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APPEARED IN BULLETIN OF THE

arXiv:math/9307230v1 [math.GT] 1 Jul 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 29, Number 1, July 1993, Pages 63-69THE GENUS-MINIMIZING PROPERTYOF ALGEBRAIC CURVESP. B. KronheimerAbstract.

A viable and still unproved conjecture states that, if X is a smoothalgebraic surface and C is a smooth algebraic curve in X, then C realizes the smallestpossible genus amongst all smoothly embedded 2-manifolds in its homology class. Aproof is announced here for this conjecture, for a large class of surfaces X, under theassumption that the normal bundle of C has positive degree.1.

IntroductionIf X is a smooth 4-manifold and ξ is a 2-dimensional homology class in X,one can always represent ξ geometrically by an oriented 2-dimensional surface Σ,smoothly embedded in the 4-manifold. Depending on X and ξ however, the genusof Σ may have to be quite large: it is not always possible to represent ξ by anembedded sphere.

It is natural to ask for a representative whose genus is as smallas possible, or at least to enquire what the genus of such a minimal representativewould be. Although not much is known about this question in general, there isan attractive conjecture concerning the case that X is the manifold underlying asmooth complex-algebraic surface.

The conjecture is best known in the case that Xis the complex projective plane CP2, in which case it is often attributed to Thom,but the statement seems still to be viable more generally [1].Conjecture 1. Let X be a smooth algebraic surface and ξ a homology class carriedby a smooth algebraic curve C in X.

Then C realizes the smallest possible genusamongst all smoothly embedded 2-manifolds representing ξ.The attractiveness of this conjecture stems from the connection to which itpoints, between low-dimensional topology and complex geometry.Through thework of Donaldson in particular, this connection is now a familiar feature of dif-ferential topology in dimension 4, and the techniques of gauge theory provide anatural starting point for an approach to the problem. The conjecture was provedin [5] for the special case that X is a K3 surface, and the results of that paper alsogave a lower bound for the genus of an embedded surface in more general complexsurfaces.

However, in all applicable cases other than K3, the lower bound provedin [5] falls short of Conjecture 1. The purpose of this present paper is to describe aresult which establishes the correctness of the conjecture for a large class of complexsurfaces.

The hypotheses of Theorem 2 still exclude the tantalizing case of CP2,but conditions (a) and (c) of the theorem admit very many (and conjecturally all)1991 Mathematics Subject Classification. Primary 57R42, 57R95.Received by the editors November 17, 1992c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1

2P. B. KRONHEIMERsimply connected surfaces X of general type and odd geometric genus.

Full detailsof the proof will appear later in [4].Theorem 2. The above conjecture holds at least under the following assumptionsconcerning X and C:(a) the surface X is simply connected;(b) the self-intersection number C · C is positive ;(c) there is a class ω ∈H2(X, C) dual to a holomorphic 2-form on X, such thatqk(ω + ω) > 0 for sufficiently large k, where qk denotes Donaldson’s polynomialinvariant.Some comments are needed concerning the third hypothesis.

Donaldson’s poly-nomial invariants [2] are homogeneous polynomial functions qk on H2(X), definedusing instanton moduli spaces for structure group SU(2); their degree depends onthe parameter k as well as the homotopy type of X.Condition (c) appears in[8], where it is shown that this condition will hold for a surface X of general typeprovided that:(i) the geometric genus pg(X) is odd; and(ii) the canonical linear system of the minimal model of X contains a smoothcurve. (This result also rests on some more technical material in [6].) The first of these twoconditions ensures that the polynomial invariants have even degree and certainly isessential as long as one considers the invariants associated with the structure groupSU(2), though the SO(3) moduli spaces can be used to treat some of the remainingcases.

The importance of condition (ii) is less clear, but it does indicate that oneshould expect (c) to be a rather general property of complex surfaces whenever thepolynomial invariants have even degree.The basic material of the proof of Theorem 2 is the same as that of the main the-orem of [5], namely, the moduli spaces of instantons on X having a singularity alongan embedded surface. The structure of the argument, however, is rather different.The difficulty of embedding 2-dimensional surfaces in four dimensions stems fromone’s inability to remove unwanted self-intersection points of an immersed surface,even when these intersections cancel algebraically in plus–minus pairs; this is thefailure of the Whitney lemma in dimension 4 and lies at the heart of 4-manifoldtopology and all its problems.

The first stage of the proof of Theorem 2 is the con-struction of invariants which measure an obstruction to the removal of such pairsof intersection points. Given an immersed surface Σ with normal crossings in a4-manifold X, we use the moduli spaces of singular instantons to define an invari-ant of the pair (X, Σ); this will be a distinguished function Rd(s) : H2(X) →R,taking the form of a homogeneous polynomial of degree d on H2(X) and a finiteLaurent series in the formal variable s. This invariant has the property that theorder of vanishing of Rd at the point s = 1 gives an upper bound on the number ofpositive-signed intersection points in Σ which can ever be removed by a homotopyof the immersion.

The second stage of the proof is to show that, in the case of analgebraic curve in a suitable algebraic surface, the invariants Rd(s) give informa-tion which is sharp enough to establish the assertion of Conjecture 1; this entailsproving a nonvanishing theorem for the value of Rd(s) at s = 1. Using ideas from[8], we shall in fact show that, if C′ is an irreducible algebraic curve with a singleordinary double point in a suitable complex surface X, then Rd(1) is positive for

GENUS-MINIMIZING CURVES3the pair (X, C′) when evaluated on a class ω + ω as in (c), so showing that theself-intersection point in C′ cannot be removed by any homotopy. Theorem 2 iseasily deduced from this.The structure of this proof is closely modeled on Donaldson’s proof of the inde-composability of complex surfaces in [2], but a closer parallel still is in [1], wherea similar strategy was used in connection with Conjecture 1.

In that paper, usewas made of the instanton moduli spaces associated with a branched double covereX →X, branched along Σ. The moduli spaces of singular instantons which weuse here can, in some circumstances, be interpreted as moduli spaces of suitablyequivariant instanton connections on such a covering manifold (equivariant, thatis, under the covering involution).

The replacing of the full moduli space on eX bythis equivariant part can be seen as the main difference in the framework of theargument between [1] and the present paper. The other main new ingredient hereis the organization of the invariants obtained from the singular instantons to formthe finite Laurent series Rd.

The remaining two sections of this paper provide somefurther details of the proof; some of this material is rather technical, particularlyin §3. It seems likely that a change of strategy in this last part of the argumentwill eventually lead to a slightly more general result than Theorem 2.2.

Obstructions to removing intersection pointsLet X be a smooth, oriented, simply connected closed 4-manifold and Σ an em-bedded (rather than just immersed) orientable surface in X. Given a Riemannianmetric on X, it was shown in [5] how one can construct moduli spaces M αk,l associ-ated to the pair; roughly speaking, M αk,l parametrizes the finite-action anti-self-dualSU(2) connections on X\Σ with the property that, near to Σ, the holonomy aroundsmall loops linking the surface is asymptotically exp 2πi( −α00α).

Here α is a realparameter in the interval (0, 1/2) and k and l are the integer topological invari-ants of such connections: the “instanton” and “monopole” numbers. For a genericchoice of Riemannian metric and away from the flat or reducible connections in themoduli space, M αk,l is a smooth manifold of dimension(1)8k + 4l −3(b+ + 1) −(2g −2)where b+ is the dimension of a maximal positive subspace for the intersection formon H2(X) and g is the genus of Σ.

In the case that b+ is odd, the dimension is even,and we write it as 2d(k, l), so that d = d(k, l) is half of (1). Following Donaldson’sdefinition [2] of the polynomial invariants qk, it was shown in [5] that the modulispaces M αk,l can be used, when b+ is odd, to define a homogeneous polynomialfunction of degree d(k, l),qk,l : H2(X) →R.

(In [5], this polynomial was defined only on the orthogonal complement of [Σ] inH2(X), and this is all we will actually need; the definition, however, can be extendedto the whole homology group. Also, a ‘homology orientation’ of X is needed to fixthe overall sign.) When b+ is at least three, the polynomial qk,l is independent ofthe parameter α and the Riemannian metric; it is an invariant of the pair (X, Σ).Because of the way k and l enter the dimension formula (1), the degree of qk−1,l+2is the same as the degree of qk,l.

It is natural to combine all the polynomials of a

4P. B. KRONHEIMERgiven degree into one Laurent series:Rd(s) =Xd(k,l)=dslqk,l.This series is actually finite in both directions, though this is not obvious a priorifrom its definition.

Note that, depending on the parity of the genus g and the valueof b+ mod 4, the invariant Rd will be defined only for d of one particular parity. Flatconnections on the complement of Σ can cause difficulties in defining the invariantsdirectly from the moduli spaces, but these can be overcome, for example, by adevice such as that described in [7].Having defined invariants for embedded surfaces, we can now define invariantsfor immersed surfaces with normal crossings.

It would seem feasible to do this bydirectly using gauge theory on the complement of the immersed surface, but a short-cut is available. We shall convert such an immersed surface Σ into an embeddedsurface by blowing up X at the intersection points.

This is the process modeledon the situation in complex geometry, where a curve Σ with a normal crossing atp is replaced by its proper transform, which is a smooth curve in the new surfaceeX = X#CP2, the blow-up of X at p. In the C∞case, the model is just the same;there are really two different cases according to the sign of the intersection point,but no essential difference in the local picture. Thus we obtain an embedded surfaceeΣ in a new manifold eX = X#nCP2.

We define the invariants qk,l and Rd for (X, Σ)to be the restriction of the invariants of ( eX, eΣ) to H2(X) ⊂H2( eX).The next stage in the argument is to see how the invariant Rd(s) changes whenthe immersion of Σ in X is changed by a homotopy. During a homotopy, doublepoints can appear and disappear in Σ in quite complicated ways, but standardtheory says that after a small perturbation any such changes can be broken downinto a combination of moves, each of which is one of six types.

One has to considerthe following three standard modifications and their inverses (see [3] for picturesand explanations of these):(a) introduce a positive double point by a twist move;(b) introduce a negative double point by a twist move;(c) introduce a cancelling plus–minus pair by a finger move. (We should emphasize that we are talking about homotopies whose starting andfinishing points are immersions; only (c) can be achieved by a homotopy throughimmersions.) The change in Rd under each of these three moves is summarized bythe following proposition.Proposition 3.

Let Σ be obtained from bΣ by one of the moves just described, andlet Rd and bRd be the invariants for (X, Σ) and (X, bΣ). Then, according to the threecases, we have :(a) Rd(s) = (1 −s−2) bRd(s);(b) Rd(s) = bRd(s);(c) Rd(s) = (1 −s−2) bRd(s).As a consequence of these relations, the order of vanishing of Rd(s) at s = 1increases by one every time a positive double point is introduced by either of themoves (a) or (c) and decreases by one every time a positive double point disappears.

GENUS-MINIMIZING CURVES5So, as was stated in the introduction, the order of vanishing of Rd at s = 1 puts anupper bound on the number of positive double points which can be removed.Note also that the invariants Rd are unable to detect subtleties of knotting: twoembedded surfaces of the same genus will have the same invariant if the embeddingsare homotopic. (There is also a simple formula for how Rd changes when the genusof Σ is increased by summing with a torus in X, but this involves aspects whichwould take us too far afield; see [4].

)The proof of Proposition 3 involves some rather simple gluing arguments. Con-sider (a) for example.

After introducing the positive double point to form Σ frombΣ, the definition of the invariants for immersed surfaces tells us to remove thedouble point by blowing up, to get ( eX, eΣ). Examining the overall effect, we findthat, up to diffeomorphism, ( eX, eΣ) is the connect sum of the pair (X, bΣ) with thepair (CP2, C), where C is a conic curve in the projective plane.

One must analysethis connected sum of pairs to show that the invariants for eΣ and bΣ are related byqk,l = bqk,l −bqk−1,l+2. (The sign here is rather subtle and crucial to the argument.

)Technical aspects of the gluing construction, as well as some aspects of the algebraicgeometry in §3, can be considerably simplified by using the fact that the modulispaces M αk,l for α = 1/4 are essentially equivalent to moduli spaces of equivariantconnections on a branched double cover, or to orbifold connections in the case thata global double covering does not exist. So, in the example above, rather than thinkof forming the connect sum along a pair (S3, S1), one may think instead of gluingacross a copy of S3, with invariance imposed under an involution.3.

The complex caseSuppose now that X1 is a smooth complex surface and C1 is a smooth algebraiccurve in X1. If the self-intersection number of C1 is positive, and we wish to proveConjecture 1 for the homology class [C1], then it turns out to be enough to tackleinstead the homology class n[C1] for any large n (see [1] or [5] for this elementaryconstruction).

So let C2 be a smooth curve in the linear system |nC1|. Taking nlarge enough, we may suppose that the linear system |C2| contains, in addition tothis smooth curve, an irreducible singular curve C′2 with a single ordinary doublepoint.

We can look at C′2 as an immersed 2-manifold with a single normal crossingof positive sign; its genus is one less than the genus of C2. Suppose the conjecturefails for the homology class of [C2].Then we can find an embedded surface Σin the this class, with the same genus as the immersed surface C′2.

Since X1 issimply connected, Σ and C′2 will be homotopic, and it follows from the results ofthe previous section that the invariant Rd(s) for (X1, C′2) vanishes at s = 1. Toobtain a contradiction and prove the conjecture, we therefore need a nonvanishingtheorem which states that Rd(1) is nonzero.

We shall in fact prove that, if the genusof C′2 is odd and its homology class is even (conditions which eventually entail noloss of generality) and if the conditions of Theorem 2 hold, then the value of Rd(1)on the class ω + ω of (c) is positive once d is sufficiently large. If we recall againthat the invariants for an immersed surface are defined in terms of the embeddedsurface obtained by blowing up, we are led to blow up (X1, C′2) at the single doublepoint of C′2 to obtain finally a smooth algebraic pair (X, C).

The following resultis therefore what is wanted.Proposition 4. Let C be a smooth curve in an algebraic surface X, satisfying theconditions of Theorem 2.

Suppose in addition that C has odd genus and that its

6P. B. KRONHEIMERhomology class has divisibility 2 in H2(X, Z).

Then for large d the value of theinvariant Rd(1)(ω + ω) for the pair (X, C) is strictly positive.Note that since Rd(1) is just the sum of the invariants qk,l of a given degree, itwill suffice to show that each of these terms is nonnegative and that at least one ofthem is positive. Under some conditions on α and the metric on X, it was shownin [5] that the moduli spaces M αk,l can be interpreted as moduli spaces of stableparabolic bundles.

It is therefore tempting to try to adapt Donaldson’s argumentin [2] to prove that each qk,l is positive when evaluated on the hyperplane class,provided that the degree d is large. (Actually this would not be quite the rightthing; one should construct a different version of qk,l which varies with α, to takeaccount of how the natural polarization of a moduli space of parabolic bundleschanges as the parabolic weight is varied.) Unfortunately, there is an obstructionto this programme.

A key technical step in the argument of [2] is to show that,once d is large, the moduli spaces of stable bundles on a complex surface havethe dimension one would naively predict from the index formula, namely, d. This“regularity” result is false in the context of parabolic bundles.To explain what is true in the way of regularity, it is convenient to introducethe “magnetic charges” m1 = k and m2 = k + l −(Σ · Σ)/4 in place of k andl. The naively expected complex dimension of the moduli space M αk,l is then d ∼2m1 + 2m2, according to the formula (1).

In order for the moduli space to havethis expected dimension, it is not enough that d alone be large—it is necessary forboth m1 and m2 to be large.Fortunately, a general vanishing theorem was proved in [5] which shows that, ifthe absolute value of the difference |m1 −m2| is larger than a quantity (KX · C)/4,then the invariants qk,l of a pair (X, C) vanish when restricted to homology classesorthogonal to [C] in H2(X). So, if we take such a homology class, then the modulispaces which might have the wrong dimension (where only one of m1 or m2 is large)will not contribute.

Taking this line forces us to abandon the idea of followingDonaldson’s argument from [2], since the hyperplane class is not orthogonal to C.Instead, we adapt O’Grady’s argument from [8].As was mentioned in the introduction, the results of [8] show that, under suitableconditions on a complex surface X and k, the value qk(ω + ω) of the ordinarypolynomial invariant is strictly positive when ω is dual to a generic holomorphic2-form. (Note that such a class is always going to be orthogonal to a holomorphiccurve such as C.) Part of the argument adapts readily to the parabolic case to showthat qk,l(ω + ω) is at least nonnegative, provided only that the moduli spaces havethe correct regularity properties.

All that remains finally is to show that at leastone of these values is nonzero. The last ingredient is another hard result from [5]which shows that, for the special value of the monopole number l = (g −1)/2, theinvariant qk,l for the pair (X, C) is equal to 2gqk when restricted to the orthogonalcomplement of C. So, for this special value of l, the nonvanishing of qk,l can bededuced from the nonvanishing of qk.AcknowledgmentThe author thanks Simon Donaldson, Bob Gompf, John Morgan, and KieranO’Grady for their help in preparing this paper, and, in particular, Tom Mrowka formany hours of discussion, out of which these results gradually emerged.

GENUS-MINIMIZING CURVES7References1. S. K. Donaldson, Complex curves and surgery, Inst.

Hautes ´Etudes Sci. Publ.

Math. 68(1988), 91–97.2., Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257–315.3.

M. H. Freedman and F. Quinn, Topology of 4-manifolds, Princeton Univ. Press, Princeton,NJ, 1990.4.

P. B. Kronheimer, papers in preparation.5. P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces.

I, II, Topology(to appear).6. J. W. Morgan, Comparison of the Donaldson invariants of algebraic surfaces with theiralgebro-geometric analogues, Topology (to appear).7.

J. W. Morgan and T. S. Mrowka, A note on Donaldson’s polynomial invariants, Internat.Math. Res.

Notices, no. 10 (1992), 223–230.8.

K. G. O’Grady, Algebro-geometric analogues of Donaldson’s polynomials, Invent. Math.107 (1992), 351–395.Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LBE-mail address: kronheim@maths.oxford.ac.uk

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