \({\mathbb{C}}P^ 2\) with homology class \([\Sigma]=d.h\), \(d>0\), we have \(genus(\Sigma)\geq (d-1)(d-2).\)

An extension of this conjecture is:

Conjecture 2. For any smooth complex algebraic curve C in a smooth complex projective surface S and smoothly embedded, oriented surface \(\Sigma\) homologous to C, we have genus\((\Sigma)\geq\) genus\((C).\)

Many of the early techniques in the study of 4-manifolds were a result of attempts to find the minimal genus of surfaces representing a given homology class. M. A. Kervaire and J. W. Milnor [Proc. Natl. Acad. Sci. USA 47, 1651-1657 (1961; Zbl 0107.403)] showed that in \({\mathbb{C}}P^ 2\), 3h is not represented by a sphere, thus verifying Conjecture 1 for \(d\leq 3\). W. C. Hsiang and R. H. Szczarba [Proc. Symp. Pure Math. 22, 97-103 (1971; Zbl 0234.57009)] and V. A. Rokhlin [Funkts. Anal. Prilozh. 5, No.1, 48-60 (1971; Zbl 0268.57019)] used the G-signature theorem to produce lower bounds for the minimal genus. More recently, techniques from Yang-Mills theory have been applied to these problems by K. Kuga [Topology 23, 133-137 (1984; Zbl 0551.57019)], R. Fintushel and R. J. Stern [Ann. Math., II. Ser. 122, 335-364 (1985; Zbl 0602.57013)], and Furuta [On pseudofree self-dual connections on orbifolds, Preprint] to show that certain homology classes cannot be represented by spheres. Utilizing Freedman’s results, L. Rudolph [Comment. Math. Helv. 59, 592-599 (1984; Zbl 0575.57003)] has shown that Conjecture 1 is false if one replaces “smoothly embedded” with “topologically locally flat” surfaces.

One way to reduce the genus of a surface \(\Sigma\) representing a given homology class in a 4-manifold S is to surger a loop \(\delta\) in \(\Sigma\) with [\(\delta\) ] non-zero in \(H_ 1(\Sigma)\); i.e. suppose there is an embedded 2-disk in S with boundary \(\delta\) whose interior is disjoint from \(\Sigma\) and such that the non-zero section of the normal bundle of \(\delta\) in \(\Sigma\) extends over this disk. Then there is a pair of disjoint parallel disks that can be used to surger \(\Sigma\) to obtain a surface of lower genus representing the same homology class as \(\Sigma\). The main result of the paper under review is to show that one cannot obtain a counterexample to Conjecture 2 via this obvious approach. More precisely:

Theorem. Let S be a simply connected complex projective surface and C a complex curve in S with associated holomorphic bundle being ample. Then it is impossible to perform smooth surgery within S on any loop \(\delta\) in C (not homologous to zero).

This theorem is shown to follow from the author’s amazing theorem [Topology 29, No.3, 257-315 (1990)]: Theorem. Let Z be a simply connected complex projective surface. If Z can be decomposed as a smooth connected sum \(Z=X\#(S^ 2\times S^ 2)\), then the 4-manifold X has a negative definite intersection form.

Reviewer: R.Stern

### MSC:

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

57R40 | Embeddings in differential topology |

### Keywords:

minimal genus; closed surface; homology class; smooth 4-manifold; smooth algebraic curve; Yang-Mills theory; negative definite intersection form### Citations:

Zbl 0107.403; Zbl 0234.57009; Zbl 0268.57019; Zbl 0551.57019; Zbl 0602.57013; Zbl 0575.57003### References:

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