such that for each i, Ciâ1 and Ci are separated by a single wall Wi = WÎ¶i of type. (w, p) which ... also [21]), the difference Î³w,p(X; Ciâ1)âÎ...

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arXiv:alg-geom/9404010v1 22 Apr 1994

Robert Friedman and Zhenbo Qin Introduction The goal of this paper is to prove the following: Theorem 0.1. Let X be a complex surface of general type. Then X is not diffeomorphic to a rational surface. Using the results from [13], we obtain the following corollary, which settles a problem raised by Severi: Corollary 0.2. If X is a complex surface diffeomorphic to a rational surface, then X is a rational surface. Thus, up to deformation equivalence, there is a unique 2 complex structure on the smooth 4-manifolds S 2 × S 2 and P2 #nP . In addition, as discussed in the book [15], Theorem 0.1 is the last step in the proof of the following, which was conjectured by Van de Ven [37] (see also [14,15]): Corollary 0.3. If X1 and X2 are two diffeomorphic complex surfaces, then κ(X1 ) = κ(X2 ), where κ(Xi ) denotes the Kodaira dimension of Xi . The first major step in proving that every complex surface diffeomorphic to a rational surface is rational was Yau’s theorem [40] that every complex surface of the same homotopy type as P2 is biholomorphic to P2 . After this case, however, the problem takes on a different character: there do exist nonrational complex surfaces with the same oriented homotopy type as rational surfaces, and the issue is to show that they are not in fact diffeomorphic to rational surfaces. The only known techniques for dealing with this question involve gauge theory and date back to Donaldson’s seminal paper [9] on the failure of the h-cobordism theorem in dimension 4. In this paper, Donaldson introduced analogues of polynomial invariants for 4-manifolds M with b+ 2 (M ) = 1 and special SU (2)-bundles. These invariants depend in an explicit way on a chamber structure in the positive cone in H 2 (M ; R). Using these invariants, he showed that a certain elliptic surface (the Dolgachev surface with multiple fibers of multiplicities 2 and 3) was not diffeomorphic to a rational surface. In [13], this result was generalized to cover all Dolgachev surfaces and their blowups (the case of minimal Dolgachev surfaces was also treated in [28]) The first author was partially supported by NSF grant DMS-9203940. The second author was partially supported by a grant from ORAU Junior Faculty Enhancement Award Program. Typeset by AMS-TEX 1

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and Donaldson’s methods were also used to study self-diffeomorphisms of rational surfaces. The only remaining complex surfaces which are homotopy equivalent (and thus homeomorphic) to rational surfaces are then of general type, and a single example of such surfaces, the Barlow surface, is known to exist [2]. In 1989, Kotschick [18], as well as Okonek-Van de Ven [29], using Donaldson polynomials associated to SO(3)-bundles, showed that the Barlow surface was not diffeomorphic to a rational surface. Subsequently Pidstrigach [30] showed that no complex surface of general type which has the same homotopy type as the Barlow surface was diffeomorphic to a rational surface, and Kotschick [20] has outlined an approach to showing that no blowup of such a surface is diffeomorphic to a rational surface. All of these approaches use SO(3)-invariants or SU (2)-invariants for small values of the (absolute value of) the first Pontrjagin class p1 of the SO(3)-bundle, so that the dependence on chamber structure can be controlled in a quite explicit way. In [33], the second author showed that no surface X of general type could be diffeomorphic to P1 ×P1 or to F1 , the blowup of P2 at one point. Here the main tool is the study of SO(3)-invariants for large values of −p1 , as defined and analyzed in [19] and [21]. These invariants also depend on a chamber structure, in a rather complicated and not very explicitly described fashion. In [34], these methods are used to analyze minimal surfaces X of general type under certain assumptions concerning the nonexistence of rational curves, which are always satisfied if X has the same homotopy type as P1 × P1 or F1 , by a theorem of Miyaoka on the number of rational curves of negative self-intersection on a minimal surface of general type. The main idea of the proof is to show the following: Let X be a minimal surface of general type, and suppose that {E0 , . . . , En } is an orthogonal basis for H 2 (X; Z) P with E02 = 1, Ei2 = −1 for i ≥ 1, and [KX ] = 3E0 − i≥1 Ei . Finally suppose that the divisor E0 − Ei is nef for some i ≥ 1. Then the class E0 − Ei cannot be represented by a smoothly embedded 2-sphere. (Actually, in [34], the proof shows that an appropriate Donaldson polynomial is not zero whereas it must be zero if X is diffeomorphic to a rational surface. However, using [26], one can also show that if E0 − Ei is represented by a smoothly embedded 2-sphere, then the Donaldson polynomial is zero.) At the same time, building on ideas of Donaldson, Pidstrigach and Tyurin [31], using Spin polynomial invariants, showed that no minimal surface of general type is diffeomorphic to a rational surface. We now discuss the contents of this paper and the general strategy for the proof of Theorem 0.1. The bulk of this paper is devoted to giving a new proof of the results of Pidstrigach and Tyurin concerning minimal surfaces X. Here our methods apply as well to minimal simply connected algebraic surfaces of general type with pg arbitrary. Instead of looking at embedded 2-spheres of self-intersection 0 as in [34], we consider those of self-intersection −1. We show in fact the following in Theorem 1.10 (which includes a generalization for blowups): Theorem 0.4. Let X be a minimal simply connected algebraic surface of general type, and let E ∈ H 2 (X; Z) be a class satisfying E 2 = −1, E · [KX ] = 1. Then the class E cannot be represented by a smoothly embedded 2-sphere. In particular, if pg (X) = 0, then X cannot be diffeomorphic to a rational surface. The method of proof of Theorem 0.4 is to show that a certain value of a Donaldson polynomial invariant for X is nonzero (Theorem 1.5), while it is a result of Kotschick that if the class E is represented by a smoothly embedded 2-sphere, then the value of the Donaldson polynomial must be zero (Proposition 1.1). In case pg (X) = 0,

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once we have have found a polynomial invariant which distinguishes X from a rational surface, it follows in a straightforward way from the characterization of self-diffeomorphisms of rational surfaces given in [13] that no blowup of X can be diffeomorphic to a rational surface either (see Theorem 1.7). This part of the argument could also be used with the result of Pidstrigach and Tyurin to give a proof of Theorem 0.1. Let us now discuss how to show that certain Donaldson polynomials do not vanish on certain classes. The prototype for such results is the nonvanishing theorem of Donaldson [10]: if S is an algebraic surface with pg (S) > 0 and H is an ample line bundle on S, then for all choices of w and all p ≪ 0, the SO(3)-invariant γw,p (H, . . . , H) 6= 0. We give a generalization of this result in Theorem 1.4 to certain cases where H is no longer ample, but satisfies: H k has no base points for ¯ and where k ≫ 0 and defines a birational morphism from X to a normal surface X, pg (X) is also allowed to be zero (for an appropriate choice of chamber). Here we must assume that there is no exceptional curve C such that H · C = 0, as well as ¯ they should the following additional assumption concerning the singularities of X: be rational or minimally elliptic in the terminology of [22]. The proof of Theorem 1.4 is a straightforward generalization of Donaldson’s original proof, together with methods developed by J. Li in [23, 24]. Given the generalized nonvanishing theorem, the problem becomes one of constructing divisors M such that M is orthogonal to a class E of square −1 and moreover such that M is eventually base point free. (Here we recall that a divisor M is eventually base point free if the complete linear system |kM | has no base points for all k ≫ 0.) There are various methods for finding base point free linear systems on an algebraic surface. For example, the well-studied method of Reider [35] implies that, if X is a minimal surface of general type and D is a nef and big divisor on X, then M = KX + D is eventually base point free. There is also a technical generalization of this result due to Kawamata [16]. However, the methods which we shall need are essentially elementary. The general outline of the construction is as follows. Let E be a class of square −1 with KX · E = 1. It is known that, if E is the class of a smoothly embedded 2-sphere, then E is of type (1, 1) [6]. Thus KX + E is a divisor orthogonal to E. If KX + E is ample we are done. If KX + E is nef but not ample, then there exist curves D with (KX + E) · D = 0, and the intersection matrix of the set of all such curves is negative definite. Thus we may contract the set of all such curves to obtain a normal surface X ′ . If X ′ has only rational singularities, then the divisor KX + E induces a Cartier divisor on X ′ which is ample, by the Nakai-Moishezon criterion, and so some multiple of KX + E is base point free. Next suppose that X ′ has a nonrational singular point p and let D1 , . . . , Dt be the irreducible curves on X mapped to p. Then we give a dual form of Artin’s criterion [1] for a rational singularity, which says the following: the point p is a nonrational singularity if and only P if there exist nonnegative integers ni , with at least one ni > 0, such that (KX + i ni Di ) · Dj ≥ 0 for all j. Moreover there is a choice of the ni such that either the inequality is strict for every j or the contraction of the Dj with nj 6= 0 is a minimally elliptic singularity. In this case, P provided that KX is itself nef, it is easy to show that KX + i ni Di is nef and big and eventually base point free, and defines the desired contraction. The remaining case is when KX + E is not nef. In this case, by considering the curves D with (KX + E) · D < 0, it is easy to find a Q-divisor of the form KX + λD, where D is an irreducible curve and λ ∈ Q+ , which is nef and big and such that some multiple

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is eventually base point free, and which is orthogonal to E. The details are given in Section 3. These methods can also handle the case of elliptic surfaces (the case where κ(X) = 1), but of course there are more elementary and direct arguments here which prove a more precise result. We have included an appendix giving a proof, due to the first author, R. Miranda, and J.W. Morgan, of a result characterizing the canonical class of a rational surface up to isometry. This result seems to be well-known to specialists but we were unable to find an explicit statement in the literature. It follows from work of Eichler and Kneser on the number of isomorphism classes of indefinite quadratic forms of rank at least 3 within a given genus (see e.g. [17]) together with some calculation. However the proof in the appendix is an elementary argument. The methods in this paper are able to rule out the possibility of embedded 2spheres whose associated class E satisfies E 2 = −1, E · [KX ] = 1. However, in case pg (X) = 0 and b2 (X) ≥ 3, there are infinitely many classes E of square −1 which satisfy |E · KX | ≥ 3. It is natural to hope that these classes also cannot be represented by smoothly embedded 2-spheres. More generally we would like to show that the surface X is strongly minimal in the sense of [15]. Likewise, in case pg (X) > 0, we have only dealt with the first case of the “(−1)-curve conjecture” (see [6]). Acknowledgements: We would like to thank Sheldon Katz, Dieter Kotschick, and Jun Li for valuable help and stimulating discussions. 1. Statement of results and overview of the proof 1.1. Generalities on SO(3)-invariants Let X be a smooth simply connected 4-manifold with b+ 2 (X) = 1, and fix an SO(3)-bundle P over X with w2 (P ) = w and p1 (P ) = p. Recall that a wall of type (w, p) for X is a class ζ ∈ H 2 (S; Z) such that ζ ≡ w mod 2 and p ≤ ζ 2 < 0. Let ΩX = {x ∈ H 2 (X; R) : x2 > 0}. Let W ζ = ΩX ∩ (ζ)⊥ . A chamber of type (w, p) for X is a connected component of the set [ ΩX − {W ζ : ζ is a wall of type (w, p) }. Let C be a chamber of type (w, p) for X and let γw,p (X; C) denote the associated Donaldson polynomial, defined via [19] and [21]. Here γw,p (X; C) is only defined up to ±1, depending on the choice of an integral lift for w, corresponding to a choice of orientation for the moduli space. The actual choice of sign will not matter, since we shall only care if a certain value of γw,p (X; C) is nonzero. In the complex case we shall always assume for convenience that the choice has been made so that the orientation of the moduli space agrees with the complex orientation. Via Poincar´e duality, we shall view γw,p (X; C) as a function on either homology or cohomology classes. Given a class M , we use the notation γw,p (X; C)(M d ) for the evaluation γw,p (X; C)(M, . . . , M ) on the class M repeated d times, where d = −p − 3 is the expected dimension of the moduli space. We then have the following vanishing result for γw,p (C), due to Kotschick [19, (6.13)]:

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Proposition 1.1. Let E ∈ H 2 (X; Z) be the cohomology class of a smoothly embedded S 2 in X with E 2 = −1. Let w be the second Stiefel-Whitney class of X, or more generally any class in H 2 (X, Z/2Z) such that w · E 6= 0. Suppose that M ∈ H2 (X; Z) satisfies M 2 > 0 and M · E = 0. Then, for every chamber C of type (w, p) such that the wall W E corresponding to E passes through the interior of C, γw,p (X; C)(M d ) = 0.

Note that if w is the second Stiefel-Whitney class of X, then W E is a wall of type (w, p) (and so does not pass through the interior of any chamber) if and only if E ⊥ 2 is even. This case arises, for example, if X has the homotopy type of (S 2 × S 2 )#P 2 and E is the standard generator of H 2 (P ; Z) ⊆ H 2 (X; Z). For the proof of Theorem 0.1, the result of (1.1) is sufficient. However, for the slightly more general result of Theorem 1.10, we will also need the following variant of (1.1): Theorem 1.2. Let E ∈ H 2 (X; Z) be the cohomology class of a smoothly embedded S 2 in X with E 2 = −1. Let w be a class in H 2 (X, Z/2Z) such that w · E 6= 0. Suppose that M ∈ H2 (X; Z) satisfies M 2 > 0 and M · E = 0. Then, for every chamber C of type (w, p) containing M in its closure, γw,p (X; C)(M d ) = 0, unless p = −5 and w is the mod 2 reduction of E. Thus, except in this last case, γw,p (X; C) is divisible by E. Proof. If W E is not a wall of type (w, p) we are done by (1.1). Otherwise, E defines a wall of type (w, p) containing M . Next let us assume that E ⊥ ∩C is a codimension one face of the closure C of C. We have an induced decomposition of X: 2

X = X0 #P . Identify H2 (X0 ; Z) with the subspace E ⊥ of H2 (X; Z), and let C 0 = E ⊥ ∩ C. Then C 0 is the closure of some chamber C0 of type (w − e, p + 1) on X0 , where e is the mod 2 reduction of E. Choose a generic Riemannian metric g0 on X0 such that the cohomology class ω0 of the self-dual harmonic 2-form associated to g0 lies in the interior of C0 . By the results in [39], there is a family of metrics ht on the connected 2 sum X0 #CP which converge in an appropriate sense to g0 ∐ g1 , where g1 is the 2 Fubini-Study metric on P , and such that the cohomology classes of the self-dual harmonic 2-forms associated to ht lie in the interior of C and converge to ω0 . Standard gluing and compactness arguments (see for example [15], Appendix to Chapter 6) and dimension counts show that the restriction of the invariant γw,p (X; C) to H2 (X0 ; Z) vanishes. Consider now the general case where W E is a wall of type (w, p) and the closure of C contains M but where W E ∩ C is not necessarily a codimension one face of C. Since W E is a wall of type (w, p) and M ∈ E ⊥ , there exists a chamber C ′ of type ′ (w, p) whose closure contains M such that W E is a codimension one face of C . By the previous argument, γw,p (X; C ′ )(M d ) = 0 and so it will suffice to show that γw,p (X; C)(M d ) = γw,p (X; C ′ )(M d ).

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Note that C and C ′ are separated by finitely many walls of type (w, p) all of which contain the class M . Thus, we have a sequence of chambers of type (w, p): C = C1 , C2 , . . . , Ck−1 , Ck = C ′ such that for each i, Ci−1 and Ci are separated by a single wall Wi = W ζi of type (w, p) which contains M . Since Wi contains M , M · ζi = 0. By [19, (3.2)(3)] (see also [21]), the difference γw,p (X; Ci−1 )−γw,p (X; Ci ) is divisible by the class ζi except in the case where p = −5 and w is the mod 2 reduction of E. It follows that, except in this last case, for each i, γw,p (X; Ci−1 )(M d ) = γw,p (X; Ci )(M d ). Hence γw,p (X; C)(M d ) = γw,p (X; C ′ )(M d ) = 0. We shall also need the following “easy” blowup formula: 2

Lemma 1.3. Let X#P be a blowup of X, and identify H2 (X; Z) with a subspace 2 of H2 (X#P ; Z) in the natural way. Given w ∈ H 2 (X; Z/2Z), let C˜ be a chamber 2 of type (w, p) for X#P containing the chamber C in its closure. Then 2

˜ 2 (X; Z) = ±γw,p (X; C). γw,p (X#P ; C)|H Proof. Choose a generic Riemannian metric g on X such that the cohomology class ω of the self-dual harmonic 2-form associated to g lies in the interior of C. We again use the results in [39] to choose a family of metrics ht on the connected sum 2 X#P which converge in an appropriate sense to g ∐ g ′ , where g ′ is the Fubini2 Study metric on P , and such that the cohomology classes of the self-dual harmonic 2-forms associated to ht lie in the interior of C˜ and converge to ω. Standard gluing and compactness arguments (see e.g. [15], Chapter 6, proof of Theorem 6.2(i)) 2 ˜ to H2 (X; Z) (with the appropriate now show that the restriction of γw,p (X#P ; C) orientation conventions) is just γw,p (X; C). 1.2. The case of a minimal X In this subsection we shall outline the results to be proved concerning minimal surfaces of general type. One basic tool is a nonvanishing theorem for certain values of the Donaldson polynomial: Theorem 1.4. Let X be a simply connected algebraic surface with pg (X) = 0, and let M be a nef and big divisor on X which is eventually base point free. Denote ¯ the birational morphism defined by |kM | for k ≫ 0, so that X ¯ is a by ϕ : X → X ¯ normal projective surface. Suppose that X has only rational or minimally elliptic singularities, and that ϕ does not contract any exceptional curves to points. Let w ∈ H 2 (X; Z/2Z) be the mod 2 reduction of the class [KX ]. Then there exists a constant A depending only on X and M with the following property: For all integers p ≤ A, let C be a chamber of type (w, p) containing M in its closure and suppose that C has nonempty intersection with the ample cone of X. Set d = −p − 3. Then γw,p (X; C)(M d ) > 0.

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We shall prove Theorem 1.4 in Section 2, where we shall also recall the salient properties of rational and minimally elliptic singularities. The proof also works in the case where pg (X) > 0, in which case γw,p (X) does not depend on the choice of a chamber. We can now state the main result concerning minimal surfaces, which we shall prove in Section 3: Theorem 1.5. Let X be a minimal simply connected algebraic surface of general type, and let E ∈ H 2 (X; Z) be a (1, 1)-class satisfying E 2 = −1, E · KX = 1. Let w be the mod 2 reduction of [KX ]. Then there exist: (i) an integer p and (in case pg (X) = 0) a chamber C of type (w, p) and (ii) a (1, 1)-class M ∈ H 2 (X; Z) such that M ·E = 0 and γw,p (X)(M d ) 6= 0 (or, in case pg (X) = 0, γw,p (X; C)(M d ) 6= 0). The method of proof of (1.5) will be the following: we will show that there exists an orientation preserving self-diffeomorphism ψ of X with ψ ∗ [KX ] = [KX ] and a nef and big divisor M on X such that: (i) M · ψ ∗ E = 0. (ii) M is eventually base point free, and the corresponding contraction ϕ : X → ¯ maps X birationally onto a normal surface X ¯ whose only singularities X are either rational or minimally elliptic. Using the naturality of γw,p (X; C), it suffices to prove (1.5) after replacing E by ψ ∗ E. In this case, by Theorem 1.4 with w = [KX ], γw,p (X; C)(M d ) 6= 0 for all p ≪ 0. Corollary 1.6. Let X be a simply connected minimal surface of general type with pg (X) = 0. Then there exist (i) a class w ∈ H 2 (X; Z/2Z); (ii) an integer p ∈ Z; (iii) a chamber C for X of type (w, p), and (iv) a homotopy equivalence α : X → Y , where Y is either the blowup of P2 at n distinct points or Y = P1 × P1 , such that, for w′ = (α∗ )−1 (w) and C ′ = (α∗ )−1 (C), α∗ γw′ ,p (Y ; C ′ ) 6= ±γw,p (X; C). Proof. If X is homotopy equivalent to P1 × P1 then the theorem follows from 2 [33]. Otherwise X is oriented homotopy equivalent to P2 #nP , for 1 ≤ n ≤ 8, and we claim that there exists a homotopy equivalence α : X → Y such that α∗ [KY ] = −[KX ]. Indeed, every integral isometry H 2 (Y ; Z) → H 2 (X; Z) is realized by an oriented homotopy equivalence. Thus it suffices to show that every two characteristic elements of H 2 (Y ; Z) of square 9 − n are conjugate under the isometry group, which follows from the appendix to this paper. Choosing such a homotopy equivalence α, let e be the class of an exceptional curve in Y and let E = α∗ e. Then E 2 = −1 and E · [KX ] = 1. We may now apply Theorem 1.5 to the class E, noting that E is a (1, 1) class since pg (X) = 0. Let M and C be a divisor and a chamber which satisfy the conclusions of Theorem 1.5 and let m = (α∗ )−1 M . If w is the mod 2 reduction of [KX ], then w′ is the mod two reduction of [KY ], so

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that w′ is also characteristic. Now e is the class of a smoothly embedded 2-sphere in Y since it is the class of an exceptional curve. Moreover m · e = 0. By Theorem 1.2, γw′ ,p (Y ; C ′ )(md ) = 0 since e is represented by a smoothly embedded 2-sphere and w′ is characteristic. But γw,p (X; C)(M d ) 6= 0 by Theorem 1.5 and the choice of M . Thus α∗ γw′ ,p (Y ; C ′ ) 6= ±γw,p (X; C). Using the result of Wall [38] that every homotopy self-equivalence from Y to itself is realized by a diffeomorphism, the proof above shows that the conclusions of the corollary hold for every homotopy equivalence α : X → Y . 1.3. Reduction to the minimal case We begin by recalling some terminology and results from [13]. A good generic rational surface Y is a rational surface such that KY = −C where C is a smooth curve, and such that there does not exist a smooth rational curve on Y with selfintersection −2. Every rational surface is diffeomorphic to a good generic rational surface. ˜ → X be Theorem 1.7. Let X be a minimal surface of general type and let X ′ ′ a blowup of X at r distinct points. Let E1 , . . . , Er be the homology classes of the ˜ Let ψ0 : X ˜ → Y˜ be a diffeomorphism, where Y˜ is a good exceptional curves on X. ˜ → Y˜ and a good generic rational surface. Then there exist a diffeomorphism ψ : X generic rational surface Y with the following properties: (i) The surface Y˜ is the blowup of Y at r distinct points. (ii) If e1 , . . . , er are the classes of the exceptional curves in H 2 (Y˜ ; Z) for the blowup Y˜ → Y , then possibly after renumbering ψ ∗ (ei ) = Ei′ for all i. ˜ and H 2 (Y ) with a subgroup (iii) Identifying H 2 (X) with a subgroup of H 2 (X) 2 ˜ ∗ of H (Y ) in the obvious way, we have ψ (H 2 (Y )) = H 2 (X). Moreover, for every choice of an isometry τ from H 2 (Y ) to H 2 (X), there exists a choice of a diffeomorphism ψ satisfying (i)–(iii) above and such that ψ ∗ |H 2 (Y ) = τ . Proof. Let e′i ∈ H 2 (Y˜ ; Z) satisfy ψ0∗ (e′i ) = Ei′ . Thus the Poincar´e dual of e′i is represented by a smoothly embedded 2-sphere in Y˜ . It follows that reflection re′i in e′i is realized by an orientation-preserving self-diffeomorphism of Y˜ . To see what this says about e′i , we shall recall the following terminology from [13]. Let H(Y˜ ) be the set { x ∈ H 2 (Y˜ ; R) | x2 = 1 } and let K(Y˜ ) ⊂ H 2 (Y˜ ; R) be intersection of the closure of the K¨ ahler cone of Y˜ with H(Y˜ ). Then K(Y˜ ) is a convex subset of H(Y˜ ) whose walls consist of the classes of exceptional curves on ˜ Y˜ together with [−KY˜ ] if b− 2 (Y ) ≥ 10, which is confusingly called the exceptional ˜ wall of K(Y ). Let R be the group generated by the reflections in the walls of K(Y˜ ) defined by exceptional classes and define the super P -cell S = S(P ) by [ S= γ · K(Y˜ ). γ∈R

By Theorem 10A on p. 355 of [13], for an integral isometry ϕ of H 2 (Y˜ ; R), there exists a diffeomorphism of Y˜ inducing ϕ if and only if ϕ(S) = ±S. (Here, if ˜ b− 2 (Y ) ≤ 9, S = H and the result reduces to a result of C.T.C. Wall [38].) Note that H(Y˜ ) has two connected components, and reflection re in a class e of square −1 preserves the set of connected components. Thus if re (S) = ±S, then necessarily re (S) = S. Next we have the following purely algebraic lemma:

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Lemma 1.8. Let e be a class of square −1 in H 2 (Y˜ ; Z) such that the reflection re satisfies re (S) = S. Then there is an isometry ϕ of H 2 (Y˜ ; Z) preserving S which sends e to the class of an exceptional curve. Proof. We first claim that, if W is the wall corresponding to e, then W meets the interior of S. Indeed, the interior int S of S is connected, by Corollary 5.5 of [13] p. 340. If W does not meet int S, then the sets { x ∈ int S | e · x > 0 } and { x ∈ int S | e · x < 0 } are disjoint open sets covering int S which are exchanged under the reflection re . Since at least one is nonempty, they are both nonempty, contradicting the fact that int S is connected. Thus W must meet int S. Now let C be a chamber for the walls of square −1 which has W as a wall. It follows from Lemma 5.3(b) on p. 339 of [13] that C ∩ S is a P -cell P and that W defines a wall of P which is not the exceptional wall. By Lemma 5.3(e) of [13], S is the unique super P -cell containing P , and the reflection group generated by the elements of square −1 defining the walls of P acts simply transitively on the P -cells in S. There is thus an element ϕ in this reflection group which preserves S and sends P to K(Y˜ ) and W to a wall of K(Y˜ ) which is not an exceptional wall. It follows that ϕ(e) is the class of an exceptional curve on Y˜ . Returning to the proof of Theorem 1.7, apply the previous lemma to the reflection in e′r . There is thus an isometry ϕ preserving S such that ϕ(e′r ) = er , where er is the class of an exceptional curve on Y˜ . Moreover ϕ is realized by a diffeomorphism. Thus after composing with the diffeomorphism inducing ϕ, we can assume that e′r = er , or equivalently that ψ0∗ er = [Er′ ]. Let Y˜ → Y˜1 be the blowing down of the exceptional curve whose class is er . Then Y˜1 is again a good generic surface by [13] p. 312 Lemma 2.3. Since e′1 , . . . , e′r−1 are orthogonal to er , they lie in the subset H 2 (Y˜1 ) of H 2 (Y˜ ). For i 6= r, the reflection in e′i preserves W ∩S, where W = (er )⊥ . Now W is just H 2 (Y˜1 ) and K(Y˜ ) ∩ H 2 (Y˜1 ) = K(Y˜1 ) by [13] p. 331 Proposition 3.5. The next lemma relates the corresponding super P -cells: Lemma 1.9. W ∩ S is the super P -cell S1 for Y˜1 containing K(Y˜1 ). Proof. Trivially S1 ⊆ W ∩ S, and both sets are convex subsets with nonempty interiors. If they are not equal, then there is a P -cell P ′ ⊂ S1 and an exceptional wall of P ′ which passes through the interior of S ∩ W . If κ(P ′ ) is the exceptional wall meeting S ∩ W , then, by [13] p. 335 Lemma 4.6, κ(P ′ ) − er is an exceptional wall of P for a well-defined P -cell in S, and κ(P ′ ) − er must pass through the interior of S. This is a contradiction. Hence S ∩ W = S1 is a super P -cell of Y˜1 , and we have seen that it contains K(Y˜1 ). Returning to the proof of Theorem 1.7, reflection in e′r−1 preserves S1 . Applying Lemma 1.8, there is a diffeomorphism of Y˜1 which sends e′r−1 to the class of an exceptional curve er−1 . Of course, there is an induced diffeomorphism of Y˜ which fixes er . Now we can clearly proceed by induction on r. The above shows that after replacing ψ0 by a diffeomorphism ψ we can find Y as above so that (i) and (ii) of the statement of Theorem 3 hold. Clearly ψ ∗ (H 2 (Y )) =

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H 2 (X). By the theorem of C.T.C. Wall mentioned above, there is a diffeomorphism of Y realizing every integral isometry of H 2 (Y ). So after further modifying by a diffeomorphism of Y , which extends to a diffeomorphism of Y˜ fixing the classes of the exceptional curves, we can assume that the diffeomorphism ψ restricts to τ for any given isometry from H 2 (Y ) to H 2 (X). We can now give a proof of Theorem 0.1: Theorem 0.1. No complex surface of general type is diffeomorphic to a rational surface. ˜ →X Proof. Suppose that X is a minimal surface of general type and that ρ : X ˜ is is a blowup of X diffeomorphic to a rational surface. We may assume that X ′ ˜ ˜ diffeomorphic via ψ to a good generic rational surface Y , and that ρ : Y → Y is a blow up of Y˜ such that Y and ψ satisfy (i)–(iii) of Theorem 1.7. Choose w, p, α, C for X such that the conclusions of Corollary 1.6 hold, with C ′ the corresponding chamber on Y , and let C˜′ be any chamber for Y˜ containing C ′ in its closure. Then ˜ containing C in its closure. Using the last sentence ψ ∗ C˜′ = C˜ is a chamber on X of Theorem 1.7, we may assume that ψ ∗ |H 2 (Y ) = α∗ . Thus ψ ∗ (ρ′ )∗ = ρ∗ α∗ . By the functorial properties of Donaldson polynomials, and viewing H 2 (X; Z/2Z) as a ˜ Z/2Z), and similarly for Y˜ , we have subset of H 2 (X; ˜ C) ˜ = ±γw,p (X, ˜ C). ˜ ψ ∗ γw′ ,p (Y˜ , C˜′ ) = ±γψ∗ w′ ,p (X, Restricting each side to ψ ∗ H2 (Y ) = H2 (X), we obtain by repeated application of Lemma 1.3 that α∗ γw′ ,p (Y ; C ′ ) = ±γw,p (X; C). But this contradicts Corollary 1.6. Using Theorem 1.5, we have the following generalization of Theorem 0.4 in the introduction to the case of nonminimal algebraic surfaces: Theorem 1.10. Let X be a minimal simply connected surface of general type, and ˜ be a blowup of X. let E ∈ H 2 (X; Z) satisfy E 2 = −1 and E · KX = 1. Let X ˜ Z), the class E is not represented by Then, viewing H 2 (X; Z) as a subset of H 2 (X; ˜ a smoothly embedded 2-sphere in X. Proof. Suppose instead that E is represented by a smoothly embedded 2-sphere. If pg (X) > 0, then it follows from the results of [6] that E is a (1, 1)-class, i.e. E lies in the image of Pic X inside H 2 (X; Z). Of course, this is automatically true if pg (X) = 0. Next assume that pg (X) = 0. By Theorem 1.5, there exists a w ∈ H 2 (X; Z/2Z), an integer p, and a chamber C of type (w, p), such that γw,p (X; C)(M d ) 6= 0, where M is a class in the closure of C and M · E = 0. Consider the Donaldson polynomial ˜ C), ˜ where we view w as an element of H 2 (X; ˜ Z/2Z) in the natural way and γw,p (X; ˜ containing C in its closure. Then C˜ also contains C˜ is a chamber of type (w, p) on X d ˜ C)(M ˜ M in its closure. Thus, by Theorem 1.2, γw,p (X; ) = 0. On the other hand, d d ˜ ˜ by Lemma 1.3, γw,p (X; C)(M ) = ±γw,p (X; C)(M ) 6= 0. This is a contradiction. The case where pg (X) > 0 is similar. We also have the following corollary, which works under the assumptions of Theorem 1.10 for surfaces with pg > 0:

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Corollary 1.11. Let X be a simply connected surface of general type with pg (X) > 0, not necessarily minimal, and let E ∈ H 2 (X; Z) satisfy E 2 = −1 and E·KX = −1. Suppose that E is represented by a smoothly embedded 2-sphere. Then E is the cohomology class associated to an exceptional curve. Proof. Using [15] and [6], we see that if E is not the cohomology class associated to an exceptional curve, then E ∈ H 2 (Xmin ; Z), where Xmin is the minimal model of X and we have the natural inclusion H 2 (Xmin ; Z) ⊆ H 2 (X; Z). We may then apply Theorem 1.10 to conclude that −E cannot be represented by a smoothly embedded 2-sphere, and thus that E cannot be so represented, a contradiction. 2. A generalized nonvanishing theorem 2.1. Statement of the theorem and the first part of the proof In this section, we shall prove Theorem 1.4. We first recall its statement: Theorem 1.4. Let X be a simply connected algebraic surface with pg (X) = 0, and let M be a nef and big divisor on X which is eventually base point free. Denote ¯ the birational morphism defined by |kM | for k ≫ 0, so that X ¯ is a by ϕ : X → X ¯ normal projective surface. Suppose that X has only rational or minimally elliptic singularities, and that ϕ does not contract any exceptional curves to points. Let w ∈ H 2 (X; Z/2Z) be the mod 2 reduction of the class [KX ]. Then there exists a constant A depending only on X and M with the following property: For all integers p ≤ A, let C be a chamber of type (w, p) containing M in its closure and suppose that C has nonempty intersection with the ample cone of X. Set d = −p − 3. Then γw,p (X; C)(M d ) > 0. A similar conclusion holds if pg (X) > 0. Proof. We begin by fixing some notation. For L an ample line bundle on X, given a divisor D on X and an integer c, let ML (D, c) denote the moduli space of isomorphism classes of L-stable rank two holomorphic vector bundles on X with c1 (V ) = D and c2 (V ) = c. Let w be the mod 2 reduction of D and let p = D2 − 4c. Then we also denote ML (D, c) by ML (w, p), the moduli space of equivalence classes of L-stable rank two holomorphic vector bundles on X corresponding to the choice of (w, p). Here we recall that two vector bundles V and V ′ are equivalent if there exists a holomorphic line bundle F such that V ′ = V ⊗ F . The invariants w and p only depend on the equivalence class of V . Let ML (w, p) denote the Gieseker compactification of ML (w, p), i.e. the Gieseker compactification ML (D, c) of ML (D, c). Thus ML (w, p) is a projective variety. We now fix a compact neighborhood N of M inside the positive cone ΩX of X. Note that, since M is nef, such a neighborhood has nontrivial intersection with the ample cone of X. Using a straightforward extension of the theorem of Donaldson [10] on the dimension of the moduli space (see e.g. [12] Chapter 8, [32], [24]), there exist constants A and A′ such that, for all ample line bundles L such that c1 (L) ∈ N , the following holds: (1) If p ≤ A, then the moduli space ML (w, p) is good, in other words it is generically reduced of the correct dimension −p − 3; (2) ML (w, p) is a dense open subset of ML (w, p) and the generic point of ML (w, p) − ML (w, p) correspond to a torsion free sheaf V such that the

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length of V ∨∨ /V is one and such that the support of V ∨∨ /V is a generic point of X; (3) For all p′ ≥ A, the dimension dim ML (w, p′ ) ≤ A′ . ¯ We shall need to make one more assumption on the integer p. Let ϕ : X → X be the contraction morphism associated to M . For each connected component E of the set of exceptional fibers of ϕ, fix a (possibly nonreduced) curve Z on X whose support is exactly E. In practice we shall always take Z to be the fundamental cycle of the singularity, to be defined in Subsection 2.3 below. A slight generalization ([12], Chapter 8) of Donaldson’s theorem on the dimension of the moduli space then shows the following: after possibly modifying the constant A, (4) The generic V ∈ ML (w, p) satisfies: the natural map H 1 (X; ad V ) → H 1 (Z; ad V |Z) is surjective. In other words, the local universal deformation of V is versal when viewed as a deformation of V |Z (keeping the determinant fixed). We now assume that p ≤ A. Let L be an ample line bundle which is not separated from M by any wall of type (w, p) (or equivalently of type (D, c)), and moreover does not lie on any wall of type (w, p). Thus by assumption, none of the points of ML (D, c) corresponds to a strictly semistable sheaf. Let C ⊂ X be a smooth curve of genus g. Suppose that C · D = 2a is even. Choosing a line bundle θ of degree g − 1 − a on C, we can form the determinant line bundle L(C, θ) on the moduli functor associated to torsion free sheaves corresponding to the values w and p ([15], Chapter 5). Using Proposition 1.7 in [23], this line bundle descends to a line bundle on ML (w, p), which we shall continue to denote by L(C, θ). Moreover, by the method of proof of Theorem 2 of [23], the line bundle L(C, θ) depends only on the linear equivalence class of C, in the sense that if C and C ′ are linearly equivalent and θ′ is a line bundle of degree g − 1 − a on C ′ , then L(C, θ) ∼ = L(C ′ , θ′ ). Next we shall use the following result, whose proof is deferred to the next subsection: Lemma 2.1. In the above notation, if k ≫ 0 and C ∈ |kM | is a smooth curve, then, for all N ≫ 0, the linear system associated to L(C, θ)N has no base points and defines a generically finite morphism from ML (w, p) to its image. In particular, if d = dim ML (w, p), then c1 (L(C, θ))d > 0. It follows by applying an easy adaptation of Theorem 6 in [23] or the results of [25] to the case pg (X) = 0 that, since the spaces ML (w, p′ ) have the expected dimension for an appropriate range of p′ ≥ p, c1 (L(C, θ))d is exactly the value k d γw,p (X; C)(M d ). Thus we have proved Theorem 1.4, modulo the proof of Lemma 2.1. This proof will be given below. 2.2. A generalization of a result of Bogomolov We keep the notation of the preceding subsection. Thus M is a nef and big divisor such that the complete linear system |kM | is base point free whenever k ≫ 0. Throughout, we shall further assume that M is divisible by 2 in Pic X. Moreover w and p are now fixed and L is an ample line bundle such that c1 (L) ∈ N is not separated from M by a wall of type (w, p) and moreover that c1 (L) does not

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lie on a wall of type (w, p). In particular the determinant line bundle L(C, θ) is defined for all smooth C in |kM | for all k ≫ 0. We then have the following generalization of a restriction theorem due to Bogomolov [4]: Lemma 2.2. With the above notation, there exists a constant k0 depending only on w, p, M , and L, such that for all k ≥ k0 and all smooth curves C ∈ |kM |, the following holds: for all c′ ≤ c and V ∈ ML (D, c′ ), either V |C is semistable or there exists a divisor G on X, a zero-dimensional subscheme Z and an exact sequence 0 → OX (G) → V → OX (D − G) ⊗ IZ → 0, where 2G − D defines a wall of type (w, p) containing M and C ∩ Supp Z = 6 ∅. Proof. The proof follows closely the original proof of Bogomolov’s theorem [4] or [15] Section 5.2. Choose k0 ≥ −p and assume also that there exists a smooth curve C in |kM | for all k ≥ k0 . Suppose that V |C is not semistable. Then there exists a surjection V |C → F , where F is a line bundle on C with deg F = f < (D · C)/2. Let W be the kernel of the induced surjection V → F . Thus W is locally free and there is an exact sequence 0 → W → V → F → 0. A calculation gives p1 (ad W ) = p1 (ad V ) + 2D · C + (C)2 − 4f > p + k 2 (M )2 ≥ p + p2 ≥ 0. By Bogomolov’s inequality, W is unstable with respect to every ample line bundle on X. Thus there exists a divisor G0 and an injection OX (G0 ) → W (which we may assume to have torsion free cokernel) such that 2(L · G0 ) > L · (D − C), i.e. L · (2G0 − D + C) > 0. By hypothesis there is an exact sequence 0 → OX (G0 ) → W → OX (−G0 + D − C) ⊗ IZ0 → 0. Thus 0 < p1 (ad W ) = (2G0 − D + C)2 − 4ℓ(Z0 ) ≤ (2G0 − D + C)2 . It follows that (2G0 −D+C)2 > 0. As L·(2G0 −D+C) > 0 and (2G0 −D+C)2 > 0, M · (2G0 − D + C) ≥ 0 as well, i.e. −(M · (2G0 − D)) ≤ k(M )2 . On the other hand, since V is L-stable, L · (2G0 − D) < 0. Since L and M are not separated by any wall of type (w, p), it follows that M · (2G0 − D) ≤ 0. Finally using p1 (ad W ) = (2G0 − D + C)2 − 4ℓ(Z0 ) = p1 (ad V ) + 2D · C + (C)2 − 4f > p + k 2 (M )2 , we obtain (2G0 − D)2 + 2k(2G0 − D) · M > p.

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Let m = −(2G0 − D) · M . As we have seen above m ≤ kM 2 and m ≥ 0. The above inequality can be rewritten as 2km < (2G0 − D)2 − p. We claim that m = 0. Otherwise 2k <

p (2G0 − D)2 − . m m 2

By the Hodge index theorem (2G0 − D)2 M 2 ≤ [(2G0 − D) · M ] = m2 , so that (2G0 − D)2 ≤ m2 /M 2 . Plugging this into the inequality above, using −p ≥ 0, gives 2k <

m p − ≤ k − p, M2 m

i.e. k < −p, contradicting our choice of k. Thus m = −(2G0 − D) · M = 0. Now the inclusions OX (G0 ) ⊂ W ⊂ V define an inclusion OX (G0 ) ⊂ V . Thus there is an effective divisor E and an inclusion OX (G0 + E) → V with torsion free cokernel. Let G = G0 + E. Thus there is an exact sequence 0 → OX (G) → V → OX (−G + D) ⊗ IZ → 0. We claim that (2G − D) · M = 0. Since V is L-stable, (2G − D) · L < 0, and since L and M are not separated by a wall of type (w, p), (2G − D) · M ≤ 0. On the other hand, (2G − D) · M = (2G0 − D) · M + 2(E · M ) = −m + 2(E · M ) = 2(E · M ). As E is effective and M is nef, 2(E · M ) ≥ 0. Thus (2G − D) · M = 0. As M 2 > 0, we must have (2G − D)2 < 0. Using p = (2G − D)2 − 4ℓ(Z) ≤ (2G − D)2 , we see that 2G − D is a wall of type (w, p). Finally note that Supp Z ∩ C 6= ∅, for otherwise we would have V |C semistable. This concludes the proof of Lemma 2.2. Returning to the proof of Lemma 2.1, we claim first that, given k ≫ 0 and C ∈ |kM |, for all N sufficiently large the sections of L(C, θ)N define a base point free linear series on ML (w, p). To see this, we first claim that, for k ≫ 0, and for a generic C ∈ |kM |, the restriction map V 7→ V |C defines a rational map rC : ML (w, p) 99K M(C), where M(C) is the moduli space of equivalence classes of semistable rank two bundles on C such that the parity of the determinant is even. It suffices to prove that, for every component N of ML (w, p) there is one V ∈ N and one C ∈ |kM | such that V |C is semistable, for then the same will hold for a Zariski open subset of |kM |. Now given V , choose a fixed C0 ∈ |kM |. If V |C0 is not semistable, then by Lemma 2.2 there is an exact sequence 0 → OX (G) → V → OX (−G + D) ⊗ IZ → 0, where Z is a zero-dimensional subscheme of X meeting C0 . Choosing C to be a curve in |kM | disjoint from Z, which is possible since |kM | is base point free, it follows that the restriction V |C is semistable.

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For C fixed, let BC = { V ∈ ML (w, p) : either V is not locally free over some point of C or V |C is not semistable }. By the openness of stability and local freeness, the set BC is a closed subset of ML (w, p) and rC defines a morphism from ML (w, p) − BC to M(C). Standard estimates (cf. [10], [12], [32], [24], [27]) show that, possibly after modifying the constant A introduced at the beginning of the proof of Theorem 1.4, the codimension of BC is at least two in ML (w, p) provided that p ≤ A (where as usual A is independent of k and depends only on X and M ). Indeed the set of bundles V which fit into an exact sequence 0 → OX (G) → V → OX (D − G) ⊗ IZ → 0, where G is a divisor such that p (2G − D) · M = 0, may be parametrized by a 3 scheme of dimension − 4 p + O( |p|) by e.g. [12], Theorem 8.18. Moreover the p constant implicit in the notation O( |p|) can be chosen uniformly over N . The case of nonlocally free V is taken care of by assumption (2) in the discussion of the constant A: it follows from standard deformation theory (see again [12], [24]) that at a generic point of the locus of nonlocally free sheaves corresponding to the semistable torsion free sheaf V the deformations of V are versal for the local deformations of the singularities of V . Thus for a general nonlocally free V , V has just one singular point which is at a general point of X and so does not lie on C. Thus the set of V which are not locally free at some point of C has codimension at least two (in fact exactly two) in ML (w, p). Let LC be the determinant line bundle on M(C) associated to the line bundle θ (see for instance [15] Chapter 5 Section 2). Then by definition the pullback via rC of LC is the restriction of L(C, θ) to ML (w, p) − BC . Since BC has codimension N two, the sections of LN on ML (w, p). Since C pull back to sections of L(C, θ) LC is ample, given V ∈ ML (w, p) − BC , there exists an N and a section of LN C not vanishing at rC (V ), and thus there is a section of L(C, θ)N not vanishing at V . Moreover by [23], for all smooth C ′ ∈ |kM | and choice of an appropriate line bundle θ′ on C ′ , there is an isomorphism L(C, θ)N ∼ = L(C ′ , θ′ )N . Next we claim that, for every V ∈ ML (w, p), there exists a C such that V is locally free above C and V |C is semistable. Given V , it fails to be locally free at a finite set of points, and its double dual W is again semistable. Thus applying the above to W , and again using the fact that |kM | has no base points, we can find C such that V is locally free over C and such that V |C = W |C is semistable. Thus, given V , there exists an N and a section of L(C, θ)N which does not vanish at V . Since ML (w, p) is of finite type, there exists an N which works for all V , so that the linear system corresponding to L(C, θ)N has no base points. Finally we must show that, for k ≫ 0, the morphism induced by L(C, θ)N is in fact generically finite for N large. We claim that it suffices to show that the restriction of the rational map rC to ML (w, p) − BC is generically finite (it is here ¯ in the statement of Theorem that we must use the condition on the singularities of X 1.4). Supposing this to be the case, and fixing a V ∈ ML (w, p) − BC for which −1 rC (rC (V )) is finite, we consider the intersection of all the divisors in L(C, θ)N

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containing V , where N is chosen so that LN C is very ample. This intersection always −1 contains V and is a subset of rC (rC (V )) ∪ BC . In particular V is an isolated point of the fiber, and so the morphism defined by L(C, θ)N cannot have all fibers of purely positive dimension. Thus it is generically finite. To see that rC is generically finite, we shall show that, for generic V , the restriction map r : H 1 (X; ad V ) → H 1 (C; ad V |C)) is injective. The map r is just the differential of the map rC from ML (w, p) to M(C) at the point corresponding to V , and so if V is generic then rC is finite. Now the kernel of the map r is a quotient of H 1 (X; ad V ⊗ OX (−C)), and we need to find circumstances where this group is zero, at least if C ∈ |kM | for k sufficiently large. By Serre duality it suffices to show that H 1 (X; ad V ⊗ OX (C) ⊗ KX ) = 0 for k sufficiently large. By applying the Leray spectral sequence to the morphism ¯ it suffices to show that ϕ : X → X, ¯ R0 ϕ∗ (ad V ⊗ OX (C) ⊗ KX )) = 0 H 1 (X; and that R1 ϕ∗ (ad V ⊗ OX (C) ⊗ KX ) = 0. Now M is the pullback of an ample ¯ on X, ¯ and OX (C) is the pullback of (M ¯ )⊗k . Thus for fixed V and line bundle M k ≫ 0, ¯ R0 ϕ∗ (ad V ⊗ OX (C) ⊗ KX )) H 1 (X; ¯ R0 ϕ∗ (ad V ⊗ KX ) ⊗ (M ¯ k )) = 0. =H 1 (X; ¯ k ), so that it is Moreover R1 ϕ∗ (ad V ⊗ OX (C) ⊗ KX )) = R1 ϕ∗ (ad V ⊗ KX ) ⊗ (M 1 enough to show that R ϕ∗ (ad V ⊗ KX ) = 0. By the formal functions theorem, 1 R1 ϕ∗ (ad V ⊗ KX ) = lim ←− H (mZ; ad V ⊗ KX |mZ), m

S

where Z = Zi is the union of the connected components Zi of the one-dimensional fibers of ϕ. Thus it suffices to show that, for all i and all positive integers m, H 1 (mZi ; ad V ⊗ KX |mZi ) = 0. Now by the adjunction formula ωmZi = KX ⊗ OX (mZi )|mZi , where ωmZi is the dualizing sheaf of the Gorenstein scheme mZi . Thus KX |mZi = OX (−mZi )|mZi ⊗ ωmZi and we must show the vanishing of H 1 (mZi ; (ad V ⊗ OX (−mZi ))|mZi ⊗ ωmZi ). By Serre duality, it suffices to show that, for all m > 0, H 0 (mZi ; (ad V ⊗ OX (mZi ))|mZi ) = 0. We shall deal with this problem in the next subsection. Remark. (1) Instead of arguing that the restriction map rC was generically finite, one could also check that it was generically one-to-one by showing that for generic V1 , V2 , the restriction map H 0 (X; Hom(V1 , V2 )) → H 0 (C; Hom(V1 , V2 )|C)

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is surjective (since then an isomorphism from V1 |C to V2 |C lifts to a nonzero map from V1 to V2 , necessarily an isomorphism by stability). In turn this would have amounted to showing that H 1 (X; Hom(V1 , V2 ) ⊗ OX (−C)) = 0 for generic V1 and V2 , and this would have been essentially the same argument. ¯ is the blowup of a smooth surface X ¯ at a point (2) Suppose that ϕ : X → X ∼ ¯ x, and that M is the pullback of an ample divisor on X. Let Z = P1 be the exceptional curve. In this case, if c1 (V ) · Z is odd, say 2a + 1, then the generic behavior for V |Z is V |Z ∼ = OP1 (a) ⊕ OP1 (a + 1) and the restriction map exhibits ML (w, p) (generically) as a P1 -bundle over its image (see for instance [5]). Thus the hypothesis that ϕ contracts no exceptional curve is essential. 2.3. Restriction of stable bundles to certain curves Let us recall the basic properties of rational and minimally elliptic singularities. ¯ and let ϕ : X → X ¯ Let x be a normal singular point on a complex surface X, S ¯ Supppose that ϕ−1 (x) = D be the minimal resolution of singularities of X. i i. The singularity is a rational singularity if (R1 ϕ∗ OX )x = 0. Equivalently, by [1], x is rational if and only if, for every choice of nonnegative integers ni such that at P least one of the ni is strictly positive, if we set Z = i ni Di , the arithmetic genus pa (Z) of the effective curve Z satisfies pa (Z) ≤ 0. Here pa (Z) = 1 − χ(OZ ) = 1 − h0 (OZ ) + h1 (OZ ) ≤ h1 (OZ ); moreover we have the adjunction formula 1 pa (Z) = 1 + (KX + Z) · Z. 2 Now every minimal resolution of a normal surface singularity x has a fundamental cycle Z0 , which is an effective cycle Z0 supported in the set ϕ−1 (x) and satisfying Z0 · Di ≤ 0 and Z0 · Di < 0 for some i which is minimal with respect to the above properties. We may find Z0 as follows [22]: start with an arbitrary component A1 of ϕ−1 (x) and set Z1 = A1 . Now either Z0 = A1 or there exists another component A2 with Z1 · A2 > 0. Set Z2 = Z1 + A2 and continue P this process. Eventually we reach Zk = Z0 . Such a sequence A1 , . . . , Ak with Zi = j≤i Aj and Zi · Ai+1 > 0, Zk = Z0 is called a computation sequence. By a theorem of Artin [1], x is rational if and only if pa (Z0 ) ≤ 0, where Z0 is the fundamental cycle, if and only if pa (Z0 ) = 0. Moreover, if x is a rational singularity, then every component Di of ϕ−1 (x) is a smooth rational curve, the Di meet transversally at at most one point, and the dual graph of ϕ−1 (x) is contractible. Next we recall the properties of minimally elliptic singularities [22]. A singularity x is minimally elliptic if and only P if there exists a minimally elliptic cycle Z for (Z) = 1 and x, in other words a cycle Z = i ni Di with all ni > 0 such that paP pa (Z ′ ) ≤ 0 for all nonzero effective cycles Z ′ < Z (i.e. such that Z ′ = i n′i Di with 0 ≤ n′i ≤ ni and Z ′ 6= Z). In this case it follows that Z = Z0 is the fundamental cycle for x, and (KX + Z0 ) · Di = 0 for every component Di of ϕ−1 (x). If Z0 is reduced, i.e. if ni = 1 for all i, then the possibilities for x are as follows: (1) ϕ−1 (x) is an irreducible curve of arithmetic genus one, and thus is either a smooth elliptic curve or a singular rational curve with either a node or a cusp; St (2) ϕ−1 (x) = i=1 Di is a cycle of t ≥ 2 smooth rational curves meeting transversally, i.e. Di · Di+1 = 1, Di · Dj 6= 0 if and only if i ≡ j ± 1 mod t, except for t = 2 where D1 · D2 = 2;

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(3) ϕ−1 (x) = D1 ∪ D2 , where the Di are smooth rational, D1 · D2 = 2 and D1 ∩ D2 is a single point (so that ϕ−1 (x) has a tacnode singularity) or ϕ−1 (x) = D1 ∪ D2 ∪ D3 where the Di are smooth rational, Di · Dj = 1 but D1 ∩ D2 ∩ D3 is a single point (the three curves meet at a common point). Here x is called a simple elliptic singularity in case ϕ−1 (x) is a smooth elliptic curve, a cusp singularity if ϕ−1 (x) is an irreducible rational curve with a node or a cycle as in (2), and a triangle singularity in the remaining cases. If Z0 is not reduced, then all components Di of ϕ−1 (x) are smooth rational curves meeting transversally and the dual graph of ϕ−1 (x) is contractible. With this said, and using the discussion in the previous subsection, we will complete the proof of Theorem 1.4 by showing that H 0 (mZi ; ad V ⊗OX (mZi )|mZi ) = 0 ¯ and Zi is an effective cycle for all i, where x1 , . . . , xk are the singular points of X with Supp Zi = ϕ−1 (xi ). The precise statement is as follows: ¯ be a birational morphism from X to a normal Theorem 2.3. Let ϕ : X → X ¯ projective surface X, corresponding to a nef, big, and eventually base point free divisor M . Let w be the mod 2 reduction of [KX ], and suppose that (i) ϕ contracts no exceptional curve; in other words, if E is an exceptional curve of the first kind on X, then M · E > 0. ¯ has only rational and minimally elliptic singularities. (ii) X Then there exists a constant A depending only on p and N with the following ¯ there exists an effective cycle Z with property: for every singular point x of X, −1 Supp Z = ϕ (x) such that, for all ample line bundles L in N , all p with p ≤ A, and generic bundles V in ML (w, p), H 0 (mZ; ad V ⊗ OX (mZ)|mZ) = 0 for every positive integer m. The statement of (i) may be rephrased by saying that X is the minimal resolu¯ As ad V ⊂ Hom(V, V ), it suffices to prove that H 0 (mZ; Hom(V, V ) ⊗ tion of X. OX (mZ)|mZ) = 0. We will consider the case of rational singularities and minimally elliptic singularities separately. Let us begin with the proof for rational S singularities. Let ϕ−1 (x) = i Di , where each Di is a smooth rational curve. By the assumption (4) of the previous subsection, we can assume that the constant A has been chosen so that V |Di is a generic bundle over Di ∼ = P1 for every i. Thus ∼ either there exists an integer a such that V |Di = OP1 (a) ⊕ OP1 (a + 1), if w · Di 6= 0, or there exists an a such that V |Di ∼ = OP1 (a) ⊕ OP1 (a), if w · Di = 0. Next, we have the following claim: ¯ be a resolution Claim 2.4. Suppose that x is a rational singularity. Let ϕ : X → X of x. There exist a sequence of curves B0 , . . . , Bk , such that Bi ⊆ ϕ−1 (x) for all i, with the following property: P (1) Let Ci = j≤i Bi . Then Bi · Ci ≤ Bi2 + 1. (2) Ck = Z0 , the fundamental cycle of x. Proof. Since (KX + Z0 ) · Z0 < 0, there must exist a component B (0) = Di of Supp Z0 = ϕ−1 (x) such that (KX + Z0 ) · B (0) < 0. Thus Z0 · B (0) < −KX · B (0) = (B (0) )2 + 2.

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Set Z1 = Z0 − B (0) . Suppose that Z1 is nonzero. Then Z1 is again effective, and by Artin’s criterion pa (Z1 ) ≤ 0. Thus by repeating the above argument there is a B (1) contained in the support of Z1 such that Z1 · B (1) < (B (1) )2 + 2. Continuing, we eventually find B (2) , . . . , B (k) with B (i) contained in the support of Zi , Zi+1 = Zi − B (i) and Zk = B (k)P , and such that Zi · B (i) < (B (i) )2 + 2. If we now relabel (i) B = Bk−i , then Zi = j≤n−i Bj and the curves B0 , . . . , Bk are as claimed. Returning to the proof of Theorem 2.3, we first prove that H 0 (Z0 ; Hom(V, V ) ⊗ OX (Z0 )|Z0 ) = 0. We have the exact sequence 0 → OCi−1 (Ci−1 ) → OCi (Ci ) → OBi (Ci ) → 0. Tensor this sequence by Hom(V, V ). We shall prove by induction that H 0 (Hom(V, V ) ⊗ OCi (Ci )) = 0 for all i. It suffices to show that H 0 (Hom(V, V ) ⊗ OBi (Ci )) = 0 for all i. Now OBi (Ci ) is a line bundle on the smooth rational curve Bi . If V |Bi ∼ = OP1 (a) ⊕ OP1 (a + 1), then w · Bi 6= 0 and so Bi2 is odd. Since Bi is not an exceptional curve, Bi2 ≤ −3 and so Bi · Ci ≤ −2. Thus, as Hom(V, V )|Bi = OP1 (−1) ⊕ OP1 ⊕ OP1 ⊕ OP1 (1), we see that H 0 (Hom(V, V ) ⊗ OBi (Ci )) = 0. Likewise if V |Di ∼ = OP1 (a) ⊕ OP1 (a), then using Bi · Ci ≤ −1 we again have H 0 (Hom(V, V ) ⊗ OBi (Ci )) = 0. Thus by induction H 0 (Hom(V, V ) ⊗ OCk (Ck )) = H 0 (Hom(V, V ) ⊗ OZ0 (Z0 )) = 0. The vanishing of H 0 (mZ0 ; Hom(V, V ) ⊗ OX (mZ0 )|mZ0 ) is similar, using instead the exact sequence 0 → OmZ0 +Ci−1 (mZ0 + Ci−1 ) → OmZ0 +Ci (mZ0 + Ci ) → OBi (mZ0 + Ci ) → 0. This concludes the proof in the case of a rational singularity. For minimally elliptic singularities, we shall deduce the theorem from the following more general result: ¯ be a birational morphism from X to a normal Theorem 2.5. Let ϕ : X → X ¯ projective surface X, corresponding to a nef, big, and eventually base point free divisor M . Let w be an arbitrary element of H 2 (X; Z/2Z), and suppose that (i) ϕ contracts no exceptional curve; in other words, if E is an exceptional curve of the first kind on X, then M · E > 0. (ii) If D is a component of ϕ−1 (x) such that w · D 6= 0, then Z0 · D < 0, where Z0 is the fundamental cycle of ϕ−1 (x).

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Then the conclusions of Theorem 2.3 hold for the moduli space ML (w, p) for all p ≪ 0. In particular the conclusions of Theorem 2.3 hold if ϕ−1 (x) is irreducible. Proof that (2.5) implies (2.3). We must show that every minimally elliptic singularity satisfies the hypotheses of Theorem 2.5(ii), provided that w is the mod 2 reduction of KX . Suppose that x is minimally elliptic and that w · D 6= 0. Thus KX · D is odd. Moreover if D is smooth rational then D2 6= −1 and KX · D ≥ 0 so that KX · D ≥ 1. Now (KX + Z0 ) · D = 0. Thus Z0 · D = −(KX · D) ≤ −1. Likewise if pa (D) 6= 0, so that D is not a smooth rational curve, then ϕ−1 (x) = D is an irreducible curve and (2.3) again follows. Proof of Theorem 2.5. We begin with a lemma on sections of line bundles over effective cycles supported in ϕ−1 (x), which generalizes (2.6) of [22]: Lemma 2.6. Let Z0 be the fundamental cycle of ϕ−1 (x) and let λ be a line bundle on Z0 such that deg(λ|D) ≤ 0 for each component D of the support of Z0 . Then either H 0 (Z0 ; λ) = 0 or λ = OZ0 and H 0 (Z0 ; λ) ∼ = C. Proof.PChoose a computation sequence for Z0 , say A1 , A2 , . . . , Ak . Thus, if we set Zi = j≤i Aj , then Zi · Ai+1 > 0, and Zk = Z0 . Now we have an exact sequence 0 → OAi+1 (−Zi ) → OZi+1 → OZi → 0.

Thus deg(OAi+1 (−Zi ) ⊗ λ|Ai+1 ) < 0. It follows that H 0 (OZi+1 ⊗ λ) ⊆ H 0 (OZi ⊗ λ) for all i. By induction dim H 0 (OZi ⊗ λ) ≤ 1 for all i, 1 ≤ i ≤ k. Thus dim H 0 (Z0 ; λ) ≤ 1. Moreover, if dim H 0 (Z0 ; λ) = 1, then the natural map H 0 (OZi+1 ⊗ λ) → H 0 (OZi ⊗ λ) is an isomorphism for all i, and so the induced map H 0 (Z0 ; λ) → H 0 (A1 ; λ|A1 ) is an isomorphism and dim H 0 (A1 ; λ|A1 ) = 1. Thus λ|A1 is trivial and a nonzero section of H 0 (Z0 ; λ) restricts to a generator of λ|A1 . Since we can begin a computation sequence with an arbitrary choice of A1 , we see that a nonzero section s of H 0 (Z0 ; λ) restricts to a nonvanishing section of H 0 (D; λ|D) for every D in the support of ϕ−1 (x). Thus the map OZ0 → λ defined by s is an isomorphism. Remark. The lemma is also true if λ is allowed to have degree one on some components D of Z0 with pa (D) ≥ 2, provided that λ|D is general for these components, and a slight variation holds if λ is also allowed to have degree one on some components D of Z0 with pa (D) = 1. We next construct a bundle W over Z0 with certain vanishing properties: ¯ is the minimal resolution of the normal Lemma 2.7. Suppose that ϕ : X → X surface singularity x. Let µ be a line bundle over the scheme Z0 . Suppose further that, if D is a component of ϕ−1 (x) such that deg(µ|D) is odd, then Z0 · D < 0, where Z0 is the fundamental cycle of ϕ−1 (x). Then there exists a rank two vector bundle W over Z0 with det W = µ and such that H 0 (Z0 ; Hom(W, W ) ⊗ OX (mZ0 )|Z0 ) = 0 for every m ≥ 1. S Proof. Let ϕ−1 (x) = i Di . Then there exists an integer ai such that deg µ|Di = 2ai or 2ai +1, depending on whether deg(µ|Di ) is odd or even. Since dim Z0 = 1, the

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L natural maps Pic Z0 → Pic(Z0 )red → i Pic Di are surjective. Thus we may choose a line bundle L1 over Z0 such that deg(L1 |Di ) = ai . It follows that µ ⊗ L⊗−−2 |Di 1 is a line bundle over Di of degree zero or 1, and if it is of degree 1, then Z0 · Di < 0. Hence µ ⊗ L⊗−2 ⊗ OZ0 (Z0 ) has degree at most zero on Di for every i. 1 Set L2 = µ ⊗ L−1 1 . Thus L1 ⊗ L2 = µ and deg(L2 |Di ) = ai or ai + 1 depending on whether deg(µ|Di ) is even or odd. The line bundle L1−1 ⊗ L2 = µ ⊗ L⊗−2 1 thus has degree zero on those components Di such that deg(µ|Di ) is even and 1 on the components Di such that deg(µ|Di ) is odd. Moreover deg(L−1 1 ⊗ L2 ⊗ OZ0 (mZ0 )|Di ) ≤ 0 for every i. Claim 2.8. Under the assumptions of (2.7), there exists a nonsplit extension W of L2 by L1 except in the case where x is rational, deg(µ|Di ) is odd for at most one i, and the multiplicity of Di in Z0 is one for such i, or χ(OZ0 ) = 0 and deg µ|Di is even for every i. Proof. A nonsplit extension exists if andPonly if h1 (L−1 2 ⊗ L1 ) 6= 0. Now by the Riemann-Roch theorem applied to Z0 = i ni Di , we have −1 0 h1 (Z0 ; L−1 2 ⊗ L1 ) = h (Z0 ; L2 ⊗ L1 ) −

X

ni deg(L−1 2 ⊗ L1 |Di ) − χ(OZ0 ).

i

Here deg(L−1 2 ⊗ L1 |Di ) = 0 on those Di with deg(µ|Di ) even and = −1 on the Di with deg(µ|Di ) odd. Moreover h0 (OZ0 ) = 1 by Lemma 2.6 and so χ(OZ0 ) ≤ 1, with χ(OZ0 ) = 1 if and only if x is rational. Thus h1 (Z0 ; L−1 2 ⊗ L1 ) ≥

X { ni : deg(µ|Di ) is odd } − χ(OZ0 ).

Hence if h1 (Z0 ; L−1 2 ⊗ L1 ) = 0, then either x is rational, deg(µ|Di ) is odd for at most one i, and for such i the multiplicity of Di in Z0 is one, or deg(µ|Di ) is even for all i and χ(OZ0 ) = 0. Returning to the proof of (2.7), choose W to be a nonsplit extension of L2 by L1 if such exist, and set W = L1 ⊕ L2 otherwise. To see that H 0 (Z0 ; Hom(W, W ) ⊗ OX (mZ0 )|Z0 ) = 0, we consider the two exact sequences 0 → L1 → W → L2 → 0; 0 → L1 ⊗ OZ0 (mZ0 ) → W ⊗ OZ0 (mZ0 ) → L2 ⊗ OZ0 (mZ0 ) → 0. Clearly H 0 (Z0 ; Hom(W, W ) ⊗ OX (mZ0 )|Z0 ) = 0 if −1 0 0 H 0 (L−1 1 ⊗ L2 ⊗ OZ0 (mZ0 )) = H (OZ0 (mZ0 )) = H (L2 ⊗ L1 ⊗ OZ0 (mZ0 )) = 0.

The line bundles OZ0 (mZ0 ) and L−1 2 ⊗ L1 ⊗ OZ0 (mZ0 ) have nonpositive degree on each Di and (since Z0 · Di < 0 for some i) have strictly negative degree on at least one component. Thus by Lemma 2.6 H 0 (OZ0 (mZ0 )) and H 0 (L2−1 ⊗L1 ⊗OZ0 (mZ0 )) are both zero. Let us now consider the group H 0 (L−1 1 ⊗ L2 ⊗ OZ0 (mZ0 )). By the hypothesis that Z0 · Di < 0 for each Di such that deg(µ|Di ) is odd, the line bundle L−1 1 ⊗ L2 ⊗ OZ0 (mZ0 ) has nonpositive degree on all components Di . Thus by −1 ∼ Lemma 2.6 either H 0 (L−1 1 ⊗ L2 ⊗ OZ0 (mZ0 )) = 0 or L1 ⊗ L2 ⊗ OZ0 (mZ0 ) = OZ0 . (Z ), and if L ⊗ O Clearly this last case is only possible if m = 1 and L1 ∼ = 2 0 Z0

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moreover Z0 · Di = 0 if deg(µ|Di ) is even and Z0 · Di = −1 if deg(µ|Di ) is odd. As Z0 · Di < 0 for at least one i, deg(µ|Di ) is odd for at least one i as well. In this case, if the nonzero section of L−1 1 ⊗ L2 ⊗ OZ0 (Z0 ) lifts to give a map L1 → W ⊗ OZ0 (Z0 ), then the image of L1 in W ⊗ OZ0 (Z0 ) splits the exact sequence 0 → L1 ⊗ OZ0 (Z0 ) → W ⊗ OZ0 (Z0 ) → L2 ⊗ OZ0 (Z0 ) → 0. Thus W is also a split extension. By Claim 2.8, since deg(µ|Di ) is odd for at least one i, it must therefore be the case that x is rational, deg(µ|Di ) is odd for exactly one i, and for such i the multiplicity of Di in Z0 is one. Moreover Z0 · Dj 6= 0 exactly when j = i and in this case P Z0 · Di = −1. But as the multiplicity of Di in Z0 is 1, we can write Z0 = Di + j6=i nj Dj , and thus Z02 = Z0 · Di = −1.

By a theorem of Artin [1], however, −Z02 is the multiplicity of the rational singularity x. It follows that x is a smooth point and ϕ is the contraction of a generalized exceptional curve, contrary to hypothesis. This concludes the proof of (2.7). We may now finish the proof of (2.5). Start with a generic vector bundle V0 ∈ ML (w, p) on X satisfying the condition that H 1 (X; ad V0 ) → H 1 (Z0 ; ad V0 |Z0 ) is surjective. If µ = det V0 |Z0 , note that, according to the assumptions of (2.5), µ satisfies the hypotheses of Lemma 2.7. For V ∈ ML (w, p), let H(mZ0 ) = H 0 (mZ0 ; ad V ⊗ OX (mZ0 )|mZ0 ). Using the exact sequence 0 → H((m − 1)Z0 ) → H(mZ0 ) → H 0 (Z0 ; ad V0 ⊗ OX (mZ0 )|Z0 ), we see that it suffices to show that, for a generic V , H 0 (Z0 ; ad V ⊗OX (mZ0 )|Z0 ) = 0 for all m ≥ 1. For a fixed m, the condition that H 0 (Z0 ; ad V ⊗ OX (mZ0 )|Z0 ) 6= 0 is a closed condition. Thus since the moduli space cannot be a countable union of proper subvarieties, it will suffice to show that the set of V for which H 0 (Z0 ; ad V ⊗ OX (mZ0 )|Z0 ) = 0 is nonempty for every m. Let S be the germ of the versal deformation of V0 |Z0 keeping det V0 |Z0 fixed. By the assumption that the map from the germ of the versal deformation of V0 to that of V0 |Z0 is submersive, it will suffice to show that, for each m ≥ 1, the set of W ∈ S such that H 0 (Z0 ; ad W ⊗ OZ0 (mZ0 )) = 0 is nonempty. One natural method for doing so is to exhibit a deformation from V0 |Z0 to the W constructed in the course of Lemma 2.7; roughly speaking this amounts to the claim that the “moduli space” of vector bundles on the scheme Z0 is connected. Although we shall proceed slightly differently, this is the main idea of the argument. Choose an ample line bundle λ on Z0 . After passing to some power, we may assume that both (V0 |Z0 ) ⊗ λ and W ⊗ λ are generated by their global sections. A standard argument shows that, in this case, both V0 |Z0 and W can be written as an extension of µ ⊗ λ by λ−1 : Working with W for example, we must show that there is a map λ−1 → W , corresponding to a section of W ⊗ λ, such that the quotient is again a line bundle. It suffices to show that there exists a section s ∈ H 0 (Z0 ; W ⊗ λ) such that, for each z ∈ Z0 , s(z) 6= 0 in the fiber of W ⊗ λ over z.

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Now for z fixed, the set of s ∈ H 0 (Z0 ; W ⊗ λ) such that s(z) = 0 has codimension two in H 0 (Z0 ; W ⊗ λ) since W ⊗ λ is generated by its global sections. Thus the set of s ∈ H 0 (Z0 ; W ⊗ λ) such that s(z) = 0 for some z ∈ Z0 has codimension at least one, and so there exists an s as claimed. Now let W0 = λ−1 ⊕ (µ ⊗ λ). Let (S0 , s0 ) be the germ of the versal deformation of W0 (with fixed determinant µ). As Z0 has dimension one, S0 is smooth. Both V0 |Z0 and W correspond to extension classes ξ, ξ ′ ∈ Ext1 (µ ⊗ λ, λ−1 ). Replacing, say, ξ by the class tξ, t ∈ C∗ , gives an isomorphic bundle. In this way we obtain a family of bundles V over Z0 × C, such that the restriction of V to Z0 × t is V0 |Z0 if t 6= 0 and is W0 if t = 0. Hence in the germ S0 there is a subvariety containing s0 in its closure and consisting of bundles isomorphic to V0 |Z0 , and similarly for W . As H 0 (Z0 ; ad W ⊗ OZ0 (mZ0 )) = 0, the locus of bundles U in S0 for which H 0 (Z0 ; ad U ⊗ OZ0 (mZ0 )) = 0 is a dense open subset. Since S0 is a smooth germ, it follows that there is a small deformation of V0 |Z0 to such a bundle. Thus the generic small deformation U of V0 |Z0 satisfies H 0 (Z0 ; ad U ⊗OZ0 (mZ0 )) = 0, and so the generic V ∈ ML (w, p) has the property that H 0 (Z0 ; ad V ⊗ OX (mZ0 )|Z0 ) = 0 for all m ≥ 1 as well. As we saw above, this implies the vanishing of H 0 (mZ0 ; ad V ⊗ OX (mZ0 )|mZ0 ). ¯ is a singular surface, but that ϕ : X → X ¯ is not the Remark. (1) Suppose that X minimal resolution. We may still define the fundamental cycle Z0 for the resolution ϕ. Moreover it is easy to see that Z0 · E = 0 for every component of a generalized exceptional curve contained in ϕ−1 (x). Thus the hypothesis of (ii) of Theorem 2.5 implies that w · E = 0 for such curves. (2) We have only considered contractions of a very special type, and have primarily been interested in the case where w is the mod two reduction of [KX ]. However it is natural to ask if the analogues of Theorem 2.3 and 2.5 (and thus Theorem 1.4) holds for more general contractions and choices of w, provided of course that no smooth rational curve of self-intersection −1 is contracted to a point. Clearly the proof of Theorem 2.5 applies to a much wider class of singularities. Indeed a little work shows that the proof goes over (with some modifications in case there are components of arithmetic genus one) to handle the case where we need only assume condition (ii) of (2.5) for those components D which are smooth rational curves. Another case where it is easy to check that the conclusions of (2.5) hold is where w is arbitrary and the dual graph of the singularity is of type Ak . We make the following rather natural conjecture: Conjecture 2.9. The conclusions of Theorem 1.4 hold for arbitrary choices of w and ϕ, provided that ϕ does not contract any exceptional curves of the first kind. 3. Nonexistence of embedded 2-spheres 3.1. A base point free theorem Theorem 3.1. Let π : X → X ′ be a birational morphism from the smooth surface X to a normal surface X ′ , not necessarily projective. Suppose that X is a minimal surface of general type, and that p ∈ S X ′ is an isolated singular point which is a −1 nonrational singularity. Let π (p) = i Di . Then:

(i) There P exist nonnegative integers ni with ni > 0 for at least one i such that KX + i ni Di is nef and big.

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P (ii) Suppose that q(X) = 0. Then there further exists a choice of D = i ni Di satisfying (i) with D connected and such that there exists a section of KX +D which is nowhere vanishing in a neighborhood of E=

[ { Dj : (KX + D) · Dj = 0 }.

In this case either E = ∅ or E = Supp D and D is the fundamental cycle of the minimal resolution of a minimally elliptic singularity. (iii) With D satisfying (i) and (ii), the linear system KX + D is eventually base ¯ is the associated contraction, then X ¯ is point free. Moreover, if ϕ : X → X a normal projective surface all of whose singular points are either rational or minimally elliptic. P Proof. To prove (i), consider the set of all effective cycles D = i ai Di , where the ai are nonnegative integers, not all zero, and such that h1 (OD ) 6= 0. This set is not empty by the definition of a nonrational singularity, and is partially ordered by ≤, ′ where D′ ≤ D if D set. This P− D is effective. Choose a minimal element D in the means that D = i ni Di where either ni = 1 for exactly one i and h1 (ODi ) 6= 0, or for every irreducible Di contained in the support of D, D − Di = D′ is effective and h1 (OD′ ) = 0. If D′′ is then any nonzero effective cycle with D′′ < D, then there exists an i such that OD−Di → OD′′ is surjective. By a standard argument, H 1 (OD−Di ) → H 1 (OD′′ ) is surjective and thus h1 (OD′′ ) = 0 for every nonzero effective D′′ < D. Finally note that D is connected, since otherwise we could replace D by some connected component D0 with h1 (OD0 ) 6= 0. Next we claim that KX +D is nef. Since KX is nef, it is clear that (KX +D)·C ≥ 0 for every irreducible curve C not contained in the support of D, and moreover, for such curves C, (KX + D) · C = 0 if and only if C is a smooth rational curve of selfintersection −2 disjoint from the support of D. Next suppose that Di is a curve in the support of D and consider (KX + D) · Di . If D = Di then KX + Di |Di = ωDi , the dualizing sheaf of Di , and this has degree 2pa (Di ) − 2 ≥ 0 since pa (Di ) = h1 (ODi ) > 0. Otherwise let D′ = D − Di consider the exact sequence 0 → ODi (−D′ ) → OD → OD′ → 0. Thus the natural map H 1 (ODi (−D′ )) → H 1 (OD ) is surjective since H 1 (OD′ ) = 0, and so H 1 (ODi (−D′ )) 6= 0 as H 1 (OD ) 6= 0. By duality H 0 (Di ; ωDi ⊗ODi (D′ )) 6= 0. On the other hand ωDi = KX + Di |Di , and so deg(KX + D′ + Di )|Di = (KX + D) · Di ≥ 0; moreover (KX + D) · Di = 0 only if the divisor class KX + D|Di is trivial. 2 Next, (KX + D)2 ≥ KX > 0, so that KX + D is big. In fact, 2 2 (KX + D)2 = KX · (KX + D) + (KX + D) · D ≥ KX + KX · D ≥ KX .

Thus KX + D is big. S To see (ii), let E = { Dj ⊆ Supp D : (KX + D) · Dj = 0 }. We shall also view E as a reduced divisor. We claim that OE (KX + D) = OE . First assume that E = D (and thus in particular that D is reduced); in this case we need to show that ωD = OD . By assumption D is connected. Then ωD has degree zero on every reduced irreducible component of D, and by Serre duality χ(ωD ) = −χ(OD ) = 1 1 0 0 2 (KX + D) · D = 0. As h (ωD ) = h (OD ) = 1, h (ωD ) = 1 as well. As ωD has

ON COMPLEX SURFACES DIFFEOMORPHIC TO RATIONAL SURFACES

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degree zero on every component of D, if s is a section of ωD , then the restriction of s to every component Di of D is either identically zero or nowhere vanishing. Thus if s is nonzero, since D is connected, s must be nowhere vanishing. It follows that the map OD → ωD is surjective and is thus an isomorphism. If D 6= E, we apply the argument that showed above that (KX + D) · Di ≥ 0 to each connected component E0 of the divisor E, with D′ = D − E0 , to see that there is a section of OE0 (KX + D). Since OE0 (KX + D) has degree zero on each irreducible component of E0 , the argument that worked for the case D = E also works in this case. Now let us show that, provided q(X) = 0, a nowhere zero section of OE (KX + D) = OE lifts to a section of KX + D. It suffices to show that, for every connected component E0 of E, a nowhere vanishing section of OE0 (KX + D) lifts to a section of KX + D. Let D′ = D − E0 . If D′ = 0 then D = E = E0 and we ask if the map H 0 (OX (KX + D)) → H 0 (OD (KX + D)) is surjective. The cokernel of this map lies in H 1 (KX ) = 0 since X is regular. Otherwise D′ 6= 0. Beginning with the exact sequence 0 → OD′ (−E0 ) → OD → OE0 → 0, and tensoring with OX (KX + D), we obtain the exact sequence 0 → OD′ (KX + D − E0 ) → OD (KX + D) → OE0 → 0. Now OD′ (KX + D − E0 ) = OD′ (KX + D′ ) = ωD′ and by duality h0 (ωD′ ) = h1 (OD′ ) = 0. Thus H 0 (OD (KX + D)) includes into H 0 (OE0 ) = C and so it suffices to prove that H 0 (OD (KX + D)) 6= 0, in which case it has dimension one. On the other hand, using the exact sequence 0 → OX (KX ) → OX (KX + D) → OD (KX + D) → 0, we see that h0 (OD (KX + D)) ≥ h0 (OX (KX + D)) − pg (X). Since h2 (KX + D) = h0 (−D) = 0, the Riemann-Roch theorem implies that 1 h0 (OX (KX + D)) = h1 (OX (KX + D)) + (KX + D) · D + 1 + pg (X). 2 Since all the terms are positive, we see that indeed h0 (OX (KX + D)) − pg (X) ≥ 1, and that h0 (OX (KX + D)) − pg (X) = 1 if and only if h1 (OX (KX + D)) = 0 and (KX + D) · Di = 0 for every component Di contained in the support of D. This last condition says exactly that E = Supp D, and thus, as D is connected, that E0 = E. We claim that in this last case D is minimally elliptic. Indeed, for every effective divisor D′ with 0 < D′ < D, we have pa (D′ ) = 1 − h0 (OD′ ) + h1 (OD′ ) = 1 − h0 (OD′ ) ≤ 0. Thus D is the fundamental cycle for the resolution of a minimally elliptic singularity. Finally we prove (iii). The irreducible curves C such that (KX + D) · C = 0 are the components Di of the support of D such that (KX + D) · Di = 0, as well as smooth rational curves of self-intersection −2 disjoint from Supp D. These last contribute rational double points, so that we need only study the Di such that (KX + D) · Di = 0. We have seen in (ii) that either there are no such Di , or every

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Di in the support of D satisfies (KX + D) · Di = 0 and the contraction of D is a minimally elliptic singularity. ¯ be the normal surface obtained by contracting all the irreducible curves Let X C on X such that (KX + D) · C = 0. The line bundle OX (KX + D) is trivial in a neighborhood of these curves, either because they correspond to a rational singularity or because we are in the minimally elliptic case and by (ii). So OX (KX + ¯ which is ample, by the Nakai-Moishezon criterion. D) induces a line bundle on X Thus |k(KX + D)| is base point free for all k ≫ 0. 3.2. Completion of the proof We now prove Theorem 1.5: Theorem 1.5. Let X be a minimal simply connected algebraic surface of general type, and let E ∈ H 2 (X; Z) be a (1, 1)-class satisfying E 2 = −1, E · KX = 1. Let w be the mod 2 reduction of [KX ]. Then there exist: (i) an integer p and (in case pg (X) = 0) a chamber C of type (w, p)) and (ii) a (1, 1)-class M ∈ H 2 (X; Z) such that M ·E = 0 and γw,p (X)(M d ) 6= 0 (or, in case pg (X) = 0, γw,p (X; C)(M d ) 6= 0). Proof. We begin with the following lemma: Lemma 3.2. With X and E as above, there exists an orientation preserving diffeomorphism ψ : X → X such that ψ ∗ [KX ] = [KX ] and such that ψ ∗ E · [C] ≥ 0 for every smooth rational curve C on X with C 2 = −2. Proof. Let ∆ = {[C1 ], . . . , [Ck ]} be the set of smooth rational curves on X of selfintersection −2, and let ri : H 2 (X; Z) → H 2 (X; Z) be the reflection about the class [Ci ]. Then ri is realized by an orientation-preserving self-diffeomorphism of X, ri∗ [KX ] = [KX ], and ri preserves the image of Pic X inside H 2 (X; Z). Let Γ be the finite group generated group by the ri . Since the classes [Ci ] are linearly independent, the set { x ∈ H 2 (X; R) : x · [Ci ] ≥ 0 } δ has a nonempty interior. Moreover, if ∆′ = Γ · ∆, and we set δ ⊥ for S W = δ ′ 2 δ ∈ ∆ , then the connected components of the set H (X; R) − δ∈∆′ W are the fundamental domains for the action of Γ on H 2 (X; R). Clearly at least one of these connected components lies inside { x ∈ H 2 (X; R) : x · [Ci ] ≥ 0 }. Thus given E (or indeed an arbitrary element of H 2 (X; R)), there exists a γ ∈ Γ such that γ(E) · [Ci ] ≥ 0 for all i. As every γ ∈ Γ is realized by an orientation preserving self-diffeomorphism ψ, this concludes the proof of (3.2).

Thus, to prove Theorem 1.5, it is sufficient by the naturality of the Donaldson polynomials to prove it for every class E satisfying E 2 = −1, E · KX = 1, and E · [C] ≥ 0 for every smooth rational curve C on X with C 2 = −2. We therefore make this assumption in what follows. Given Theorem 1.4, it therefore suffices to find a nef and big divisor M orthogonal to E, which is eventually base point free, such that the contraction morphism defined by |kM | has an image with at worst rational and minimally elliptic singularities (note that, since X is assumed minimal, no exceptional curves can be contracted). Thus we will be done by the following lemma:

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Lemma 3.3. There exists a nef and big divisor M which is eventually base point free and such that (1) M · E = 0. ¯ of X defined by |kM | for all k ≫ 0 has only rational (2) The contraction X and minimally elliptic singularities. Proof. To find M we proceed as follows: consider the divisor KX + E = M . As KX · E = 1 and E 2 = −1, M is orthogonal to E. Moreover M 2 = (KX + E)2 = 2 KX + 1 > 0. We now consider separately the cases where M is nef and where M is not nef. Case I: M = KX + E is nef. Consider the union of all the curves D such that M · D = 0. The intersection matrix of the D is negative definite, and so we can contract all the D on X to obtain a normal surface X ′ . If X ′ has only rational singularities, then M induces an ample divisor on X ′ and so M itself is eventually base point free. In this case we are done. Otherwise we may apply Theorem 3.1 to find a subset D1 , . . . , DP t of the curves D with M ·D = 0 and positive integers ai such that the divisor KX + i ai Di ¯ of X is nef, big, and eventually base point free, and such that the contraction X has only rational and minimally elliptic singularities, with exactly one nonrational singularity. Note that Di · E = −Di · KX ≤ 0, and Di · E = 0 if and only if Di · KX = 0, or in other words ifP and only if Di is a smooth rational curve of self-intersection −2. Setting e = − i ai (Di · E), we have e ≥ 0, and e = 0 if and ¯ has a nonrational singularity, we cannot have only if Di · E = 0 for all i. But as X Di · E = 0 for all i, for then all singularities would be rational double points. Thus 1 P ′ ai Di is a rational convex combination e > 0. Now the Q-divisor M = K + X i e P ′ of KX and KX + i ai Di , andPM ′ · E = 0. Moreover either M P is a strict convex ′ combination of KX and KX + i ai Di (if e > 1) or M = KX + i ai Di (if e = 1). In the second case, M ′ satisfies (1) of Lemma 3.3, and it is eventually base point free by (iii) of Theorem 3.1. Thus M ′ satisfies the conclusions of Lemma 3.3. In ′ the first case, M ′ is nef and big, and the only P curves C such that M · C = 0 are curves C such that KX · C = 0 and (KX + i ai Di ) · C = 0. The set of all such curves must therefore be a subset of the set of all smooth rational curves on X with self-intersection −2. Hence, if X ′′ denotes the contraction of all the curves C on X such that M ′ · C = 0, then X ′′ has only rational singularities and M ′ induces an ample Q-divisor on X ′′ . Once again some multiple of M ′ is eventually base point free and (1) and (2) of Lemma 3.3 are satisfied. Thus we have proved the lemma in case KX + E is nef. Case II: M = KX + E is not nef. Let D be an irreducible curve with M · D < 0. We claim first that in this case D2 < 0. Indeed, suppose that D2 ≥ 0. As Pic X ⊗Z R has signature (1, ρ − 1), the set Q = { x ∈ Pic X ⊗Z R : x2 ≥ 0, x 6= 0 } has two connected components, and two classes x and x′ are in the same connected component of Q if and only if x · x′ ≥ 0 (cf. [13] p. 320 Lemma 1.1). Now (KX + 2 E) · KX = (KX + E)2 = KX + 1 > 0, so that KX + E and KX lie in the same connected component of Q. Likewise, if D2 ≥ 0, then since KX · D ≥ 0, KX and D lie in the same connected component of Q. Thus D and KX + E lie in the same

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connected component of Q, so that (KX + E) · D ≥ 0. Conversely, if M · D < 0, then D2 < 0. Fix an irreducible curve D with M · D < 0, and let d = −E · D > KX · D ≥ 0. Recall that by assumption E · D ≥ 0 if D is a smooth rational curve of selfintersection −2. If pa (D) ≥ 1, then set M ′ = KX + d1 D. Then M ′ · E = 0 by construction. Moreover we claim that M ′ is nef and big. Indeed 1 1 1 1 (M ′ )2 = (KX + D)2 = KX · (KX + D) + (KX + D) · D. d d d d Thus M ′ is big if it is nef and to see that M ′ is nef it suffices to show that M ′ ·D ≥ 0. But 1 1 2 ′ D2 . M · D = KX · D + D = 2pa (D) − 2 − 1 − d d As D2 < 0, we see that M ′ · D ≥ 0, and M ′ · D = 0 if and only if pa (D) = 1 and d = 1. Suppose that pa (D) = 1 and M ′ · D = 0. Using the exact sequence 0 → OX (KX ) → OX (KX + D) → ωD → 0, and arguments as in the proof of Theorem 3.1, we see that the linear system M ′ is eventually base point free and that the associated contraction has just rational double points and a minimally elliptic singular point which is the image of D. In all other cases, M ′ · D > 0, so that the curves orthogonal to M ′ are smooth rational curves of self-intersection −2. Again, some positive multiple of M ′ is eventually base point free and the contraction has just rational singularities. Thus we may assume that pa (D) = 0 for every irreducible curve D such that M ·D < 0. By assumption D2 6= −1, −2, so that D2 ≤ −3. Thus d = −−D ·E ≥ 2. If either D2 ≤ −4 or D2 = −3 and d ≥ 3, then again let M ′ = KX + d1 D. Thus M ′ · E = 0 and 1 2 1 ′ M · D = KX · D + D = −2 − 1 − D2 ≥ 0. d d Thus M ′ is nef and big, and some multiple of M ′ is eventually base point free, and the associated contraction has just rational singularities. The remaining case is where there is a smooth rational curve D on X with self-intersection −3 and such that −D · E = 2. In this case KX · D = 1, and so D − E is orthogonal to KX . Note that D − E is not numerically trivial since D is not numerically equivalent to E. Thus, by the Hodge index theorem (D − E)2 < 0. But (D − E)2 = −3 + 4 − 1 = 0, a contradiction. Thus this last case does not arise. Appendix: On the canonical class of a rational surface Let Λn be a lattice of type (1, n), i.e. a free Z-module of rank n + 1, together with a quadratic form q : Λn → Z, such that there exists an orthogonal basis {e0 , e1 , . . . , en } of Λn with q(e0 ) = 1 and q(ei ) = −1 for all i > 0. Fix once

ON COMPLEX SURFACES DIFFEOMORPHIC TO RATIONAL SURFACES

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and for all such a basis. We shall always view Λn as included in Λn+1 in the obvious way. Let n X ei . κn = 3e0 − i=1

Then q(κn ) = 9−n and κn is characteristic, i.e. κn ·α ≡ q(α) mod 2 for all α ∈ Λn . The goal of this appendix is to give a proof, due to the first author, R. Miranda, and J.W. Morgan, of the following: Theorem A.1. Suppose that n ≤ 8 and that κ ∈ Λn is a characteristic vector satisfying q(κ) = 9 − n. Then there exists an automorphism ϕ of Λn such that ϕ(κ) = κn . A similar statement holds for n = 9 provided that κ is primitive. Proof. We shall freely use the notation and results of Chapter II of [13] and shall quote the results there by number. For the purposes of the appendix, chamber shall mean a chamber in { x ∈ Λn ⊗ R | x2 = 1 } for the set of walls defined by the set { α ∈ Λn | α2 = −1 }. Let Cn be the chamber associated to κn [13, p. 329, 2.7(a)]: the oriented walls of Cn are exactly the set { α ∈ Λn | q(α) = −1, α · κn = 1 }. Then κn lies in the interior of R+ · Cn , by [13, p. 329, 2.7(a)]. Similarly κ lies in the interior of a set of the form R+ · C for some chamber C, since κ is not orthogonal to any wall (because it is characteristic) and q(κ) > 0. But the automorphism group of Λn acts transitively on the chambers, by [13 p. 324]. Hence we may assume that κ ∈ Cn . In this case we shall prove that κ = κn . We shall refer to Cn as the fundamental chamber of Λn ⊗Z R. Let us record two lemmas about Cn . Lemma A.2. An automorphism ϕ of Λn fixes Cn if and only if it fixes κn . Proof. The oriented walls of Cn are precisely the α ∈ Λn such that q(α) = −1 and κn · α = 1. Thus, an automorphism fixing κn fixes Cn . The converse follows from [13, p. 335, 4.4]. P Lemma A.3. Let α = i αi ei be an oriented wall of Cn , where e0 , . . . , en is the standard basis of Λn . After reordering the elements e1 , . . . , en , let us assume that |α1 | ≥ |α2 | ≥ · · · ≥ |αn |. Then for n ≤ 8, the possibilities for (α0 , . . . αn ) are as follows (where we omit the αi which are zero): (1) α0 = 0, α1 = 1; (2) α0 = 1, α1 = α2 = −1 (n ≥ 2); (3) α0 = 2, α1 = α2 = α3 = α4 = α5 = −1 (n ≥ 5); (4) α0 = 3, α1 = −2, α2 = α3 = α4 = α5 = α6 = α7 = −1 (n ≥ 7); (5) α0 = 4, α1 = α2 = α3 = 2, α4 = · · · = α8 = −1 (n = 8); (6) α0 = 5, α1 = · · · = α6 = −2, α7 = α8 = −1 (n = 8); (7) α0 = 6, α1 = −3, α2 = · · · = α8 = −2 (n = 8). Proof. This statement is extremely well-known as the characterization of the lines on a del Pezzo surface (see [7], Table 3). We can give a proof as follows. It clearly suffices to prove the result for n = 8. But for n = 8, there is a bijection between

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the α defining an oriented wall of C8 and the elements γ ∈ κ⊥ 8 with q(γ) = −2. This bijection is given as follows: α defines an oriented wall of C8 if and only if q(α) = −1 and κ8 · α = 1. Map α to α − κ8 = γ. Thus, as q(κ8 ) = 1, q(γ) = −2 and γ · κ8 = 0. Conversely, if γ ∈ κ⊥ 8 satisfies q(γ) = −2, then γ + κ8 defines an oriented wall of C8 . Now the number of α listed above, after we are allowed to reorder the ei , is easily seen to be 8 8 8 8 8+ + +8·7+ + + 8 = 240. 2 5 3 2 Since this is exactly the number of vectors of square −2 in −E8 , by e.g. [36], we must have enumerated all the possible α. P Write κ = ni=0 ai ei , where ei is the standard basis of Λn given above. Since κ · ei > 0, ai < 0. After reordering the elements e1 , . . . , en , we may assume that |a1 | ≥ |a2 | ≥ · · · ≥ |an |. P By inspecting the cases in Lemma A.3, P for every α = i αi ei not of the form ei , αi ≤ 0 for all i ≥ 1. Given α = i αi ei with α 6= ei for any i, let us call α well-ordered if |α1 | ≥ |α2 | ≥ · · · ≥ |αn |. P Quite generally, given α = α0 e0 + i>0 αi ei , we define the reordering r(α) of α to be X ασ(i) ei , r(α) = α0 e0 + i>0

where σ is a permutation of {1, . . . , n} such that r(α) is well-ordered. Clearly r(α) is independent of the choice of σ. We then have the following: Claim A.4. κ ∈ Cn if and only if κ · α > 0 for every well-ordered wall α. Proof. Clearly if κ ∈ Cn , then κ·α > 0 for every α, well-ordered or not. Conversely, suppose that κ · α > 0 for every well-ordered wall α. We claim that α · κ ≥ r(α) · κ,

(∗)

which clearly implies (A.4) since r(α) is well-ordered. Now α · κ = α0 a0 −

X

αi ai .

i>0

Since αi < 0 and ai < 0, (∗) is easily reduced to the following statement about positive real numbers: if c1 ≥ · · · ≥ cn is a sequence of positive real numbers and d1 , . . . , dn is any sequence of positive real numbers, then a permutation σ of P {1, . . . , n} is such that i ci dσ(i) is maximal exactly when dσ(1) ≥ · · · ≥ dσ(n) . We leave the proof of this elementary fact to the reader. Next, we claim the following:

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Lemma A.5. View Λn ⊂ Λn+1 . Defining κn+1 and Cn+1 in the natural way for Λn+1 , suppose that κ ∈ Cn . Then κ′ = κ − en+1 ∈ Cn+1 . Proof. We have ordered our basis {e0 , . . . , en } so that |a1 | ≥ |a2 | ≥ · · · ≥ |an |. Since ai < 0 for all i, |an | ≥ 1. Thus the coefficients of κ′ are also so ordered. Note also that all coefficients of κ′ are less than zero, so that the inequalities from (1) of Lemma A.3 are automatic. Given any other wall α′ of Cn+1 , to verify that κ′ ·α′ > 0, it suffices to look at κ′ · r(α′ ), where r(α′ ) is the reordering of α′ . Expressing α′ as a linear combination of the standard basis vectors, if some coefficient is zero, then r(α′ ) ∈ Λn . Clearly, in this case, viewing r(α′ ) as an element of Λn , it is a wall of Cn . Since then κ′ · r(α′ ) = κ · r(α′ ), we have κ′ · r(α′ ) > 0 in this case. In the remaining case, r(α′ ) does not lie in Λn . This can only happen for n = 1, 4, 6, 7, with α′ one of the new types of walls corresponding to the cases (2) — (7) of Lemma A.3. Thus, the only thing we need to check is that, every time we introduce a new type of wall, we still get the inequalities as needed. Since r(α′ ) is well-ordered, we can assume that it is in fact one of the walls listed in Lemma A.3. The n = 1 case simply says that a0 > −a1 + 1. However, we can easily solve the equations a20 − a21 = 8, a1 < 0 to get a0 = 3, a1 = − − 1. Since 3 > 1 + 1, we are done in this case. P4 Next assume that n = 4. We have κ = a0 e0 + i=1 ai ei . We must show that P4 2a0 > − i=1 ai + 1. We know that a0 > −a1 − a2 , hence that a0 ≥ −a1 − a2 + 1. Moreover a0 ≥ −a3 − a4 + 1 since |a1 | ≥ |a2 | ≥ |a3 | ≥ |a4 |. Adding gives 2a0 ≥ P4 P4 P4 − i=1 ai + 2 = (− i=1 ai + 1) + 1 and therefore 2a0 > − i=1 ai + 1. The case P6 where n = 6 is similar: we must show that 3a0 > −2a1 − i=2 ai + 1. But we know P5 that 2a0 ≥ − i=1 ai + 1 and that a0 ≥ −a1 − a6 + 1. Adding gives the desired inequality. For n = 7, we have three new inequalities to check. The inequality 4a0 > −2a1 − 2a2 − 2a3 −

7 X

ai + 1

i=4

P follows by adding the inequalities 3a0 > −2a1 − 7i=2 ai and a0 > −a2 − a3 . The inequality 6 X ai − a7 + 1 5a0 > −2 i=1

follows from adding the inequalities 3a0 > −2a1 − Likewise, the last inequality 6a0 > −3a1 − 2

7 X

P7

i=2

ai and 2a0 > −

P5

i=2

ai .

ai + 2

i=2

P5 P7 follows by adding up the three inequalities 3a0 > −2a1 − i=2 ai , 2a0 > − i=1 ai , and a0 > −a6 − a7 . Thus we have established the lemma. Completion of the proof of Theorem A.1. Begin with κ. Applying Lemma A.5 P8 and induction, if n < 8, then the vector η = κ − j=n+1 ej lies in the fundamental

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chamber of Λ8 . Moreover η is a characteristic vectorPof square 1. Thus η ⊥ ∼ = −−E8 . P8 The same is true for κ8 = 3e0 − i=1 ei = κn − 8j=n+1 ej . Clearly, then, there is an automorphism ϕ of Λ8 such that η = ϕ(κ8 ). But both η and κ8 lie in the fundamental chamber for Λ8 . Since the automorphism group preserves the chamber structure, the automorphism ϕ must stabilize the fundamental chamber. By Lemma P8 P8 A.2, ϕ(κ8 ) = κ8 . Thus η = κ8 . Hence κ − j=n+1 ej = κn − j=n+1 ej . It follows that κ = κn . Note. To handle the case n = 9, we argue that every vector κ ∈ Λ9 which is primitive of square zero and characteristic is conjugate to κ9 as above. To do this, an easy argument shows that, if κ is such a class, then there is an orthogonal splitting Λ9 ∼ = hκ, δi ⊕ (−E8 ), where δ is an element of Λ9 satisfying δ · κ = 1 and q(δ) = 1. Thus clearly every two such κ are conjugate. References 1. M. Artin, On isolated rational singularities of surfaces, Amer. J. Math. 88 (1966), 129–136. 2. R. Barlow, A simply connected surface of general type with pg = 0, Invent. Math. 79 (1985), 293–301. 3. W. Barth, C. Peters, A. Van de Ven, Compact Complex Surfaces , Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge 4, Springer, Berlin Heidelberg New York Tokyo, 1984. 4. F. Bogomolov, Holomorphic tensors and vector bundles on projective varieties, Math. USSR Izvestiya 13 (1979), 499–555. 5. R. Brussee, Stable bundles on blown up surfaces, Math. Zeit. 205 (1990), 551–565. , On the (−1)-curve conjecture of Friedman and Morgan, Invent. Math. 114 (1993), 6. 219–229. 7. M. Demazure, Surfaces de del Pezzo II, S´ eminaire sur les Singularit´ es des Surfaces Palaiseau, France 1976–1977 (M. Demazure, H. Pinkham, B. Teissier, eds.), Lecture Notes in Mathematics 777, Springer, Berlin Heidelberg New York, 1980. 8. S.K. Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. Lond. Math. Soc. 50 (1985), 1–26. 9. , Irrationality and the h-cobordism conjecture, J. Differ. Geom. 26 (1987), 141–168. , Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257–315. 10. 11. S.K. Donaldson, P. Kronheimer, The Geometry of Four-Manifolds, Clarendon Press, Oxford, 1990. 12. R. Friedman, Stable Vector Bundles over Algebraic Varieties (to appear). 13. R. Friedman, J. W. Morgan, On the diffeomorphism types of certain algebraic surfaces I, J. Differ. Geom. 27 (1988), 297–369. 14. , Algebraic surfaces and 4-manifolds: some conjectures and speculations, Bull. Amer. Math. Soc. (N.S.) 18 (1988), 1–19. , Smooth Four-Manifolds and Complex Surfaces, Ergebnisse der Mathematik und ihrer 15. Grenzgebiete 3. Folge 27, Springer, Berlin Heidelberg New York, 1994. 16. Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model problem, Algebraic Geometry, Sendai Adv. Stud. Pure Math. (T. Oda, ed.), vol. 10, Kinokuniya and Amsterdam North-Holland, Tokyo, 1987, pp. 283–360. 17. M. Kneser, Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Ver¨ anderlichen, Arch. der Math. 7 (1956), 323–332. 2 18. D. Kotschick, On manifolds homeomorphic to CP 2 #8CP , Invent. Math. 95 (1989), 591–600. + , SO(3)-invariants for 4-manifolds with b2 = 1, Proc. Lond. Math. Soc. 63 (1991), 19. 426–448. 20. , Positivity versus rationality of algebraic surfaces (to appear). 21. D. Kotschick, J.W. Morgan, SO(3)-invariants for 4-manifolds with b+ 2 = 1 II, J. Differ. Geom. 39 (1994), 443–456.

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